Module mici.systems
Hamiltonian systems encapsulating energy functions and their derivatives.
Expand source code Browse git
"""Hamiltonian systems encapsulating energy functions and their derivatives."""
from abc import ABC, abstractmethod
import numpy as np
from mici.states import cache_in_state, cache_in_state_with_aux
import mici.matrices as matrices
from mici.autodiff import autodiff_fallback
class System(ABC):
r"""Base class for Hamiltonian systems.
The Hamiltonian function \(h\) is assumed to have the general form
\[ h(q, p) = h_1(q) + h_2(q, p) \]
where \(q\) and \(p\) are the position and momentum variables respectively,
and \(h_1\) and \(h_2\) Hamiltonian component functions. The exact
Hamiltonian flow for the \(h_1\) component can be always be computed as it
depends only on the position variable however depending on the form of
\(h_2\) the corresponding exact Hamiltonian flow may or may not be
simulable.
By default \(h_1\) is assumed to correspond to the negative logarithm of an
unnormalized density on the position variables with respect to the Lebesgue
measure, with the corresponding distribution on the position space being
the target distribution it is wished to draw approximate samples from.
"""
def __init__(self, neg_log_dens, grad_neg_log_dens=None):
"""
Args:
neg_log_dens (Callable[[array], float]): Function which given a
position array returns the negative logarithm of an
unnormalized probability density on the position space with
respect to the Lebesgue measure, with the corresponding
distribution on the position space being the target
distribution it is wished to draw approximate samples from.
grad_neg_log_dens (
None or Callable[[array], array or Tuple[array, float]]):
Function which given a position array returns the derivative of
`neg_log_dens` with respect to the position array argument.
Optionally the function may instead return a 2-tuple of values
with the first being the array corresponding to the derivative
and the second being the value of the `neg_log_dens` evaluated
at the passed position array. If `None` is passed (the default)
an automatic differentiation fallback will be used to attempt
to construct the derivative of `neg_log_dens` automatically.
"""
self._neg_log_dens = neg_log_dens
self._grad_neg_log_dens = autodiff_fallback(
grad_neg_log_dens, neg_log_dens, "grad_and_value", "grad_neg_log_dens"
)
@cache_in_state("pos")
def neg_log_dens(self, state):
"""Negative logarithm of unnormalized density of target distribution.
Args:
state (mici.states.ChainState): State to compute value at.
Returns:
float: Value of computed negative log density.
"""
return self._neg_log_dens(state.pos)
@cache_in_state_with_aux("pos", "neg_log_dens")
def grad_neg_log_dens(self, state):
"""Derivative of negative log density with respect to position.
Args:
state (mici.states.ChainState): State to compute value at.
Returns:
array: Value of `neg_log_dens(state)` derivative with respect to
`state.pos`.
"""
return self._grad_neg_log_dens(state.pos)
def h1(self, state):
"""Hamiltonian component depending only on position.
Args:
state (mici.states.ChainState): State to compute value at.
Returns:
float: Value of `h1` Hamiltonian component.
"""
return self.neg_log_dens(state)
def dh1_dpos(self, state):
"""Derivative of `h1` Hamiltonian component with respect to position.
Args:
state (mici.states.ChainState): State to compute value at.
Returns:
array: Value of computed `h1` derivative.
"""
return self.grad_neg_log_dens(state)
def h1_flow(self, state, dt):
"""Apply exact flow map corresponding to `h1` Hamiltonian component.
`state` argument is modified in place.
Args:
state (mici.states.ChainState): State to start flow at.
dt (float): Time interval to simulate flow for.
"""
state.mom -= dt * self.dh1_dpos(state)
@abstractmethod
def h2(self, state):
"""Hamiltonian component depending on momentum and optionally position.
Args:
state (mici.states.ChainState): State to compute value at.
Returns:
float: Value of `h2` Hamiltonian component.
"""
@abstractmethod
def dh2_dmom(self, state):
"""Derivative of `h2` Hamiltonian component with respect to momentum.
Args:
state (mici.states.ChainState): State to compute value at.
Returns:
array: Value of `h2(state)` derivative with respect to `state.pos`.
"""
def h(self, state):
"""Hamiltonian function for system.
Args:
state (mici.states.ChainState): State to compute value at.
Returns:
float: Value of Hamiltonian.
"""
return self.h1(state) + self.h2(state)
def dh_dpos(self, state):
"""Derivative of Hamiltonian with respect to position.
Args:
state (mici.states.ChainState): State to compute value at.
Returns:
array: Value of `h(state)` derivative with respect to `state.pos`.
"""
if hasattr(self, "dh2_dpos"):
return self.dh1_dpos(state) + self.dh2_dpos(state)
else:
return self.dh1_dpos(state)
def dh_dmom(self, state):
"""Derivative of Hamiltonian with respect to momentum.
Args:
state (mici.states.ChainState): State to compute value at.
Returns:
array: Value of `h(state)` derivative with respect to `state.mom`.
"""
return self.dh2_dmom(state)
@abstractmethod
def sample_momentum(self, state, rng):
"""
Sample a momentum from its conditional distribution given a position.
Args:
state (mici.states.ChainState): State defining position to
condition on.
Returns:
mom (array): Sampled momentum.
"""
class EuclideanMetricSystem(System):
r"""Hamiltonian system with a Euclidean metric on the position space.
Here Euclidean metric is defined to mean a metric with a fixed positive
definite matrix representation \(M\). The momentum variables are taken to
be independent of the position variables and with a zero-mean Gaussian
marginal distribution with covariance specified by \(M\), so that the
\(h_2\) Hamiltonian component is
\[ h_2(q, p) = \frac{1}{2} p^T M^{-1} p \]
where \(q\) and \(p\) are the position and momentum variables respectively.
The \(h_1\) Hamiltonian component function is
\[ h_1(q) = \ell(q) \]
where \(\ell(q)\) is the negative log (unnormalized) density of
the target distribution with respect to the Lebesgue measure.
"""
def __init__(self, neg_log_dens, metric=None, grad_neg_log_dens=None):
"""
Args:
neg_log_dens (Callable[[array], float]): Function which given a
position array returns the negative logarithm of an
unnormalized probability density on the position space with
respect to the Lebesgue measure, with the corresponding
distribution on the position space being the target
distribution it is wished to draw approximate samples from.
metric (None or array or PositiveDefiniteMatrix): Matrix object
corresponding to matrix representation of metric on position
space and covariance of Gaussian marginal distribution on
momentum vector. If `None` is passed (the default), the
identity matrix will be used. If a 1D array is passed then this
is assumed to specify a metric with positive diagonal matrix
representation and the array the matrix diagonal. If a 2D array
is passed then this is assumed to specify a metric with a dense
positive definite matrix representation specified by the array.
Otherwise if the value is a subclass of
`mici.matrices.PositiveDefiniteMatrix` it is assumed to
directly specify the metric matrix representation.
grad_neg_log_dens (
None or Callable[[array], array or Tuple[array, float]]):
Function which given a position array returns the derivative of
`neg_log_dens` with respect to the position array argument.
Optionally the function may instead return a 2-tuple of values
with the first being the array corresponding to the derivative
and the second being the value of the `neg_log_dens` evaluated
at the passed position array. If `None` is passed (the default)
an automatic differentiation fallback will be used to attempt
to construct the derivative of `neg_log_dens` automatically.
"""
super().__init__(neg_log_dens, grad_neg_log_dens)
if metric is None:
self.metric = matrices.IdentityMatrix()
elif isinstance(metric, np.ndarray):
if metric.ndim == 1:
self.metric = matrices.PositiveDiagonalMatrix(metric)
elif metric.ndim == 2:
self.metric = matrices.DensePositiveDefiniteMatrix(metric)
else:
raise ValueError(
"If NumPy ndarray value is used for `metric`"
" must be either 1D (diagonal matrix) or 2D "
"(dense positive definite matrix)"
)
else:
self.metric = metric
@cache_in_state("mom")
def h2(self, state):
return 0.5 * state.mom @ self.dh2_dmom(state)
@cache_in_state("mom")
def dh2_dmom(self, state):
return self.metric.inv @ state.mom
def h2_flow(self, state, dt):
"""Apply exact flow map corresponding to `h2` Hamiltonian component.
`state` argument is modified in place.
Args:
state (mici.states.ChainState): State to start flow at.
dt (float): Time interval to simulate flow for.
"""
state.pos += dt * self.dh2_dmom(state)
def dh2_flow_dmom(self, dt):
"""Derivatives of `h2_flow` flow map with respect to input momentum.
Args:
dt (float): Time interval flow simulated for.
Returns:
dpos_dmom (mici.matrices.Matrix): Matrix representing derivative
(Jacobian) of position output of `h2_flow` with respect to the
value of the momentum component of the initial input state.
dmom_dmom (mici.matrices.Matrix): Matrix representing derivative
(Jacobian) of momentum output of `h2_flow` with respect to the
value of the momentum component of the initial input state.
"""
return (dt * self.metric.inv, matrices.IdentityMatrix(self.metric.shape[0]))
def sample_momentum(self, state, rng):
return self.metric.sqrt @ rng.standard_normal(state.pos.shape)
class GaussianEuclideanMetricSystem(EuclideanMetricSystem):
r"""Euclidean Hamiltonian system with a tractable Gaussian component.
Here Euclidean metric is defined to mean a metric with a fixed positive
definite matrix representation \(M\). The momentum variables are taken to
be independent of the position variables and with a zero-mean Gaussian
marginal distribution with covariance specified by \(M\).
Additionally the target distribution on the position variables is assumed
to be defined by an unnormalized density with respect to the standard
Gaussian measure on the position space (with identity covariance and zero
mean), with the Hamiltonian component \(h_1\) corresponding to the negative
logarithm of this density rather than the density with respect to the
Lebesgue measure on the position space, i.e.
\[ h_1(q) = \ell(q) - \frac{1}{2} q^T q \]
where \(q\) is the position and \(\ell(q)\) is the negative log
(unnormalized) density of the target distribution with respect to the
Lebesgue measure at \(q\). The Hamiltonian component function \(h_2\) is
then assumed to have the form
\[ h_2(q, p) = \frac{1}{2} q^T q + \frac{1}{2} p^T M^{-1} p \]
where \(p\) is the momentum. In this case the Hamiltonian flow due to the
quadratic \(h_2\) component can be solved for analytically, allowing an
integrator to be defined using this alternative splitting of the
Hamiltonian [1].
References:
1. Shahbaba, B., Lan, S., Johnson, W.O. and Neal, R.M., 2014. Split
Hamiltonian Monte Carlo. Statistics and Computing, 24(3), pp.339-349.
"""
def __init__(self, neg_log_dens, metric=None, grad_neg_log_dens=None):
"""
Args:
neg_log_dens (Callable[[array], float]): Function which given a
position array returns the negative logarithm of an
unnormalized probability density on the position space with
respect to the standard Gaussian measure on the position space,
with the corresponding distribution on the position space being
the target distribution it is wished to draw approximate
samples from.
metric (None or array or PositiveDefiniteMatrix): Matrix object
corresponding to matrix representation of metric on position
space and covariance of Gaussian marginal distribution on
momentum vector. If `None` is passed (the default), the
identity matrix will be used. If a 1D array is passed then this
is assumed to specify a metric with positive diagonal matrix
representation and the array the matrix diagonal. If a 2D array
is passed then this is assumed to specify a metric with a dense
positive definite matrix representation specified by the array.
Otherwise if the value is a subclass of
`mici.matrices.PositiveDefiniteMatrix` it is assumed to
directly specify the metric matrix representation.
grad_neg_log_dens (
None or Callable[[array], array or Tuple[array, float]]):
Function which given a position array returns the derivative of
`neg_log_dens` with respect to the position array argument.
Optionally the function may instead return a 2-tuple of values
with the first being the array corresponding to the derivative
and the second being the value of the `neg_log_dens` evaluated
at the passed position array. If `None` is passed (the default)
an automatic differentiation fallback will be used to attempt
to construct the derivative of `neg_log_dens` automatically.
"""
super().__init__(neg_log_dens, metric, grad_neg_log_dens)
def h2(self, state):
return (
0.5 * state.pos @ state.pos + 0.5 * state.mom @ self.metric.inv @ state.mom
)
@cache_in_state("mom")
def dh2_dmom(self, state):
return self.metric.inv @ state.mom
@cache_in_state("mom")
def dh2_dpos(self, state):
return state.pos
def h2_flow(self, state, dt):
omega = 1.0 / self.metric.eigval ** 0.5
sin_omega_dt, cos_omega_dt = np.sin(omega * dt), np.cos(omega * dt)
eigvec_T_pos = self.metric.eigvec.T @ state.pos
eigvec_T_mom = self.metric.eigvec.T @ state.mom
state.pos = self.metric.eigvec @ (
cos_omega_dt * eigvec_T_pos + (sin_omega_dt * omega) * eigvec_T_mom
)
state.mom = self.metric.eigvec @ (
cos_omega_dt * eigvec_T_mom - (sin_omega_dt / omega) * eigvec_T_pos
)
def dh2_flow_dmom(self, dt):
omega = 1.0 / self.metric.eigval ** 0.5
sin_omega_dt, cos_omega_dt = np.sin(omega * dt), np.cos(omega * dt)
return (
matrices.EigendecomposedSymmetricMatrix(
self.metric.eigvec, sin_omega_dt * omega
),
matrices.EigendecomposedSymmetricMatrix(self.metric.eigvec, cos_omega_dt),
)
class ConstrainedEuclideanMetricSystem(EuclideanMetricSystem):
r"""Base class for Euclidean Hamiltonian systems subject to constraints.
The (constrained) position space is assumed to be a differentiable manifold
embedded with a \(Q\)-dimensional ambient Euclidean space. The \(Q-C\)
dimensional manifold \(\mathcal{M}\) is implicitly defined by an equation
\(\mathcal{M} = \lbrace q \in \mathbb{R}^Q : c(q) = 0 \rbrace\) with
\(c: \mathbb{R}^Q \to \mathbb{R}^C\) the *constraint function*.
The ambient Euclidean space is assumed to be equipped with a metric with
constant positive-definite matrix representation \(M\) which further
specifies the covariance of the zero-mean Gaussian distribution
\(\mathcal{N}(0, M)\) on the *unconstrained* momentum (co-)vector \(p\)
with corresponding \(h_2\) Hamiltonian component defined as
\[ h_2(q, p) = \frac{1}{2} p^T M^{-1} p. \]
The time-derivative of the constraint equation implies a further set of
constraints on the momentum \(q\) with \( \partial c(q) M^{-1} p = 0\)
at all time points, corresponding to the momentum (velocity) being in the
co-tangent space (tangent space) to the manifold.
The target distribution is either assumed to be directly specified with
unnormalized density \(\exp(-\ell(q))\) with respect to the Hausdorff
measure on the manifold (under the metric induced from the ambient metric)
with in this case the \(h_1\) Hamiltonian component then simply
\[ h_1(q) = \ell(q), \]
or alternatively it is assumed a prior distribution on the position \(q\)
with density \(\exp(-\ell(q))\) with respect to the Lebesgue measure on
the ambient space is specifed and the target distribution is the posterior
distribution on \(q\) when conditioning on the event \(c(q) = 0\). The
negative logarithm of the posterior distribution density with respect to
the Hausdorff measure (and so \(h_1\) Hamiltonian component) is then
\[
h_1(q) =
\ell(q) + \frac{1}{2} \log\left|\partial c(q)M^{-1}\partial c(q)^T\right|
\]
with an additional second *Gram matrix* determinant term to give the
correct density with respect to the Hausdorff measure on the manifold.
Due to the requirement to enforce the constraints on the position and
momentum, a constraint-preserving numerical integrator needs to be used
when simulating the Hamiltonian dynamic associated with the system, e.g.
`mici.integrators.ConstrainedLeapfrogIntegrator`.
References:
1. Lelièvre, T., Rousset, M. and Stoltz, G., 2019. Hybrid Monte Carlo
methods for sampling probability measures on submanifolds. Numerische
Mathematik, 143(2), pp.379-421.
2. Graham, M.M. and Storkey, A.J., 2017. Asymptotically exact inference
in differentiable generative models. Electronic Journal of Statistics,
11(2), pp.5105-5164.
"""
def __init__(
self,
neg_log_dens,
constr,
metric=None,
dens_wrt_hausdorff=True,
grad_neg_log_dens=None,
jacob_constr=None,
):
"""
Args:
neg_log_dens (Callable[[array], float]): Function which given a
position array returns the negative logarithm of an
unnormalized probability density on the constrained position
space with respect to the Hausdorff measure on the constraint
manifold (if `dens_wrt_hausdorff == True`) or alternatively the
negative logarithm of an unnormalized probability density on
the unconstrained (ambient) position space with respect to the
Lebesgue measure. In the former case the target distribution it
is wished to draw approximate samples from is assumed to be
directly specified by the density function on the manifold. In
the latter case the density function is instead taken to
specify a prior distribution on the ambient space with the
target distribution then corresponding to the posterior
distribution when conditioning on the (zero Lebesgue measure)
event `constr(pos) == 0`. This target posterior distribution
has support on the differentiable manifold implicitly defined
by the constraint equation, with density with respect to the
Hausdorff measure on the manifold corresponding to the ratio of
the prior density (specified by `neg_log_dens`) and the
square-root of the determinant of the Gram matrix defined by
gram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).T
where `jacob_constr` is the Jacobian of the constraint function
`constr` and `metric` is the matrix representation of the
metric on the ambient space.
constr (Callable[[array], array]): Function which given a position
array return as a 1D array the value of the (vector-valued)
constraint function, the zero level-set of which implicitly
defines the manifold the dynamic is simulated on.
metric (None or array or PositiveDefiniteMatrix): Matrix object
corresponding to matrix representation of metric on
*unconstrained* position space and covariance of Gaussian
marginal distribution on *unconstrained* momentum vector. If
`None` is passed (the default), the identity matrix will be
used. If a 1D array is passed then this is assumed to specify a
metric with positive diagonal matrix representation and the
array the matrix diagonal. If a 2D array is passed then this is
assumed to specify a metric with a dense positive definite
matrix representation specified by the array. Otherwise if the
value is a `mici.matrices.PositiveDefiniteMatrix` subclass it
is assumed to directly specify the metric matrix
representation.
dens_wrt_hausdorff (bool): Whether the `neg_log_dens` function
specifies the (negative logarithm) of the density of the target
distribution with respect to the Hausdorff measure on the
manifold directly (True) or alternatively the negative
logarithm of a density of a prior distriubtion on the
unconstrained (ambient) position space with respect to the
Lebesgue measure, with the target distribution then
corresponding to the posterior distribution when conditioning
on the event `const(pos) == 0` (False). Note that in the former
case the base Hausdorff measure on the manifold depends on the
metric defined on the ambient space, with the Hausdorff measure
being defined with respect to the metric induced on the
manifold from this ambient metric.
grad_neg_log_dens (
None or Callable[[array], array or Tuple[array, float]]):
Function which given a position array returns the derivative of
`neg_log_dens` with respect to the position array argument.
Optionally the function may instead return a 2-tuple of values
with the first being the array corresponding to the derivative
and the second being the value of the `neg_log_dens` evaluated
at the passed position array. If `None` is passed (the default)
an automatic differentiation fallback will be used to attempt
to construct a function to compute the derivative (and value)
of `neg_log_dens` automatically.
jacob_constr (
None or Callable[[array], array or Tuple[array, array]]):
Function which given a position array computes the Jacobian
(matrix / 2D array of partial derivatives) of the output of the
constraint function `c = constr(q)` with respect to the position
array argument `q`, returning the computed Jacobian as a 2D
array `jacob` with
jacob[i, j] = ∂c[i] / ∂q[j]
Optionally the function may instead return a 2-tuple of values
with the first being the array corresponding to the Jacobian and
the second being the value of `constr` evaluated at the passed
position array. If `None` is passed (the default) an automatic
differentiation fallback will be used to attempt to construct a
function to compute the Jacobian (and value) of `constr`
automatically.
"""
super().__init__(
neg_log_dens=neg_log_dens,
metric=metric,
grad_neg_log_dens=grad_neg_log_dens,
)
self._constr = constr
self.dens_wrt_hausdorff = dens_wrt_hausdorff
self._jacob_constr = autodiff_fallback(
jacob_constr, constr, "jacobian_and_value", "jacob_constr"
)
@cache_in_state("pos")
def constr(self, state):
"""Constraint function at the current position.
Args:
state (mici.states.ChainState): State to compute value at.
Returns:
array: Value of `constr(state.pos)` as 1D array.
"""
return self._constr(state.pos)
@cache_in_state_with_aux("pos", "constr")
def jacob_constr(self, state):
"""Jacobian of constraint function at the current position.
Args:
state (mici.states.ChainState): State to compute value at.
Returns:
array: Value of Jacobian of `constr(state.pos)` as 2D array.
"""
return self._jacob_constr(state.pos)
@abstractmethod
def jacob_constr_inner_product(
self, jacob_constr_1, inner_product_matrix, jacob_constr_2=None
):
"""Compute inner product of rows of constraint Jacobian matrices.
Computes `jacob_constr_1 @ inner_product_matrix @ jacob_constr_2.T`
potentially exploiting any structure / sparsity in `jacob_constr_1`,
`jacob_constr_2` and `inner_product_matrix`.
Args:
jacob_constr_1 (Matrix): First constraint Jacobian in product.
inner_product_matrix (Matrix): Positive-definite matrix defining
inner-product between rows of two constraint Jacobians.
jacob_constr_2 (None or Matrix): Second constraint Jacobian in
product. Defaults to `jacob_constr_1` if set to `None`.
Returns
Matrix: Object corresponding to computed inner products of
the constraint Jacobian rows.
"""
@cache_in_state("pos")
def gram(self, state):
"""Gram matrix at current position.
The Gram matrix as a position `q` is defined as
gram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).T
where `jacob_constr` is the Jacobian of the constraint function
`constr` and `metric` is the matrix representation of the metric on the
ambient space.
Args:
state (mici.states.ChainState): State to compute value at.
Returns:
mici.matrices.PositiveDefiniteMatrix: Gram matrix as matrix object.
"""
return self.jacob_constr_inner_product(
self.jacob_constr(state), self.metric.inv
)
def inv_gram(self, state):
"""Inverse of Gram matrix at current position.
Args:
state (mici.states.ChainState): State to compute value at.
Returns:
mici.matrices.PositiveDefiniteMatrix: Inverse of Gram matrix as
matrix object.
"""
return self.gram(state).inv
def log_det_sqrt_gram(self, state):
"""Value of (half of) log-determinant of Gram matrix."""
return 0.5 * self.gram(state).log_abs_det
@abstractmethod
def grad_log_det_sqrt_gram(self, state):
"""Derivative of (half of) log-determinant of Gram matrix wrt position.
Args:
state (mici.states.ChainState): State to compute value at.
Returns:
array: Value of `log_det_sqrt_gram(state)` derivative with respect
to `state.pos`.
"""
def h1(self, state):
if self.dens_wrt_hausdorff:
return self.neg_log_dens(state)
else:
return self.neg_log_dens(state) + self.log_det_sqrt_gram(state)
def dh1_dpos(self, state):
if self.dens_wrt_hausdorff:
return self.grad_neg_log_dens(state)
else:
return self.grad_neg_log_dens(state) + self.grad_log_det_sqrt_gram(state)
def project_onto_cotangent_space(self, mom, state):
"""Project a momentum on to the co-tangent space at a position.
Args:
mom (array): Momentum (co-)vector as 1D array to project on to
co-tangent space.
state (mici.states.ChainState): State definining position on the
manifold to project in to the co-tangent space of.
Returns:
array: Projected momentum in the co-tangent space at `state.pos`.
"""
# Use parenthesis to force right-to-left evaluation to avoid
# matrix-matrix products
mom -= self.jacob_constr(state).T @ (
self.inv_gram(state) @ (self.jacob_constr(state) @ (self.metric.inv @ mom))
)
return mom
def sample_momentum(self, state, rng):
mom = super().sample_momentum(state, rng)
mom = self.project_onto_cotangent_space(mom, state)
return mom
class DenseConstrainedEuclideanMetricSystem(ConstrainedEuclideanMetricSystem):
r"""Euclidean Hamiltonian system subject to a dense set of constraints.
See `ConstrainedEuclideanMetricSystem` for more details about constrained
systems.
"""
def __init__(
self,
neg_log_dens,
constr,
metric=None,
dens_wrt_hausdorff=True,
grad_neg_log_dens=None,
jacob_constr=None,
mhp_constr=None,
):
"""
Args:
neg_log_dens (Callable[[array], float]): Function which given a
position array returns the negative logarithm of an
unnormalized probability density on the constrained position
space with respect to the Hausdorff measure on the constraint
manifold (if `dens_wrt_hausdorff == True`) or alternatively the
negative logarithm of an unnormalized probability density on
the unconstrained (ambient) position space with respect to the
Lebesgue measure. In the former case the target distribution it
is wished to draw approximate samples from is assumed to be
directly specified by the density function on the manifold. In
the latter case the density function is instead taken to
specify a prior distribution on the ambient space with the
target distribution then corresponding to the posterior
distribution when conditioning on the (zero Lebesgue measure)
event `constr(pos) == 0`. This target posterior distribution
has support on the differentiable manifold implicitly defined
by the constraint equation, with density with respect to the
Hausdorff measure on the manifold corresponding to the ratio of
the prior density (specified by `neg_log_dens`) and the
square-root of the determinant of the Gram matrix defined by
gram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).T
where `jacob_constr` is the Jacobian of the constraint function
`constr` and `metric` is the matrix representation of the
metric on the ambient space.
constr (Callable[[array], array]): Function which given a position
array return as a 1D array the value of the (vector-valued)
constraint function, the zero level-set of which implicitly
defines the manifold the dynamic is simulated on.
metric (None or array or PositiveDefiniteMatrix): Matrix object
corresponding to matrix representation of metric on
*unconstrained* position space and covariance of Gaussian
marginal distribution on *unconstrained* momentum vector. If
`None` is passed (the default), the identity matrix will be
used. If a 1D array is passed then this is assumed to specify a
metric with positive diagonal matrix representation and the
array the matrix diagonal. If a 2D array is passed then this is
assumed to specify a metric with a dense positive definite
matrix representation specified by the array. Otherwise if the
value is a `mici.matrices.PositiveDefiniteMatrix` subclass it
is assumed to directly specify the metric matrix
representation.
dens_wrt_hausdorff (bool): Whether the `neg_log_dens` function
specifies the (negative logarithm) of the density of the target
distribution with respect to the Hausdorff measure on the
manifold directly (True) or alternatively the negative
logarithm of a density of a prior distriubtion on the
unconstrained (ambient) position space with respect to the
Lebesgue measure, with the target distribution then
corresponding to the posterior distribution when conditioning
on the event `const(pos) == 0` (False). Note that in the former
case the base Hausdorff measure on the manifold depends on the
metric defined on the ambient space, with the Hausdorff measure
being defined with respect to the metric induced on the
manifold from this ambient metric.
grad_neg_log_dens (
None or Callable[[array], array or Tuple[array, float]]):
Function which given a position array returns the derivative of
`neg_log_dens` with respect to the position array argument.
Optionally the function may instead return a 2-tuple of values
with the first being the array corresponding to the derivative
and the second being the value of the `neg_log_dens` evaluated
at the passed position array. If `None` is passed (the default)
an automatic differentiation fallback will be used to attempt
to construct a function to compute the derivative (and value)
of `neg_log_dens` automatically.
jacob_constr (
None or Callable[[array], array or Tuple[array, array]]):
Function which given a position array computes the Jacobian
(matrix / 2D array of partial derivatives) of the output of the
constraint function `c = constr(q)` with respect to the
position array argument `q`, returning the computed Jacobian as
a 2D array `jacob` with
jacob[i, j] = ∂c[i] / ∂q[j]
Optionally the function may instead return a 2-tuple of values
with the first being the array corresponding to the Jacobian
and the second being the value of `constr` evaluated
at the passed position array. If `None` is passed (the default)
an automatic differentiation fallback will be used to attempt
to construct a function to compute the Jacobian (and value) of
`neg_log_dens` automatically.
mhp_constr (None or
Callable[[array], Callable[[array], array]] or
Callable[[array], Tuple[Callable, array, array]]):
Function which given a position array returns another function
which takes a 2D array as an argument and returns the
*matrix-Hessian-product* (MHP) of the constraint function
`constr` with respect to the position array argument. The MHP
is here defined as a function of a `(dim_constr, dim_pos)`
shaped 2D array `m`
mhp(m) = sum(m[:, :, None] * hess[:, :, :], axis=(0, 1))
where `hess` is the `(dim_constr, dim_pos, dim_pos)` shaped
vector-Hessian of `c = constr(q)` with respect to `q` i.e. the
array of second-order partial derivatives of such that
hess[i, j, k] = ∂²c[i] / (∂q[j] ∂q[k])
Optionally the function may instead return a 3-tuple of values
with the first a function to compute a MHP of `constr`, the
second a 2D array corresponding to the Jacobian of `constr`,
and the third the value of `constr`, all evaluated at the
passed position array. If `None` is passed (the default) an
automatic differentiation fallback will be used to attempt to
construct a function which calculates the MHP (and Jacobian and
value) of `constr` automatically.
"""
super().__init__(
neg_log_dens=neg_log_dens,
constr=constr,
metric=metric,
dens_wrt_hausdorff=dens_wrt_hausdorff,
grad_neg_log_dens=grad_neg_log_dens,
jacob_constr=jacob_constr,
)
if not dens_wrt_hausdorff:
self._mhp_constr = autodiff_fallback(
mhp_constr, constr, "mhp_jacobian_and_value", "mhp_constr"
)
@cache_in_state_with_aux("pos", ("jacob_constr", "constr"))
def mhp_constr(self, state):
return self._mhp_constr(state.pos)
def jacob_constr_inner_product(
self, jacob_constr_1, inner_product_matrix, jacob_constr_2=None
):
if jacob_constr_2 is None or jacob_constr_2 is jacob_constr_1:
return matrices.DensePositiveDefiniteMatrix(
jacob_constr_1 @ (inner_product_matrix @ jacob_constr_1.T)
)
else:
return matrices.DenseSquareMatrix(
jacob_constr_1 @ (inner_product_matrix @ jacob_constr_2.T)
)
@cache_in_state("pos")
def grad_log_det_sqrt_gram(self, state):
# Evaluate MHP of constraint function before Jacobian as Jacobian value
# will potentially be computed in 'forward' pass and cached
mhp_constr = self.mhp_constr(state)
return mhp_constr(
self.inv_gram(state) @ self.jacob_constr(state) @ self.metric.inv
)
class GaussianDenseConstrainedEuclideanMetricSystem(
GaussianEuclideanMetricSystem, DenseConstrainedEuclideanMetricSystem
):
r"""Gaussian Euclidean Hamiltonian system st. a dense set of constraints.
See `ConstrainedEuclideanMetricSystem` for more details about constrained
systems and `GaussianEuclideanMetricSystem` for Gaussian Euclidean metric
systems.
"""
def __init__(
self,
neg_log_dens,
constr,
metric=None,
grad_neg_log_dens=None,
jacob_constr=None,
mhp_constr=None,
):
"""
Args:
neg_log_dens (Callable[[array], float]): Function which given a
position array returns the negative logarithm of an
unnormalized probability density on the unconstrained (ambient)
position space with respect to the standard Gaussian measure.
The density function is taken to specify a prior distribution
on the ambient space with the target distribution then
corresponding to the posterior distribution when conditioning
on the (zero Lebesgue measure) event `constr(pos) == 0`. This
target posterior distribution has support on the differentiable
manifold implicitly defined by the constraint equation, with
density with respect to the Hausdorff measure on the manifold
corresponding to the ratio of the prior density (specified by
`neg_log_dens`) and the square-root of the determinant of the
Gram matrix defined by
gram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).T
where `jacob_constr` is the Jacobian of the constraint function
`constr` and `metric` is the matrix representation of the
metric on the ambient space.
constr (Callable[[array], array]): Function which given a position
array return as a 1D array the value of the (vector-valued)
constraint function, the zero level-set of which implicitly
defines the manifold the dynamic is simulated on.
metric (None or array or PositiveDefiniteMatrix): Matrix object
corresponding to matrix representation of metric on
*unconstrained* position space and covariance of Gaussian
marginal distribution on *unconstrained* momentum vector. If
`None` is passed (the default), the identity matrix will be
used. If a 1D array is passed then this is assumed to specify a
metric with positive diagonal matrix representation and the
array the matrix diagonal. If a 2D array is passed then this is
assumed to specify a metric with a dense positive definite
matrix representation specified by the array. Otherwise if
a subclass of `mici.matrices.PositiveDefiniteMatrix` it is
assumed to directly specify the metric matrix representation.
grad_neg_log_dens (
None or Callable[[array], array or Tuple[array, float]]):
Function which given a position array returns the derivative of
`neg_log_dens` with respect to the position array argument.
Optionally the function may instead return a 2-tuple of values
with the first being the array corresponding to the derivative
and the second being the value of the `neg_log_dens` evaluated
at the passed position array. If `None` is passed (the default)
an automatic differentiation fallback will be used to attempt
to construct a function to compute the derivative (and value)
of `neg_log_dens` automatically.
jacob_constr (
None or Callable[[array], array or Tuple[array, array]]):
Function which given a position array computes the Jacobian
(matrix / 2D array of partial derivatives) of the output of the
constraint function `c = constr(q)` with respect to the
position array argument `q`, returning the computed Jacobian as
a 2D array `jacob` with
jacob[i, j] = ∂c[i] / ∂q[j]
Optionally the function may instead return a 2-tuple of values
with the first being the array corresponding to the Jacobian
and the second being the value of `constr` evaluated
at the passed position array. If `None` is passed (the default)
an automatic differentiation fallback will be used to attempt
to construct a function to compute the Jacobian (and value) of
`neg_log_dens` automatically.
mhp_constr (None or
Callable[[array], Callable[[array], array]] or
Callable[[array], Tuple[Callable, array, array]]):
Function which given a position array returns another function
which takes a 2D array as an argument and returns the
*matrix-Hessian-product* (MHP) of the constraint function
`constr` with respect to the position array argument. The MHP
is here defined as a function of a `(dim_constr, dim_pos)`
shaped 2D array `m`
mhp(m) = sum(m[:, :, None] * hess[:, :, :], axis=(0, 1))
where `hess` is the `(dim_constr, dim_pos, dim_pos)` shaped
vector-Hessian of `c = constr(q)` with respect to `q` i.e. the
array of second-order partial derivatives of such that
hess[i, j, k] = ∂²c[i] / (∂q[j] ∂q[k])
Optionally the function may instead return a 3-tuple of values
with the first a function to compute a MHP of `constr`, the
second a 2D array corresponding to the Jacobian of `constr`,
and the third the value of `constr`, all evaluated at the
passed position array. If `None` is passed (the default) an
automatic differentiation fallback will be used to attempt to
construct a function which calculates the MHP (and Jacobian and
value) of `constr` automatically.
"""
DenseConstrainedEuclideanMetricSystem.__init__(
self,
neg_log_dens=neg_log_dens,
constr=constr,
metric=metric,
dens_wrt_hausdorff=False,
grad_neg_log_dens=grad_neg_log_dens,
jacob_constr=jacob_constr,
mhp_constr=mhp_constr,
)
def jacob_constr_inner_product(
self, jacob_constr_1, inner_product_matrix, jacob_constr_2=None
):
if jacob_constr_2 is None or jacob_constr_2 is jacob_constr_1:
return matrices.DenseSymmetricMatrix(
jacob_constr_1 @ (inner_product_matrix @ jacob_constr_1.T)
)
else:
return matrices.DenseSquareMatrix(
jacob_constr_1 @ (inner_product_matrix @ jacob_constr_2.T)
)
class RiemannianMetricSystem(System):
r"""Riemannian Hamiltonian system with a position-dependent metric.
This class allows for metric matrix representations of any generic type.
In most cases a specialized subclass such as `DenseRiemannianMetricSystem`,
`CholeskyFactoredRiemannianMetricSystem`, `DiagonalRiemannianMetricSystem`,
`ScalarRiemannianMetricSystem` or `SoftAbsRiemannianMetricSystem` will
provide a simpler method of constructng a system with a metric matrix
representation of a specific type.
The position space is assumed to be a Riemannian manifold with a metric
with position-dependent positive definite matrix-representation \(M(q)\)
where \(q\) is a position vector. The momentum \(p\) is then taken to have
a zero-mean Gaussian conditional distribution given the position \(q\),
with covariance \(M(q)\), i.e. \(p \sim \mathcal{N}(0, M(q))\) [1].
The \(h_1\) Hamiltonian component is then
\[ h_1(q) = \ell(q) + \frac{1}{2}\log\left|M(q)\right| \]
where \(\ell(q)\) is the negative log (unnormalized) density of the target
distribution with respect to the Lebesgue measure at \(q\). The \(h_2\)
Hamiltonian component is
\[ h_2(q, p) = \frac{1}{2} p^T (M(q))^{-1} p. \]
Due to the coupling between the position and momentum variables in \(h_2\),
the Hamiltonian system is non-separable, requiring use of a numerical
integrator with implicit steps when simulating the Hamiltonian dynamic
associated with the system, e.g.
`mici.integrators.ImplicitLeapfrogIntegrator`.
References:
1. Girolami, M. and Calderhead, B., 2011. Riemann manifold Langevin and
Hamiltonian Monte Varlo methods. Journal of the Royal Statistical
Society: Series B (Statistical Methodology), 73(2), pp.123-214.
"""
def __init__(
self,
neg_log_dens,
metric_matrix_class,
metric_func,
vjp_metric_func=None,
grad_neg_log_dens=None,
metric_kwargs=None,
):
"""
Args:
neg_log_dens (Callable[[array], float]): Function which given a
position array returns the negative logarithm of an
unnormalized probability density on the position space with
respect to the Lebesgue measure, with the corresponding
distribution on the position space being the target
distribution it is wished to draw approximate samples from.
metric_matrix_class (type[PositiveDefiniteMatrix]): Class (or
factory function returning an instance of the class) which
defines type of matrix representation of metric. The class
initializer should take a single positional argument which will
be passed the array outputted by `metric_func`, and which is
assumed to be a parameter which fully defines the resulting
matrix (e.g. the diagonal of a `mici.matrices.DiagonalMatrix`).
The class initializer may also optionally take one or more
keyword arguments, with the `metric_kwargs` argument used to
specify the value of these, if any. Together this means the
metric matrix representation at a position `pos` is constructed
as
metric = metric_matrix_class(
metric_func(pos), **metric_kwargs)
The `mici.matrices.PositiveDefiniteMatrix` subclass should as a
minimum define `inv`, `log_abs_det`, `grad_log_abs_det`,
`grad_quadratic_form_inv`, `__matmul__` and `__rmatmul__`
methods / properties (see documentation of
`mici.matrices.PositiveDefiniteMatrix` and
`mici.matrices.DifferentiableMatrix` for definitions of the
expected behaviour of these methods).
metric_func (Callable[[array], array]): Function which given a
position array returns an array containing the parameter value
of the metric matrix representation passed as the single
positional argument to the `metric_matrix_class` initializer.
vjp_metric_func (None or
Callable[[array], Callable[[array], array]] or
Callable[[array], Tuple[Callable[[array], array], array]]):
Function which given a position array returns another function
which takes an array as an argument and returns the
*vector-Jacobian-product* (VJP) of `metric_func` with respect
to the position array argument. The VJP is here defined as a
function of an array `v` (of the same shape as the output of
`metric_func`) corresponding to
vjp(v) = sum(v[..., None] * jacob, tuple(range(v.ndim))
where `jacob` is the Jacobian of `m = metric_func(q)` wrt `q`
i.e. the array of partial derivatives of the function such that
jacob[..., i] = ∂m[...] / ∂q[i]
Optionally the function may instead return a 2-tuple of values
with the first a function to compute a VJP of `metric_func` and
the second an array containing the value of `metric_func`, both
evaluated at the passed position array. If `None` is passed
(the default) an automatic differentiation fallback will be
used to attempt to construct a function which calculates the
VJP (and value) of `metric_func` automatically.
grad_neg_log_dens (
None or Callable[[array], array or Tuple[array, float]]):
Function which given a position array returns the derivative of
`neg_log_dens` with respect to the position array argument.
Optionally the function may instead return a 2-tuple of values
with the first being the array corresponding to the derivative
and the second being the value of the `neg_log_dens` evaluated
at the passed position array. If `None` is passed (the default)
an automatic differentiation fallback will be used to attempt
to construct the derivative of `neg_log_dens` automatically.
metric_kwargs (None or Dict[str, object]): An optional dictionary
of any additional keyword arguments to the initializer of
`metric_matrix_class`.
"""
self._metric_matrix_class = metric_matrix_class
self._metric_func = metric_func
self._vjp_metric_func = autodiff_fallback(
vjp_metric_func, metric_func, "vjp_and_value", "vjp_metric_func"
)
self._metric_kwargs = {} if metric_kwargs is None else metric_kwargs
super().__init__(neg_log_dens, grad_neg_log_dens)
@cache_in_state("pos")
def metric_func(self, state):
"""
Function computing the parameter of the metric matrix representation.
Args:
state (mici.states.ChainState): State to compute value at.
Returns:
array: Value of `metric_func(state.pos)`.
"""
return self._metric_func(state.pos)
@cache_in_state_with_aux("pos", "metric_func")
def vjp_metric_func(self, state):
"""
Function constructing a vector-Jacobian-product for `metric_func`.
The vector-Jacobian-product is here defined as a function of an array
`v` (of the same shape as the output of `metric_func`) corresponding to
vjp(v) = sum(v[..., None] * jacob, axis=tuple(range(v.ndim))
where `jacob` is the Jacobian of `m = metric_func(q)` wrt `q` i.e.
the array of partial derivatives of the function such that
jacob[..., i] = ∂m[...] / ∂q[i]
Args:
state (mici.states.ChainState): State to compute VJP at.
Returns:
Callable[[array], array]: Vector-Jacobian-product function.
"""
return self._vjp_metric_func(state.pos)
@cache_in_state("pos")
def metric(self, state):
"""
Function computing the metric matrix representation.
The returned type of this function is that specified by the
`metric_matrix_class` argument to the initializer.
Args:
state (mici.states.ChainState): State to compute value at.
Returns:
mici.matrices.PositiveDefiniteMatrix: Metric matrix representation.
"""
return self._metric_matrix_class(self.metric_func(state), **self._metric_kwargs)
def h(self, state):
return self.h1(state) + self.h2(state)
def h1(self, state):
return self.neg_log_dens(state) + 0.5 * self.metric(state).log_abs_det
def dh1_dpos(self, state):
# Evaluate VJP of metric function before metric as metric value will
# potentially be computed in forward pass and cached
vjp_metric = self.vjp_metric_func(state)
return self.grad_neg_log_dens(state) + 0.5 * vjp_metric(
self.metric(state).grad_log_abs_det
)
def h2(self, state):
return 0.5 * state.mom @ self.metric(state).inv @ state.mom
def dh2_dpos(self, state):
# Evaluate VJP of metric function before metric as metric value will
# potentially be computed in forward pass and cached
vjp_metric = self.vjp_metric_func(state)
return 0.5 * vjp_metric(self.metric(state).grad_quadratic_form_inv(state.mom))
def dh2_dmom(self, state):
return self.metric(state).inv @ state.mom
def sample_momentum(self, state, rng):
return self.metric(state).sqrt @ rng.normal(size=state.pos.shape)
class ScalarRiemannianMetricSystem(RiemannianMetricSystem):
"""Riemannian-metric system with scaled identity matrix representation.
Hamiltonian system with a position dependent scaled identity metric matrix
representation which is specified by a scalar function
`metric_scalar_function` of the position `q` which outputs a strictly
positive scalar `s = metric_scalar_func(q)` with the the metric matrix
representation then taken to be `s * identity(q.shape[0])`.
See documentation of `RiemannianMetricSystem` for more general details
about Riemannian-metric Hamiltonian systems.
"""
def __init__(
self,
neg_log_dens,
metric_scalar_func,
vjp_metric_scalar_func=None,
grad_neg_log_dens=None,
):
"""
Args:
neg_log_dens (Callable[[array], float]): Function which given a
position array returns the negative logarithm of an
unnormalized probability density on the position space with
respect to the Lebesgue measure, with the corresponding
distribution on the position space being the target
distribution it is wished to draw approximate samples from.
metric_scalar_func (Callable[[array], float]): Function which
given a position array returns a strictly positive scalar
corresponding to the parameter value of the scaled identity
metric matrix representation.
vjp_metric_scalar_func (None or
Callable[[array], Callable[[array], float]] or
Callable[[array], Tuple[Callable[[array, float]], float]]):
Function which given a position array returns another function
which takes a scalar as an argument and returns the
*vector-Jacobian-product* (VJP) of `metric_scalar_func` with
respect to the position array argument. The VJP is here defined
as a function of a scalar `v`
vjp(v) = v * grad
where `grad` is the `(dim_pos,)` shaped Jacobian (gradient) of
`s = metric_scalar_func(q)` with respect to `q` i.e. the array
of partial derivatives of the function such that
grad[i] = ∂s / ∂q[i]
Optionally the function may instead return a 2-tuple of values
with the first a function to compute a VJP of
`metric_scalar_func` and the second a float containing the
value of `metric_scalar_func`, both evaluated at the passed
position array. If `None` is passed (the default) an automatic
differentiation fallback will be used to attempt to construct a
function which calculates the VJP (and value) of
`metric_scalar_func` automatically.
grad_neg_log_dens (
None or Callable[[array], array or Tuple[array, float]]):
Function which given a position array returns the derivative of
`neg_log_dens` with respect to the position array argument.
Optionally the function may instead return a 2-tuple of values
with the first being the array corresponding to the derivative
and the second being the value of the `neg_log_dens` evaluated
at the passed position array. If `None` is passed (the default)
an automatic differentiation fallback will be used to attempt
to construct the derivative of `neg_log_dens` automatically.
"""
super().__init__(
neg_log_dens,
matrices.PositiveScaledIdentityMatrix,
metric_scalar_func,
vjp_metric_scalar_func,
grad_neg_log_dens,
)
@cache_in_state("pos")
def metric(self, state):
return self._metric_matrix_class(
self.metric_func(state), size=state.pos.shape[0]
)
class DiagonalRiemannianMetricSystem(RiemannianMetricSystem):
"""Riemannian-metric system with diagonal matrix representation.
Hamiltonian system with a position dependent diagonal metric matrix
representation which is specified by a vector-valued function
`metric_diagonal_func` of the position `q` which outputs a 1D array with
strictly positive elements `d = metric_diagonal_func(q)` with the metric
matrix representation then taken to be `diag(d)`.
See documentation of `RiemannianMetricSystem` for more general details
about Riemannian-metric Hamiltonian systems.
"""
def __init__(
self,
neg_log_dens,
metric_diagonal_func,
vjp_metric_diagonal_func=None,
grad_neg_log_dens=None,
):
"""
Args:
neg_log_dens (Callable[[array], float]): Function which given a
position array returns the negative logarithm of an
unnormalized probability density on the position space with
respect to the Lebesgue measure, with the corresponding
distribution on the position space being the target
distribution it is wished to draw approximate samples from.
metric_diagonal_func (Callable[[array], array]): Function which
given a position array returns a 1D array with strictly
positive values corresponding to the diagonal values
(left-to-right) of the diagonal metric matrix representation.
vjp_metric_diagonal_func (None or
Callable[[array], Callable[[array], array]] or
Callable[[array], Tuple[Callable[[array], array], array]]):
Function which given a position array returns another function
which takes a 1D array as an argument and returns the
*vector-Jacobian-product* (VJP) of `metric_diagonal_func` with
respect to the position array argument. The VJP is here defined
as a function of a 1D array `v`
vjp(v) = sum(v[:, None] * jacob[:, :], axis=0)
where `jacob` is the `(dim_pos, dim_pos)` shaped Jacobian of
`d = metric_diagonal_func(q)` with respect to `q` i.e. the
array of partial derivatives of the function such that
jacob[i, j] = ∂d[i] / ∂q[j]
Optionally the function may instead return a 2-tuple of values
with the first a function to compute a VJP of
`metric_diagonal_func` and the second a 1D array containing the
value of `metric_diagonal_func`, both evaluated at the passed
position array. If `None` is passed (the default) an automatic
differentiation fallback will be used to attempt to construct a
function which calculates the VJP (and value) of
`metric_diagonal_func` automatically.
grad_neg_log_dens (
None or Callable[[array], array or Tuple[array, float]]):
Function which given a position array returns the derivative of
`neg_log_dens` with respect to the position array argument.
Optionally the function may instead return a 2-tuple of values
with the first being the array corresponding to the derivative
and the second being the value of the `neg_log_dens` evaluated
at the passed position array. If `None` is passed (the default)
an automatic differentiation fallback will be used to attempt
to construct the derivative of `neg_log_dens` automatically.
"""
super().__init__(
neg_log_dens,
matrices.PositiveDiagonalMatrix,
metric_diagonal_func,
vjp_metric_diagonal_func,
grad_neg_log_dens,
)
class CholeskyFactoredRiemannianMetricSystem(RiemannianMetricSystem):
"""Riemannian-metric system with Cholesky-factored matrix representation.
Hamiltonian system with a position dependent metric matrix representation
which is specified by its Cholesky factor by a matrix function
`metric_chol_func` of the position `q` which outputs a lower-triangular
matrix `L = metric_chol_func(q)` with the metric matrix representation then
taken to be `L @ L.T`.
See documentation of `RiemannianMetricSystem` for more general details
about Riemannian-metric Hamiltonian systems.
"""
def __init__(
self,
neg_log_dens,
metric_chol_func,
vjp_metric_chol_func=None,
grad_neg_log_dens=None,
):
"""
Args:
neg_log_dens (Callable[[array], float]): Function which given a
position array returns the negative logarithm of an
unnormalized probability density on the position space with
respect to the Lebesgue measure, with the corresponding
distribution on the position space being the target
distribution it is wished to draw approximate samples from.
metric_chol_func (Callable[[array], array]): Function which given
a position array returns a 2D array with zeros above the
diagonal corresponding to the lower-triangular Cholesky-factor
of the positive definite metric matrix representation.
vjp_metric_chol_func (None or
Callable[[array], Callable[[array], array]] or
Callable[[array], Tuple[Callable[[array], array], array]]):
Function which given a position array returns another function
which takes a lower-triangular 2D array as an argument (any
values in the array above the diagonal are ignored) and returns
the *vector-Jacobian-product* (VJP) of `metric_chol_func` with
respect to the position array argument. The VJP is here defined
as a function of a 2D array `v`
vjp(v) = sum(v[:, :, None] * jacob[:, :, :], axis=(0, 1))
where `jacob` is the `(dim_pos, dim_pos, dim_pos)` shaped
Jacobian of `L = metric_chol_func(q)` with respect to `q` i.e.
the array of partial derivatives of the function such that
jacob[i, j, k] = ∂L[i, j] / ∂q[k]
Optionally the function may instead return a 2-tuple of values
with the first a function to compute a VJP of
`metric_chol_func` and the second a 2D array containing the
value of `metric_chol_func`, both evaluated at the passed
position array. If `None` is passed (the default) an automatic
differentiation fallback will be used to attempt to construct a
function which calculates the VJP (and value) of
`metric_chol_func` automatically.
grad_neg_log_dens (
None or Callable[[array], array or Tuple[array, float]]):
Function which given a position array returns the derivative of
`neg_log_dens` with respect to the position array argument.
Optionally the function may instead return a 2-tuple of values
with the first being the array corresponding to the derivative
and the second being the value of the `neg_log_dens` evaluated
at the passed position array. If `None` is passed (the default)
an automatic differentiation fallback will be used to attempt
to construct the derivative of `neg_log_dens` automatically.
"""
super().__init__(
neg_log_dens,
matrices.TriangularFactoredPositiveDefiniteMatrix,
metric_chol_func,
vjp_metric_chol_func,
grad_neg_log_dens,
metric_kwargs={"factor_is_lower": True},
)
class DenseRiemannianMetricSystem(RiemannianMetricSystem):
"""Riemannian-metric system with dense matrix representation.
Hamiltonian system with a position dependent metric matrix representation
which is specified to be a dense matrix function `metric_func` of the
position `q` which is guaranteed to be positive definite almost-everywhere,
with `M = metric_func(q)` then the metric matrix representation.
See documentation of `RiemannianMetricSystem` for more general details
about Riemannian-metric Hamiltonian systems.
"""
def __init__(
self, neg_log_dens, metric_func, vjp_metric_func=None, grad_neg_log_dens=None
):
"""
Args:
neg_log_dens (Callable[[array], float]): Function which given a
position array returns the negative logarithm of an
unnormalized probability density on the position space with
respect to the Lebesgue measure, with the corresponding
distribution on the position space being the target
distribution it is wished to draw approximate samples from.
metric_func (Callable[[array], array]): Function which given a
position array returns a 2D array corresponding to the positive
definite metric matrix representation. The returned matrices
(2D arrays) are assumed to be positive-definite for all input
positions and a `LinAlgError` exception may be raised if this
fails to be the case.
vjp_metric_func (None or
Callable[[array], Callable[[array], array]] or
Callable[[array], Tuple[Callable[[array], array], array]]):
Function which given a position array returns another function
which takes a 2D array as an argument and returns the
*vector-Jacobian-product* (VJP) of `metric_func` with respect
to the position array argument. The VJP is here defined as a
function of a 2D array `v`
vjp(v) = sum(v[:, :, None] * jacob[:, :, :], axis=(0, 1))
where `jacob` is the `(dim_pos, dim_pos, dim_pos)` shaped
Jacobian of `M = metric_func(q)` with respect to `q` i.e. the
array of partial derivatives of the function such that
jacob[i, j, k] = ∂M[i, j] / ∂q[k]
Optionally the function may instead return a 2-tuple of values
with the first a function to compute a VJP of `metric_func` and
the second a 2D array containing the value of `metric_func`,
both evaluated at the passed position array. If `None` is
passed (the default) an automatic differentiation fallback will
be used to attempt to construct a function which calculates the
VJP (and value) of `metric_func` automatically.
grad_neg_log_dens (
None or Callable[[array], array or Tuple[array, float]]):
Function which given a position array returns the derivative of
`neg_log_dens` with respect to the position array argument.
Optionally the function may instead return a 2-tuple of values
with the first being the array corresponding to the derivative
and the second being the value of the `neg_log_dens` evaluated
at the passed position array. If `None` is passed (the default)
an automatic differentiation fallback will be used to attempt
to construct the derivative of `neg_log_dens` automatically.
"""
super().__init__(
neg_log_dens,
matrices.DensePositiveDefiniteMatrix,
metric_func,
vjp_metric_func,
grad_neg_log_dens,
)
class SoftAbsRiemannianMetricSystem(RiemannianMetricSystem):
"""SoftAbs Riemmanian metric Hamiltonian system.
Hamiltonian system with a position dependent metric matrix representation
which is specified to be a dense matrix function `metric_func` of the
position `q` which is guaranteed to be positive definite almost-everywhere,
with `M = metric_func(q)` then the metric matrix representation.
Hamiltonian system with a position dependent metric matrix representation
which is specified to be an eigenvalue-regularized transformation of the
Hessian of the negative log density function (the symmetric matrix of
second derivatives the negative log density function with respect to the
position array components. Specifically if `hess_neg_log_dens` is a
symmetric 2D square array valued function of the position `q`, with
`H = hess_neg_log_dens(q)` then if an eigenvalue decomposition of `H` is
computed, i.e. `eigval, eigvec = eigh(H)`, with `eigval` a 1D array of
real eigenvalues, and `eigvec` the corresponding 2D array (orthogonal
matrix) with eigenvectors as columns, then the resulting positive-definite
metric matrix representation `M` is computed as
M = eigvec @ diag(softabs(eigval, softabs_coeff)) @ eigvec.T
with `softabs(x, softabs_coeff) = x / tanh(x * softabs_coeff)` an
elementwise function which acts as a smooth approximation to the absolute
function (ensuring all the eigenvalues of `M` are strictly positive) with
the additional scalar parameter `softabs_coeff` controlling the smoothness
of the approximation, with `softabs` tending to the piecewise linear `abs`
function as `softabs_coeff` tends to infinity, and becoming increasingly
smooth as `softabs_coeff` tends to zero.
See documentation of `RiemannianMetricSystem` for more general details
about Riemannian-metric Hamiltonian systems.
References:
1. Betancourt, M., 2013. A general metric for Riemannian manifold
Hamiltonian Monte Carlo. In Geometric science of information
(pp. 327-334).
"""
def __init__(
self,
neg_log_dens,
grad_neg_log_dens=None,
hess_neg_log_dens=None,
mtp_neg_log_dens=None,
softabs_coeff=1.0,
):
"""
Args:
neg_log_dens (Callable[[array], float]): Function which given a
position array returns the negative logarithm of an
unnormalized probability density on the position space with
respect to the Lebesgue measure, with the corresponding
distribution on the position space being the target
distribution it is wished to draw approximate samples from.
grad_neg_log_dens (
None or Callable[[array], array or Tuple[array, float]]):
Function which given a position array returns the derivative of
`neg_log_dens` with respect to the position array argument.
Optionally the function may instead return a 2-tuple of values
with the first being the array corresponding to the derivative
and the second being the value of the `neg_log_dens` evaluated
at the passed position array. If `None` is passed (the default)
an automatic differentiation fallback will be used to attempt
to construct the derivative of `neg_log_dens` automatically.
hess_neg_log_dens (None or
Callable[[array], array or Tuple[array, array, float]]):
Function which given a position array returns the Hessian of
`neg_log_dens` with respect to the position array argument as a
2D array. Optionally the function may instead return a 3-tuple
of values with the first a 2D array containting the Hessian of
`neg_log_dens`, the second a 1D array containing the gradient
of `neg_log_dens` and the third the value of `neg_log_dens`,
all evaluated at the passed position array. If `None` is passed
(the default) an automatic differentiation fallback will be
used to attempt to construct a function which calculates the
Hessian (and gradient and value) of `neg_log_dens`
automatically.
mtp_neg_log_dens (None or
Callable[[array], Callable[[array], array]] or
Callable[[array], Tuple[Callable, array, array, float]]):
Function which given a position array returns another function
which takes a 2D array (matrix) as an argument and returns the
*matrix-Tressian-product* (MTP) of `neg_log_dens` with respect
to the position array argument. The MTP is here defined as a
function of a matrix `m` corresponding to
mtp(m) = sum(m[:, :, None] * tress[:, :, :], axis=(0, 1))
where `tress` is the 'Tressian' of `f = neg_log_dens(q)` wrt
`q` i.e. the 3D array of third-order partial derivatives of the
scalar-valued function such that
tress[i, j, k] = ∂³f / (∂q[i] ∂q[j] ∂q[k])
Optionally the function may instead return a 4-tuple of values
with the first a function to compute a MTP of `neg_log_dens`,
the second a 2D array containing the Hessian of `neg_log_dens`,
the third a 1D array containing the gradient of `neg_log_dens`
and the fourth the value of `neg_log_dens`, all evaluated at
the passed position array. If `None` is passed (the default) an
automatic differentiation fallback will be used to attempt to
construct a function which calculates the MTP (and Hesisan and
gradient and value) of `neg_log_dens` automatically.
softabs_coeff (float): Positive regularisation coefficient for
smooth approximation to absolute value used to regularize
Hessian eigenvalues in metric matrix representation. As the
value tends to infinity the approximation becomes increasingly
close to the absolute function.
"""
self._hess_neg_log_dens = autodiff_fallback(
hess_neg_log_dens, neg_log_dens, "hessian_grad_and_value", "neg_log_dens"
)
self._mtp_neg_log_dens = autodiff_fallback(
mtp_neg_log_dens,
neg_log_dens,
"mtp_hessian_grad_and_value",
"mtp_neg_log_dens",
)
super().__init__(
neg_log_dens,
matrices.SoftAbsRegularizedPositiveDefiniteMatrix,
self._hess_neg_log_dens,
self._mtp_neg_log_dens,
grad_neg_log_dens,
metric_kwargs={"softabs_coeff": softabs_coeff},
)
def metric_func(self, state):
return self.hess_neg_log_dens(state)
def vjp_metric_func(self, state):
return self.mtp_neg_log_dens(state)
@cache_in_state_with_aux("pos", ("grad_neg_log_dens", "neg_log_dens"))
def hess_neg_log_dens(self, state):
"""Hessian of negative log density with respect to position.
Args:
state (mici.states.ChainState): State to compute value at.
Returns:
hessian (array): 2D array of `neg_log_dens(state)` second
derivatives with respect to `state.pos`, with `hessian[i, j]`
the second derivative of `neg_log_dens(state)` with respect to
`state.pos[i]` and `state.pos[j]`.
"""
return self._hess_neg_log_dens(state.pos)
@cache_in_state_with_aux(
"pos", ("hess_neg_log_dens", "grad_neg_log_dens", "neg_log_dens")
)
def mtp_neg_log_dens(self, state):
"""Generate MTP of negative log density with respect to position.
The matrix-Tressian-product (MTP) is here defined as a function of a
matrix `m` corresponding to
mtp(m) = sum(m[:, :, None] * tress[:, :, :], axis=(0, 1))
where `tress` is the 'Tressian' of `f = neg_log_dens(q)` with respect
to `q = state.pos` i.e. the 3D array of third-order partial derivatives
of the scalar-valued function such that
tress[i, j, k] = ∂³f / (∂q[i] ∂q[j] ∂q[k])
Args:
state (mici.states.ChainState): State to compute value at.
Returns:
mtp (Callable[[array], array]): Function which accepts a 2D array
of shape `(state.pos.shape[0], state.pos.shape[0])` as an
argument and returns an array of shape `state.pos.shape`
containing the computed MTP value.
"""
return self._mtp_neg_log_dens(state.pos)Classes
class System (neg_log_dens, grad_neg_log_dens=None)-
Base class for Hamiltonian systems.
The Hamiltonian function h is assumed to have the general form
h(q, p) = h_1(q) + h_2(q, p)
where q and p are the position and momentum variables respectively, and h_1 and h_2 Hamiltonian component functions. The exact Hamiltonian flow for the h_1 component can be always be computed as it depends only on the position variable however depending on the form of h_2 the corresponding exact Hamiltonian flow may or may not be simulable.
By default h_1 is assumed to correspond to the negative logarithm of an unnormalized density on the position variables with respect to the Lebesgue measure, with the corresponding distribution on the position space being the target distribution it is wished to draw approximate samples from.
Args
neg_log_dens:Callable[[array],float]- Function which given a position array returns the negative logarithm of an unnormalized probability density on the position space with respect to the Lebesgue measure, with the corresponding distribution on the position space being the target distribution it is wished to draw approximate samples from.
grad_neg_log_dens:NoneorCallable[[array],arrayorTuple[array,float]]-
Function which given a position array returns the derivative of
neg_log_denswith respect to the position array argument. Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the derivative and the second being the value of theneg_log_densevaluated at the passed position array. IfNoneis passed (the default) an automatic differentiation fallback will be used to attempt to construct the derivative ofneg_log_densautomatically.
Expand source code Browse git
class System(ABC): r"""Base class for Hamiltonian systems. The Hamiltonian function \(h\) is assumed to have the general form \[ h(q, p) = h_1(q) + h_2(q, p) \] where \(q\) and \(p\) are the position and momentum variables respectively, and \(h_1\) and \(h_2\) Hamiltonian component functions. The exact Hamiltonian flow for the \(h_1\) component can be always be computed as it depends only on the position variable however depending on the form of \(h_2\) the corresponding exact Hamiltonian flow may or may not be simulable. By default \(h_1\) is assumed to correspond to the negative logarithm of an unnormalized density on the position variables with respect to the Lebesgue measure, with the corresponding distribution on the position space being the target distribution it is wished to draw approximate samples from. """ def __init__(self, neg_log_dens, grad_neg_log_dens=None): """ Args: neg_log_dens (Callable[[array], float]): Function which given a position array returns the negative logarithm of an unnormalized probability density on the position space with respect to the Lebesgue measure, with the corresponding distribution on the position space being the target distribution it is wished to draw approximate samples from. grad_neg_log_dens ( None or Callable[[array], array or Tuple[array, float]]): Function which given a position array returns the derivative of `neg_log_dens` with respect to the position array argument. Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the derivative and the second being the value of the `neg_log_dens` evaluated at the passed position array. If `None` is passed (the default) an automatic differentiation fallback will be used to attempt to construct the derivative of `neg_log_dens` automatically. """ self._neg_log_dens = neg_log_dens self._grad_neg_log_dens = autodiff_fallback( grad_neg_log_dens, neg_log_dens, "grad_and_value", "grad_neg_log_dens" ) @cache_in_state("pos") def neg_log_dens(self, state): """Negative logarithm of unnormalized density of target distribution. Args: state (mici.states.ChainState): State to compute value at. Returns: float: Value of computed negative log density. """ return self._neg_log_dens(state.pos) @cache_in_state_with_aux("pos", "neg_log_dens") def grad_neg_log_dens(self, state): """Derivative of negative log density with respect to position. Args: state (mici.states.ChainState): State to compute value at. Returns: array: Value of `neg_log_dens(state)` derivative with respect to `state.pos`. """ return self._grad_neg_log_dens(state.pos) def h1(self, state): """Hamiltonian component depending only on position. Args: state (mici.states.ChainState): State to compute value at. Returns: float: Value of `h1` Hamiltonian component. """ return self.neg_log_dens(state) def dh1_dpos(self, state): """Derivative of `h1` Hamiltonian component with respect to position. Args: state (mici.states.ChainState): State to compute value at. Returns: array: Value of computed `h1` derivative. """ return self.grad_neg_log_dens(state) def h1_flow(self, state, dt): """Apply exact flow map corresponding to `h1` Hamiltonian component. `state` argument is modified in place. Args: state (mici.states.ChainState): State to start flow at. dt (float): Time interval to simulate flow for. """ state.mom -= dt * self.dh1_dpos(state) @abstractmethod def h2(self, state): """Hamiltonian component depending on momentum and optionally position. Args: state (mici.states.ChainState): State to compute value at. Returns: float: Value of `h2` Hamiltonian component. """ @abstractmethod def dh2_dmom(self, state): """Derivative of `h2` Hamiltonian component with respect to momentum. Args: state (mici.states.ChainState): State to compute value at. Returns: array: Value of `h2(state)` derivative with respect to `state.pos`. """ def h(self, state): """Hamiltonian function for system. Args: state (mici.states.ChainState): State to compute value at. Returns: float: Value of Hamiltonian. """ return self.h1(state) + self.h2(state) def dh_dpos(self, state): """Derivative of Hamiltonian with respect to position. Args: state (mici.states.ChainState): State to compute value at. Returns: array: Value of `h(state)` derivative with respect to `state.pos`. """ if hasattr(self, "dh2_dpos"): return self.dh1_dpos(state) + self.dh2_dpos(state) else: return self.dh1_dpos(state) def dh_dmom(self, state): """Derivative of Hamiltonian with respect to momentum. Args: state (mici.states.ChainState): State to compute value at. Returns: array: Value of `h(state)` derivative with respect to `state.mom`. """ return self.dh2_dmom(state) @abstractmethod def sample_momentum(self, state, rng): """ Sample a momentum from its conditional distribution given a position. Args: state (mici.states.ChainState): State defining position to condition on. Returns: mom (array): Sampled momentum. """Ancestors
- abc.ABC
Subclasses
Methods
def neg_log_dens(self, state)-
Negative logarithm of unnormalized density of target distribution.
Args
state:ChainState- State to compute value at.
Returns
float- Value of computed negative log density.
Expand source code Browse git
@cache_in_state("pos") def neg_log_dens(self, state): """Negative logarithm of unnormalized density of target distribution. Args: state (mici.states.ChainState): State to compute value at. Returns: float: Value of computed negative log density. """ return self._neg_log_dens(state.pos) def grad_neg_log_dens(self, state)-
Derivative of negative log density with respect to position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
neg_log_dens(state)derivative with respect tostate.pos.
Expand source code Browse git
@cache_in_state_with_aux("pos", "neg_log_dens") def grad_neg_log_dens(self, state): """Derivative of negative log density with respect to position. Args: state (mici.states.ChainState): State to compute value at. Returns: array: Value of `neg_log_dens(state)` derivative with respect to `state.pos`. """ return self._grad_neg_log_dens(state.pos) def h1(self, state)-
Hamiltonian component depending only on position.
Args
state:ChainState- State to compute value at.
Returns
float- Value of
h1Hamiltonian component.
Expand source code Browse git
def h1(self, state): """Hamiltonian component depending only on position. Args: state (mici.states.ChainState): State to compute value at. Returns: float: Value of `h1` Hamiltonian component. """ return self.neg_log_dens(state) def dh1_dpos(self, state)-
Derivative of
h1Hamiltonian component with respect to position.Args
state:ChainState- State to compute value at.
Returns
array- Value of computed
h1derivative.
Expand source code Browse git
def dh1_dpos(self, state): """Derivative of `h1` Hamiltonian component with respect to position. Args: state (mici.states.ChainState): State to compute value at. Returns: array: Value of computed `h1` derivative. """ return self.grad_neg_log_dens(state) def h1_flow(self, state, dt)-
Apply exact flow map corresponding to
h1Hamiltonian component.stateargument is modified in place.Args
state:ChainState- State to start flow at.
dt:float- Time interval to simulate flow for.
Expand source code Browse git
def h1_flow(self, state, dt): """Apply exact flow map corresponding to `h1` Hamiltonian component. `state` argument is modified in place. Args: state (mici.states.ChainState): State to start flow at. dt (float): Time interval to simulate flow for. """ state.mom -= dt * self.dh1_dpos(state) def h2(self, state)-
Hamiltonian component depending on momentum and optionally position.
Args
state:ChainState- State to compute value at.
Returns
float- Value of
h2Hamiltonian component.
Expand source code Browse git
@abstractmethod def h2(self, state): """Hamiltonian component depending on momentum and optionally position. Args: state (mici.states.ChainState): State to compute value at. Returns: float: Value of `h2` Hamiltonian component. """ def dh2_dmom(self, state)-
Derivative of
h2Hamiltonian component with respect to momentum.Args
state:ChainState- State to compute value at.
Returns
array- Value of
h2(state)derivative with respect tostate.pos.
Expand source code Browse git
@abstractmethod def dh2_dmom(self, state): """Derivative of `h2` Hamiltonian component with respect to momentum. Args: state (mici.states.ChainState): State to compute value at. Returns: array: Value of `h2(state)` derivative with respect to `state.pos`. """ def h(self, state)-
Hamiltonian function for system.
Args
state:ChainState- State to compute value at.
Returns
float- Value of Hamiltonian.
Expand source code Browse git
def h(self, state): """Hamiltonian function for system. Args: state (mici.states.ChainState): State to compute value at. Returns: float: Value of Hamiltonian. """ return self.h1(state) + self.h2(state) def dh_dpos(self, state)-
Derivative of Hamiltonian with respect to position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
h(state)derivative with respect tostate.pos.
Expand source code Browse git
def dh_dpos(self, state): """Derivative of Hamiltonian with respect to position. Args: state (mici.states.ChainState): State to compute value at. Returns: array: Value of `h(state)` derivative with respect to `state.pos`. """ if hasattr(self, "dh2_dpos"): return self.dh1_dpos(state) + self.dh2_dpos(state) else: return self.dh1_dpos(state) def dh_dmom(self, state)-
Derivative of Hamiltonian with respect to momentum.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
h(state)derivative with respect tostate.mom.
Expand source code Browse git
def dh_dmom(self, state): """Derivative of Hamiltonian with respect to momentum. Args: state (mici.states.ChainState): State to compute value at. Returns: array: Value of `h(state)` derivative with respect to `state.mom`. """ return self.dh2_dmom(state) def sample_momentum(self, state, rng)-
Sample a momentum from its conditional distribution given a position.
Args
state:ChainState- State defining position to condition on.
Returns
mom:array- Sampled momentum.
Expand source code Browse git
@abstractmethod def sample_momentum(self, state, rng): """ Sample a momentum from its conditional distribution given a position. Args: state (mici.states.ChainState): State defining position to condition on. Returns: mom (array): Sampled momentum. """
class EuclideanMetricSystem (neg_log_dens, metric=None, grad_neg_log_dens=None)-
Hamiltonian system with a Euclidean metric on the position space.
Here Euclidean metric is defined to mean a metric with a fixed positive definite matrix representation M. The momentum variables are taken to be independent of the position variables and with a zero-mean Gaussian marginal distribution with covariance specified by M, so that the h_2 Hamiltonian component is
h_2(q, p) = \frac{1}{2} p^T M^{-1} p
where q and p are the position and momentum variables respectively.
The h_1 Hamiltonian component function is
h_1(q) = \ell(q)
where \ell(q) is the negative log (unnormalized) density of the target distribution with respect to the Lebesgue measure.
Args
neg_log_dens:Callable[[array],float]- Function which given a position array returns the negative logarithm of an unnormalized probability density on the position space with respect to the Lebesgue measure, with the corresponding distribution on the position space being the target distribution it is wished to draw approximate samples from.
metric:NoneorarrayorPositiveDefiniteMatrix- Matrix object
corresponding to matrix representation of metric on position
space and covariance of Gaussian marginal distribution on
momentum vector. If
Noneis passed (the default), the identity matrix will be used. If a 1D array is passed then this is assumed to specify a metric with positive diagonal matrix representation and the array the matrix diagonal. If a 2D array is passed then this is assumed to specify a metric with a dense positive definite matrix representation specified by the array. Otherwise if the value is a subclass ofPositiveDefiniteMatrixit is assumed to directly specify the metric matrix representation. grad_neg_log_dens:NoneorCallable[[array],arrayorTuple[array,float]]-
Function which given a position array returns the derivative of
neg_log_denswith respect to the position array argument. Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the derivative and the second being the value of theneg_log_densevaluated at the passed position array. IfNoneis passed (the default) an automatic differentiation fallback will be used to attempt to construct the derivative ofneg_log_densautomatically.
Expand source code Browse git
class EuclideanMetricSystem(System): r"""Hamiltonian system with a Euclidean metric on the position space. Here Euclidean metric is defined to mean a metric with a fixed positive definite matrix representation \(M\). The momentum variables are taken to be independent of the position variables and with a zero-mean Gaussian marginal distribution with covariance specified by \(M\), so that the \(h_2\) Hamiltonian component is \[ h_2(q, p) = \frac{1}{2} p^T M^{-1} p \] where \(q\) and \(p\) are the position and momentum variables respectively. The \(h_1\) Hamiltonian component function is \[ h_1(q) = \ell(q) \] where \(\ell(q)\) is the negative log (unnormalized) density of the target distribution with respect to the Lebesgue measure. """ def __init__(self, neg_log_dens, metric=None, grad_neg_log_dens=None): """ Args: neg_log_dens (Callable[[array], float]): Function which given a position array returns the negative logarithm of an unnormalized probability density on the position space with respect to the Lebesgue measure, with the corresponding distribution on the position space being the target distribution it is wished to draw approximate samples from. metric (None or array or PositiveDefiniteMatrix): Matrix object corresponding to matrix representation of metric on position space and covariance of Gaussian marginal distribution on momentum vector. If `None` is passed (the default), the identity matrix will be used. If a 1D array is passed then this is assumed to specify a metric with positive diagonal matrix representation and the array the matrix diagonal. If a 2D array is passed then this is assumed to specify a metric with a dense positive definite matrix representation specified by the array. Otherwise if the value is a subclass of `mici.matrices.PositiveDefiniteMatrix` it is assumed to directly specify the metric matrix representation. grad_neg_log_dens ( None or Callable[[array], array or Tuple[array, float]]): Function which given a position array returns the derivative of `neg_log_dens` with respect to the position array argument. Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the derivative and the second being the value of the `neg_log_dens` evaluated at the passed position array. If `None` is passed (the default) an automatic differentiation fallback will be used to attempt to construct the derivative of `neg_log_dens` automatically. """ super().__init__(neg_log_dens, grad_neg_log_dens) if metric is None: self.metric = matrices.IdentityMatrix() elif isinstance(metric, np.ndarray): if metric.ndim == 1: self.metric = matrices.PositiveDiagonalMatrix(metric) elif metric.ndim == 2: self.metric = matrices.DensePositiveDefiniteMatrix(metric) else: raise ValueError( "If NumPy ndarray value is used for `metric`" " must be either 1D (diagonal matrix) or 2D " "(dense positive definite matrix)" ) else: self.metric = metric @cache_in_state("mom") def h2(self, state): return 0.5 * state.mom @ self.dh2_dmom(state) @cache_in_state("mom") def dh2_dmom(self, state): return self.metric.inv @ state.mom def h2_flow(self, state, dt): """Apply exact flow map corresponding to `h2` Hamiltonian component. `state` argument is modified in place. Args: state (mici.states.ChainState): State to start flow at. dt (float): Time interval to simulate flow for. """ state.pos += dt * self.dh2_dmom(state) def dh2_flow_dmom(self, dt): """Derivatives of `h2_flow` flow map with respect to input momentum. Args: dt (float): Time interval flow simulated for. Returns: dpos_dmom (mici.matrices.Matrix): Matrix representing derivative (Jacobian) of position output of `h2_flow` with respect to the value of the momentum component of the initial input state. dmom_dmom (mici.matrices.Matrix): Matrix representing derivative (Jacobian) of momentum output of `h2_flow` with respect to the value of the momentum component of the initial input state. """ return (dt * self.metric.inv, matrices.IdentityMatrix(self.metric.shape[0])) def sample_momentum(self, state, rng): return self.metric.sqrt @ rng.standard_normal(state.pos.shape)Ancestors
- System
- abc.ABC
Subclasses
Methods
def h2(self, state)-
Hamiltonian component depending on momentum and optionally position.
Args
state:ChainState- State to compute value at.
Returns
float- Value of
h2Hamiltonian component.
Expand source code Browse git
@cache_in_state("mom") def h2(self, state): return 0.5 * state.mom @ self.dh2_dmom(state) def dh2_dmom(self, state)-
Derivative of
h2Hamiltonian component with respect to momentum.Args
state:ChainState- State to compute value at.
Returns
array- Value of
h2(state)derivative with respect tostate.pos.
Expand source code Browse git
@cache_in_state("mom") def dh2_dmom(self, state): return self.metric.inv @ state.mom def h2_flow(self, state, dt)-
Apply exact flow map corresponding to
h2Hamiltonian component.stateargument is modified in place.Args
state:ChainState- State to start flow at.
dt:float- Time interval to simulate flow for.
Expand source code Browse git
def h2_flow(self, state, dt): """Apply exact flow map corresponding to `h2` Hamiltonian component. `state` argument is modified in place. Args: state (mici.states.ChainState): State to start flow at. dt (float): Time interval to simulate flow for. """ state.pos += dt * self.dh2_dmom(state) def dh2_flow_dmom(self, dt)-
Derivatives of
h2_flowflow map with respect to input momentum.Args
dt:float- Time interval flow simulated for.
Returns
dpos_dmom:Matrix- Matrix representing derivative
(Jacobian) of position output of
h2_flowwith respect to the value of the momentum component of the initial input state. dmom_dmom:Matrix- Matrix representing derivative
(Jacobian) of momentum output of
h2_flowwith respect to the value of the momentum component of the initial input state.
Expand source code Browse git
def dh2_flow_dmom(self, dt): """Derivatives of `h2_flow` flow map with respect to input momentum. Args: dt (float): Time interval flow simulated for. Returns: dpos_dmom (mici.matrices.Matrix): Matrix representing derivative (Jacobian) of position output of `h2_flow` with respect to the value of the momentum component of the initial input state. dmom_dmom (mici.matrices.Matrix): Matrix representing derivative (Jacobian) of momentum output of `h2_flow` with respect to the value of the momentum component of the initial input state. """ return (dt * self.metric.inv, matrices.IdentityMatrix(self.metric.shape[0])) def sample_momentum(self, state, rng)-
Sample a momentum from its conditional distribution given a position.
Args
state:ChainState- State defining position to condition on.
Returns
mom:array- Sampled momentum.
Expand source code Browse git
def sample_momentum(self, state, rng): return self.metric.sqrt @ rng.standard_normal(state.pos.shape) def neg_log_dens(self, state)-
Negative logarithm of unnormalized density of target distribution.
Args
state:ChainState- State to compute value at.
Returns
float- Value of computed negative log density.
def grad_neg_log_dens(self, state)-
Derivative of negative log density with respect to position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
neg_log_dens(state)derivative with respect tostate.pos.
def h1(self, state)-
Hamiltonian component depending only on position.
Args
state:ChainState- State to compute value at.
Returns
float- Value of
h1Hamiltonian component.
def dh1_dpos(self, state)-
Derivative of
h1Hamiltonian component with respect to position.Args
state:ChainState- State to compute value at.
Returns
array- Value of computed
h1derivative.
def h1_flow(self, state, dt)-
Apply exact flow map corresponding to
h1Hamiltonian component.stateargument is modified in place.Args
state:ChainState- State to start flow at.
dt:float- Time interval to simulate flow for.
def h(self, state)-
Hamiltonian function for system.
Args
state:ChainState- State to compute value at.
Returns
float- Value of Hamiltonian.
def dh_dpos(self, state)-
Derivative of Hamiltonian with respect to position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
h(state)derivative with respect tostate.pos.
def dh_dmom(self, state)-
Derivative of Hamiltonian with respect to momentum.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
h(state)derivative with respect tostate.mom.
class GaussianEuclideanMetricSystem (neg_log_dens, metric=None, grad_neg_log_dens=None)-
Euclidean Hamiltonian system with a tractable Gaussian component.
Here Euclidean metric is defined to mean a metric with a fixed positive definite matrix representation M. The momentum variables are taken to be independent of the position variables and with a zero-mean Gaussian marginal distribution with covariance specified by M.
Additionally the target distribution on the position variables is assumed to be defined by an unnormalized density with respect to the standard Gaussian measure on the position space (with identity covariance and zero mean), with the Hamiltonian component h_1 corresponding to the negative logarithm of this density rather than the density with respect to the Lebesgue measure on the position space, i.e.
h_1(q) = \ell(q) - \frac{1}{2} q^T q
where q is the position and \ell(q) is the negative log (unnormalized) density of the target distribution with respect to the Lebesgue measure at q. The Hamiltonian component function h_2 is then assumed to have the form
h_2(q, p) = \frac{1}{2} q^T q + \frac{1}{2} p^T M^{-1} p
where p is the momentum. In this case the Hamiltonian flow due to the quadratic h_2 component can be solved for analytically, allowing an integrator to be defined using this alternative splitting of the Hamiltonian [1].
References
- Shahbaba, B., Lan, S., Johnson, W.O. and Neal, R.M., 2014. Split Hamiltonian Monte Carlo. Statistics and Computing, 24(3), pp.339-349.
Args
neg_log_dens:Callable[[array],float]- Function which given a position array returns the negative logarithm of an unnormalized probability density on the position space with respect to the standard Gaussian measure on the position space, with the corresponding distribution on the position space being the target distribution it is wished to draw approximate samples from.
metric:NoneorarrayorPositiveDefiniteMatrix- Matrix object
corresponding to matrix representation of metric on position
space and covariance of Gaussian marginal distribution on
momentum vector. If
Noneis passed (the default), the identity matrix will be used. If a 1D array is passed then this is assumed to specify a metric with positive diagonal matrix representation and the array the matrix diagonal. If a 2D array is passed then this is assumed to specify a metric with a dense positive definite matrix representation specified by the array. Otherwise if the value is a subclass ofPositiveDefiniteMatrixit is assumed to directly specify the metric matrix representation. grad_neg_log_dens:NoneorCallable[[array],arrayorTuple[array,float]]-
Function which given a position array returns the derivative of
neg_log_denswith respect to the position array argument. Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the derivative and the second being the value of theneg_log_densevaluated at the passed position array. IfNoneis passed (the default) an automatic differentiation fallback will be used to attempt to construct the derivative ofneg_log_densautomatically.
Expand source code Browse git
class GaussianEuclideanMetricSystem(EuclideanMetricSystem): r"""Euclidean Hamiltonian system with a tractable Gaussian component. Here Euclidean metric is defined to mean a metric with a fixed positive definite matrix representation \(M\). The momentum variables are taken to be independent of the position variables and with a zero-mean Gaussian marginal distribution with covariance specified by \(M\). Additionally the target distribution on the position variables is assumed to be defined by an unnormalized density with respect to the standard Gaussian measure on the position space (with identity covariance and zero mean), with the Hamiltonian component \(h_1\) corresponding to the negative logarithm of this density rather than the density with respect to the Lebesgue measure on the position space, i.e. \[ h_1(q) = \ell(q) - \frac{1}{2} q^T q \] where \(q\) is the position and \(\ell(q)\) is the negative log (unnormalized) density of the target distribution with respect to the Lebesgue measure at \(q\). The Hamiltonian component function \(h_2\) is then assumed to have the form \[ h_2(q, p) = \frac{1}{2} q^T q + \frac{1}{2} p^T M^{-1} p \] where \(p\) is the momentum. In this case the Hamiltonian flow due to the quadratic \(h_2\) component can be solved for analytically, allowing an integrator to be defined using this alternative splitting of the Hamiltonian [1]. References: 1. Shahbaba, B., Lan, S., Johnson, W.O. and Neal, R.M., 2014. Split Hamiltonian Monte Carlo. Statistics and Computing, 24(3), pp.339-349. """ def __init__(self, neg_log_dens, metric=None, grad_neg_log_dens=None): """ Args: neg_log_dens (Callable[[array], float]): Function which given a position array returns the negative logarithm of an unnormalized probability density on the position space with respect to the standard Gaussian measure on the position space, with the corresponding distribution on the position space being the target distribution it is wished to draw approximate samples from. metric (None or array or PositiveDefiniteMatrix): Matrix object corresponding to matrix representation of metric on position space and covariance of Gaussian marginal distribution on momentum vector. If `None` is passed (the default), the identity matrix will be used. If a 1D array is passed then this is assumed to specify a metric with positive diagonal matrix representation and the array the matrix diagonal. If a 2D array is passed then this is assumed to specify a metric with a dense positive definite matrix representation specified by the array. Otherwise if the value is a subclass of `mici.matrices.PositiveDefiniteMatrix` it is assumed to directly specify the metric matrix representation. grad_neg_log_dens ( None or Callable[[array], array or Tuple[array, float]]): Function which given a position array returns the derivative of `neg_log_dens` with respect to the position array argument. Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the derivative and the second being the value of the `neg_log_dens` evaluated at the passed position array. If `None` is passed (the default) an automatic differentiation fallback will be used to attempt to construct the derivative of `neg_log_dens` automatically. """ super().__init__(neg_log_dens, metric, grad_neg_log_dens) def h2(self, state): return ( 0.5 * state.pos @ state.pos + 0.5 * state.mom @ self.metric.inv @ state.mom ) @cache_in_state("mom") def dh2_dmom(self, state): return self.metric.inv @ state.mom @cache_in_state("mom") def dh2_dpos(self, state): return state.pos def h2_flow(self, state, dt): omega = 1.0 / self.metric.eigval ** 0.5 sin_omega_dt, cos_omega_dt = np.sin(omega * dt), np.cos(omega * dt) eigvec_T_pos = self.metric.eigvec.T @ state.pos eigvec_T_mom = self.metric.eigvec.T @ state.mom state.pos = self.metric.eigvec @ ( cos_omega_dt * eigvec_T_pos + (sin_omega_dt * omega) * eigvec_T_mom ) state.mom = self.metric.eigvec @ ( cos_omega_dt * eigvec_T_mom - (sin_omega_dt / omega) * eigvec_T_pos ) def dh2_flow_dmom(self, dt): omega = 1.0 / self.metric.eigval ** 0.5 sin_omega_dt, cos_omega_dt = np.sin(omega * dt), np.cos(omega * dt) return ( matrices.EigendecomposedSymmetricMatrix( self.metric.eigvec, sin_omega_dt * omega ), matrices.EigendecomposedSymmetricMatrix(self.metric.eigvec, cos_omega_dt), )Ancestors
- EuclideanMetricSystem
- System
- abc.ABC
Subclasses
Methods
def h2(self, state)-
Hamiltonian component depending on momentum and optionally position.
Args
state:ChainState- State to compute value at.
Returns
float- Value of
h2Hamiltonian component.
Expand source code Browse git
def h2(self, state): return ( 0.5 * state.pos @ state.pos + 0.5 * state.mom @ self.metric.inv @ state.mom ) def dh2_dmom(self, state)-
Derivative of
h2Hamiltonian component with respect to momentum.Args
state:ChainState- State to compute value at.
Returns
array- Value of
h2(state)derivative with respect tostate.pos.
Expand source code Browse git
@cache_in_state("mom") def dh2_dmom(self, state): return self.metric.inv @ state.mom def dh2_dpos(self, state)-
Expand source code Browse git
@cache_in_state("mom") def dh2_dpos(self, state): return state.pos def h2_flow(self, state, dt)-
Apply exact flow map corresponding to
h2Hamiltonian component.stateargument is modified in place.Args
state:ChainState- State to start flow at.
dt:float- Time interval to simulate flow for.
Expand source code Browse git
def h2_flow(self, state, dt): omega = 1.0 / self.metric.eigval ** 0.5 sin_omega_dt, cos_omega_dt = np.sin(omega * dt), np.cos(omega * dt) eigvec_T_pos = self.metric.eigvec.T @ state.pos eigvec_T_mom = self.metric.eigvec.T @ state.mom state.pos = self.metric.eigvec @ ( cos_omega_dt * eigvec_T_pos + (sin_omega_dt * omega) * eigvec_T_mom ) state.mom = self.metric.eigvec @ ( cos_omega_dt * eigvec_T_mom - (sin_omega_dt / omega) * eigvec_T_pos ) def dh2_flow_dmom(self, dt)-
Derivatives of
h2_flowflow map with respect to input momentum.Args
dt:float- Time interval flow simulated for.
Returns
dpos_dmom:Matrix- Matrix representing derivative
(Jacobian) of position output of
h2_flowwith respect to the value of the momentum component of the initial input state. dmom_dmom:Matrix- Matrix representing derivative
(Jacobian) of momentum output of
h2_flowwith respect to the value of the momentum component of the initial input state.
Expand source code Browse git
def dh2_flow_dmom(self, dt): omega = 1.0 / self.metric.eigval ** 0.5 sin_omega_dt, cos_omega_dt = np.sin(omega * dt), np.cos(omega * dt) return ( matrices.EigendecomposedSymmetricMatrix( self.metric.eigvec, sin_omega_dt * omega ), matrices.EigendecomposedSymmetricMatrix(self.metric.eigvec, cos_omega_dt), ) def sample_momentum(self, state, rng)-
Sample a momentum from its conditional distribution given a position.
Args
state:ChainState- State defining position to condition on.
Returns
mom:array- Sampled momentum.
def neg_log_dens(self, state)-
Negative logarithm of unnormalized density of target distribution.
Args
state:ChainState- State to compute value at.
Returns
float- Value of computed negative log density.
def grad_neg_log_dens(self, state)-
Derivative of negative log density with respect to position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
neg_log_dens(state)derivative with respect tostate.pos.
def h1(self, state)-
Hamiltonian component depending only on position.
Args
state:ChainState- State to compute value at.
Returns
float- Value of
h1Hamiltonian component.
def dh1_dpos(self, state)-
Derivative of
h1Hamiltonian component with respect to position.Args
state:ChainState- State to compute value at.
Returns
array- Value of computed
h1derivative.
def h1_flow(self, state, dt)-
Apply exact flow map corresponding to
h1Hamiltonian component.stateargument is modified in place.Args
state:ChainState- State to start flow at.
dt:float- Time interval to simulate flow for.
def h(self, state)-
Hamiltonian function for system.
Args
state:ChainState- State to compute value at.
Returns
float- Value of Hamiltonian.
def dh_dpos(self, state)-
Derivative of Hamiltonian with respect to position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
h(state)derivative with respect tostate.pos.
def dh_dmom(self, state)-
Derivative of Hamiltonian with respect to momentum.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
h(state)derivative with respect tostate.mom.
class ConstrainedEuclideanMetricSystem (neg_log_dens, constr, metric=None, dens_wrt_hausdorff=True, grad_neg_log_dens=None, jacob_constr=None)-
Base class for Euclidean Hamiltonian systems subject to constraints.
The (constrained) position space is assumed to be a differentiable manifold embedded with a Q-dimensional ambient Euclidean space. The Q-C dimensional manifold \mathcal{M} is implicitly defined by an equation \mathcal{M} = \lbrace q \in \mathbb{R}^Q : c(q) = 0 \rbrace with c: \mathbb{R}^Q \to \mathbb{R}^C the constraint function.
The ambient Euclidean space is assumed to be equipped with a metric with constant positive-definite matrix representation M which further specifies the covariance of the zero-mean Gaussian distribution \mathcal{N}(0, M) on the unconstrained momentum (co-)vector p with corresponding h_2 Hamiltonian component defined as
h_2(q, p) = \frac{1}{2} p^T M^{-1} p.
The time-derivative of the constraint equation implies a further set of constraints on the momentum q with \partial c(q) M^{-1} p = 0 at all time points, corresponding to the momentum (velocity) being in the co-tangent space (tangent space) to the manifold.
The target distribution is either assumed to be directly specified with unnormalized density \exp(-\ell(q)) with respect to the Hausdorff measure on the manifold (under the metric induced from the ambient metric) with in this case the h_1 Hamiltonian component then simply
h_1(q) = \ell(q),
or alternatively it is assumed a prior distribution on the position q with density \exp(-\ell(q)) with respect to the Lebesgue measure on the ambient space is specifed and the target distribution is the posterior distribution on q when conditioning on the event c(q) = 0. The negative logarithm of the posterior distribution density with respect to the Hausdorff measure (and so h_1 Hamiltonian component) is then
h_1(q) = \ell(q) + \frac{1}{2} \log\left|\partial c(q)M^{-1}\partial c(q)^T\right|
with an additional second Gram matrix determinant term to give the correct density with respect to the Hausdorff measure on the manifold.
Due to the requirement to enforce the constraints on the position and momentum, a constraint-preserving numerical integrator needs to be used when simulating the Hamiltonian dynamic associated with the system, e.g.
ConstrainedLeapfrogIntegrator.References
- Lelièvre, T., Rousset, M. and Stoltz, G., 2019. Hybrid Monte Carlo methods for sampling probability measures on submanifolds. Numerische Mathematik, 143(2), pp.379-421.
- Graham, M.M. and Storkey, A.J., 2017. Asymptotically exact inference in differentiable generative models. Electronic Journal of Statistics, 11(2), pp.5105-5164.
Args
neg_log_dens:Callable[[array],float]-
Function which given a position array returns the negative logarithm of an unnormalized probability density on the constrained position space with respect to the Hausdorff measure on the constraint manifold (if
dens_wrt_hausdorff == True) or alternatively the negative logarithm of an unnormalized probability density on the unconstrained (ambient) position space with respect to the Lebesgue measure. In the former case the target distribution it is wished to draw approximate samples from is assumed to be directly specified by the density function on the manifold. In the latter case the density function is instead taken to specify a prior distribution on the ambient space with the target distribution then corresponding to the posterior distribution when conditioning on the (zero Lebesgue measure) eventconstr(pos) == 0. This target posterior distribution has support on the differentiable manifold implicitly defined by the constraint equation, with density with respect to the Hausdorff measure on the manifold corresponding to the ratio of the prior density (specified byneg_log_dens) and the square-root of the determinant of the Gram matrix defined bygram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).Twhere
jacob_constris the Jacobian of the constraint functionconstrandmetricis the matrix representation of the metric on the ambient space. constr:Callable[[array],array]- Function which given a position array return as a 1D array the value of the (vector-valued) constraint function, the zero level-set of which implicitly defines the manifold the dynamic is simulated on.
metric:NoneorarrayorPositiveDefiniteMatrix- Matrix object
corresponding to matrix representation of metric on
unconstrained position space and covariance of Gaussian
marginal distribution on unconstrained momentum vector. If
Noneis passed (the default), the identity matrix will be used. If a 1D array is passed then this is assumed to specify a metric with positive diagonal matrix representation and the array the matrix diagonal. If a 2D array is passed then this is assumed to specify a metric with a dense positive definite matrix representation specified by the array. Otherwise if the value is aPositiveDefiniteMatrixsubclass it is assumed to directly specify the metric matrix representation. dens_wrt_hausdorff:bool- Whether the
neg_log_densfunction specifies the (negative logarithm) of the density of the target distribution with respect to the Hausdorff measure on the manifold directly (True) or alternatively the negative logarithm of a density of a prior distriubtion on the unconstrained (ambient) position space with respect to the Lebesgue measure, with the target distribution then corresponding to the posterior distribution when conditioning on the eventconst(pos) == 0(False). Note that in the former case the base Hausdorff measure on the manifold depends on the metric defined on the ambient space, with the Hausdorff measure being defined with respect to the metric induced on the manifold from this ambient metric. grad_neg_log_dens:NoneorCallable[[array],arrayorTuple[array,float]]-
Function which given a position array returns the derivative of
neg_log_denswith respect to the position array argument. Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the derivative and the second being the value of theneg_log_densevaluated at the passed position array. IfNoneis passed (the default) an automatic differentiation fallback will be used to attempt to construct a function to compute the derivative (and value) ofneg_log_densautomatically. jacob_constr:NoneorCallable[[array],arrayorTuple[array,array]]-
Function which given a position array computes the Jacobian (matrix / 2D array of partial derivatives) of the output of the constraint function
c = constr(q)with respect to the position array argumentq, returning the computed Jacobian as a 2D arrayjacobwithjacob[i, j] = ∂c[i] / ∂q[j]Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the Jacobian and the second being the value of
constrevaluated at the passed position array. IfNoneis passed (the default) an automatic differentiation fallback will be used to attempt to construct a function to compute the Jacobian (and value) ofconstrautomatically.
Expand source code Browse git
class ConstrainedEuclideanMetricSystem(EuclideanMetricSystem): r"""Base class for Euclidean Hamiltonian systems subject to constraints. The (constrained) position space is assumed to be a differentiable manifold embedded with a \(Q\)-dimensional ambient Euclidean space. The \(Q-C\) dimensional manifold \(\mathcal{M}\) is implicitly defined by an equation \(\mathcal{M} = \lbrace q \in \mathbb{R}^Q : c(q) = 0 \rbrace\) with \(c: \mathbb{R}^Q \to \mathbb{R}^C\) the *constraint function*. The ambient Euclidean space is assumed to be equipped with a metric with constant positive-definite matrix representation \(M\) which further specifies the covariance of the zero-mean Gaussian distribution \(\mathcal{N}(0, M)\) on the *unconstrained* momentum (co-)vector \(p\) with corresponding \(h_2\) Hamiltonian component defined as \[ h_2(q, p) = \frac{1}{2} p^T M^{-1} p. \] The time-derivative of the constraint equation implies a further set of constraints on the momentum \(q\) with \( \partial c(q) M^{-1} p = 0\) at all time points, corresponding to the momentum (velocity) being in the co-tangent space (tangent space) to the manifold. The target distribution is either assumed to be directly specified with unnormalized density \(\exp(-\ell(q))\) with respect to the Hausdorff measure on the manifold (under the metric induced from the ambient metric) with in this case the \(h_1\) Hamiltonian component then simply \[ h_1(q) = \ell(q), \] or alternatively it is assumed a prior distribution on the position \(q\) with density \(\exp(-\ell(q))\) with respect to the Lebesgue measure on the ambient space is specifed and the target distribution is the posterior distribution on \(q\) when conditioning on the event \(c(q) = 0\). The negative logarithm of the posterior distribution density with respect to the Hausdorff measure (and so \(h_1\) Hamiltonian component) is then \[ h_1(q) = \ell(q) + \frac{1}{2} \log\left|\partial c(q)M^{-1}\partial c(q)^T\right| \] with an additional second *Gram matrix* determinant term to give the correct density with respect to the Hausdorff measure on the manifold. Due to the requirement to enforce the constraints on the position and momentum, a constraint-preserving numerical integrator needs to be used when simulating the Hamiltonian dynamic associated with the system, e.g. `mici.integrators.ConstrainedLeapfrogIntegrator`. References: 1. Lelièvre, T., Rousset, M. and Stoltz, G., 2019. Hybrid Monte Carlo methods for sampling probability measures on submanifolds. Numerische Mathematik, 143(2), pp.379-421. 2. Graham, M.M. and Storkey, A.J., 2017. Asymptotically exact inference in differentiable generative models. Electronic Journal of Statistics, 11(2), pp.5105-5164. """ def __init__( self, neg_log_dens, constr, metric=None, dens_wrt_hausdorff=True, grad_neg_log_dens=None, jacob_constr=None, ): """ Args: neg_log_dens (Callable[[array], float]): Function which given a position array returns the negative logarithm of an unnormalized probability density on the constrained position space with respect to the Hausdorff measure on the constraint manifold (if `dens_wrt_hausdorff == True`) or alternatively the negative logarithm of an unnormalized probability density on the unconstrained (ambient) position space with respect to the Lebesgue measure. In the former case the target distribution it is wished to draw approximate samples from is assumed to be directly specified by the density function on the manifold. In the latter case the density function is instead taken to specify a prior distribution on the ambient space with the target distribution then corresponding to the posterior distribution when conditioning on the (zero Lebesgue measure) event `constr(pos) == 0`. This target posterior distribution has support on the differentiable manifold implicitly defined by the constraint equation, with density with respect to the Hausdorff measure on the manifold corresponding to the ratio of the prior density (specified by `neg_log_dens`) and the square-root of the determinant of the Gram matrix defined by gram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).T where `jacob_constr` is the Jacobian of the constraint function `constr` and `metric` is the matrix representation of the metric on the ambient space. constr (Callable[[array], array]): Function which given a position array return as a 1D array the value of the (vector-valued) constraint function, the zero level-set of which implicitly defines the manifold the dynamic is simulated on. metric (None or array or PositiveDefiniteMatrix): Matrix object corresponding to matrix representation of metric on *unconstrained* position space and covariance of Gaussian marginal distribution on *unconstrained* momentum vector. If `None` is passed (the default), the identity matrix will be used. If a 1D array is passed then this is assumed to specify a metric with positive diagonal matrix representation and the array the matrix diagonal. If a 2D array is passed then this is assumed to specify a metric with a dense positive definite matrix representation specified by the array. Otherwise if the value is a `mici.matrices.PositiveDefiniteMatrix` subclass it is assumed to directly specify the metric matrix representation. dens_wrt_hausdorff (bool): Whether the `neg_log_dens` function specifies the (negative logarithm) of the density of the target distribution with respect to the Hausdorff measure on the manifold directly (True) or alternatively the negative logarithm of a density of a prior distriubtion on the unconstrained (ambient) position space with respect to the Lebesgue measure, with the target distribution then corresponding to the posterior distribution when conditioning on the event `const(pos) == 0` (False). Note that in the former case the base Hausdorff measure on the manifold depends on the metric defined on the ambient space, with the Hausdorff measure being defined with respect to the metric induced on the manifold from this ambient metric. grad_neg_log_dens ( None or Callable[[array], array or Tuple[array, float]]): Function which given a position array returns the derivative of `neg_log_dens` with respect to the position array argument. Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the derivative and the second being the value of the `neg_log_dens` evaluated at the passed position array. If `None` is passed (the default) an automatic differentiation fallback will be used to attempt to construct a function to compute the derivative (and value) of `neg_log_dens` automatically. jacob_constr ( None or Callable[[array], array or Tuple[array, array]]): Function which given a position array computes the Jacobian (matrix / 2D array of partial derivatives) of the output of the constraint function `c = constr(q)` with respect to the position array argument `q`, returning the computed Jacobian as a 2D array `jacob` with jacob[i, j] = ∂c[i] / ∂q[j] Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the Jacobian and the second being the value of `constr` evaluated at the passed position array. If `None` is passed (the default) an automatic differentiation fallback will be used to attempt to construct a function to compute the Jacobian (and value) of `constr` automatically. """ super().__init__( neg_log_dens=neg_log_dens, metric=metric, grad_neg_log_dens=grad_neg_log_dens, ) self._constr = constr self.dens_wrt_hausdorff = dens_wrt_hausdorff self._jacob_constr = autodiff_fallback( jacob_constr, constr, "jacobian_and_value", "jacob_constr" ) @cache_in_state("pos") def constr(self, state): """Constraint function at the current position. Args: state (mici.states.ChainState): State to compute value at. Returns: array: Value of `constr(state.pos)` as 1D array. """ return self._constr(state.pos) @cache_in_state_with_aux("pos", "constr") def jacob_constr(self, state): """Jacobian of constraint function at the current position. Args: state (mici.states.ChainState): State to compute value at. Returns: array: Value of Jacobian of `constr(state.pos)` as 2D array. """ return self._jacob_constr(state.pos) @abstractmethod def jacob_constr_inner_product( self, jacob_constr_1, inner_product_matrix, jacob_constr_2=None ): """Compute inner product of rows of constraint Jacobian matrices. Computes `jacob_constr_1 @ inner_product_matrix @ jacob_constr_2.T` potentially exploiting any structure / sparsity in `jacob_constr_1`, `jacob_constr_2` and `inner_product_matrix`. Args: jacob_constr_1 (Matrix): First constraint Jacobian in product. inner_product_matrix (Matrix): Positive-definite matrix defining inner-product between rows of two constraint Jacobians. jacob_constr_2 (None or Matrix): Second constraint Jacobian in product. Defaults to `jacob_constr_1` if set to `None`. Returns Matrix: Object corresponding to computed inner products of the constraint Jacobian rows. """ @cache_in_state("pos") def gram(self, state): """Gram matrix at current position. The Gram matrix as a position `q` is defined as gram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).T where `jacob_constr` is the Jacobian of the constraint function `constr` and `metric` is the matrix representation of the metric on the ambient space. Args: state (mici.states.ChainState): State to compute value at. Returns: mici.matrices.PositiveDefiniteMatrix: Gram matrix as matrix object. """ return self.jacob_constr_inner_product( self.jacob_constr(state), self.metric.inv ) def inv_gram(self, state): """Inverse of Gram matrix at current position. Args: state (mici.states.ChainState): State to compute value at. Returns: mici.matrices.PositiveDefiniteMatrix: Inverse of Gram matrix as matrix object. """ return self.gram(state).inv def log_det_sqrt_gram(self, state): """Value of (half of) log-determinant of Gram matrix.""" return 0.5 * self.gram(state).log_abs_det @abstractmethod def grad_log_det_sqrt_gram(self, state): """Derivative of (half of) log-determinant of Gram matrix wrt position. Args: state (mici.states.ChainState): State to compute value at. Returns: array: Value of `log_det_sqrt_gram(state)` derivative with respect to `state.pos`. """ def h1(self, state): if self.dens_wrt_hausdorff: return self.neg_log_dens(state) else: return self.neg_log_dens(state) + self.log_det_sqrt_gram(state) def dh1_dpos(self, state): if self.dens_wrt_hausdorff: return self.grad_neg_log_dens(state) else: return self.grad_neg_log_dens(state) + self.grad_log_det_sqrt_gram(state) def project_onto_cotangent_space(self, mom, state): """Project a momentum on to the co-tangent space at a position. Args: mom (array): Momentum (co-)vector as 1D array to project on to co-tangent space. state (mici.states.ChainState): State definining position on the manifold to project in to the co-tangent space of. Returns: array: Projected momentum in the co-tangent space at `state.pos`. """ # Use parenthesis to force right-to-left evaluation to avoid # matrix-matrix products mom -= self.jacob_constr(state).T @ ( self.inv_gram(state) @ (self.jacob_constr(state) @ (self.metric.inv @ mom)) ) return mom def sample_momentum(self, state, rng): mom = super().sample_momentum(state, rng) mom = self.project_onto_cotangent_space(mom, state) return momAncestors
- EuclideanMetricSystem
- System
- abc.ABC
Subclasses
Methods
def constr(self, state)-
Constraint function at the current position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
constr(state.pos)as 1D array.
Expand source code Browse git
@cache_in_state("pos") def constr(self, state): """Constraint function at the current position. Args: state (mici.states.ChainState): State to compute value at. Returns: array: Value of `constr(state.pos)` as 1D array. """ return self._constr(state.pos) def jacob_constr(self, state)-
Jacobian of constraint function at the current position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of Jacobian of
constr(state.pos)as 2D array.
Expand source code Browse git
@cache_in_state_with_aux("pos", "constr") def jacob_constr(self, state): """Jacobian of constraint function at the current position. Args: state (mici.states.ChainState): State to compute value at. Returns: array: Value of Jacobian of `constr(state.pos)` as 2D array. """ return self._jacob_constr(state.pos) def jacob_constr_inner_product(self, jacob_constr_1, inner_product_matrix, jacob_constr_2=None)-
Compute inner product of rows of constraint Jacobian matrices.
Computes
jacob_constr_1 @ inner_product_matrix @ jacob_constr_2.Tpotentially exploiting any structure / sparsity injacob_constr_1,jacob_constr_2andinner_product_matrix.Args
jacob_constr_1:Matrix- First constraint Jacobian in product.
inner_product_matrix:Matrix- Positive-definite matrix defining inner-product between rows of two constraint Jacobians.
jacob_constr_2:NoneorMatrix- Second constraint Jacobian in
product. Defaults to
jacob_constr_1if set toNone. Returns- Matrix: Object corresponding to computed inner products of the constraint Jacobian rows.
Expand source code Browse git
@abstractmethod def jacob_constr_inner_product( self, jacob_constr_1, inner_product_matrix, jacob_constr_2=None ): """Compute inner product of rows of constraint Jacobian matrices. Computes `jacob_constr_1 @ inner_product_matrix @ jacob_constr_2.T` potentially exploiting any structure / sparsity in `jacob_constr_1`, `jacob_constr_2` and `inner_product_matrix`. Args: jacob_constr_1 (Matrix): First constraint Jacobian in product. inner_product_matrix (Matrix): Positive-definite matrix defining inner-product between rows of two constraint Jacobians. jacob_constr_2 (None or Matrix): Second constraint Jacobian in product. Defaults to `jacob_constr_1` if set to `None`. Returns Matrix: Object corresponding to computed inner products of the constraint Jacobian rows. """ def gram(self, state)-
Gram matrix at current position.
The Gram matrix as a position
qis defined asgram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).Twhere
jacob_constris the Jacobian of the constraint functionconstrandmetricis the matrix representation of the metric on the ambient space.Args
state:ChainState- State to compute value at.
Returns
mici.matrices.PositiveDefiniteMatrix: Gram matrix as matrix object.
Expand source code Browse git
@cache_in_state("pos") def gram(self, state): """Gram matrix at current position. The Gram matrix as a position `q` is defined as gram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).T where `jacob_constr` is the Jacobian of the constraint function `constr` and `metric` is the matrix representation of the metric on the ambient space. Args: state (mici.states.ChainState): State to compute value at. Returns: mici.matrices.PositiveDefiniteMatrix: Gram matrix as matrix object. """ return self.jacob_constr_inner_product( self.jacob_constr(state), self.metric.inv ) def inv_gram(self, state)-
Inverse of Gram matrix at current position.
Args
state:ChainState- State to compute value at.
Returns
PositiveDefiniteMatrix:InverseofGrammatrixas- matrix object.
Expand source code Browse git
def inv_gram(self, state): """Inverse of Gram matrix at current position. Args: state (mici.states.ChainState): State to compute value at. Returns: mici.matrices.PositiveDefiniteMatrix: Inverse of Gram matrix as matrix object. """ return self.gram(state).inv def log_det_sqrt_gram(self, state)-
Value of (half of) log-determinant of Gram matrix.
Expand source code Browse git
def log_det_sqrt_gram(self, state): """Value of (half of) log-determinant of Gram matrix.""" return 0.5 * self.gram(state).log_abs_det def grad_log_det_sqrt_gram(self, state)-
Derivative of (half of) log-determinant of Gram matrix wrt position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
log_det_sqrt_gram(state)derivative with respect tostate.pos.
Expand source code Browse git
@abstractmethod def grad_log_det_sqrt_gram(self, state): """Derivative of (half of) log-determinant of Gram matrix wrt position. Args: state (mici.states.ChainState): State to compute value at. Returns: array: Value of `log_det_sqrt_gram(state)` derivative with respect to `state.pos`. """ def h1(self, state)-
Hamiltonian component depending only on position.
Args
state:ChainState- State to compute value at.
Returns
float- Value of
h1Hamiltonian component.
Expand source code Browse git
def h1(self, state): if self.dens_wrt_hausdorff: return self.neg_log_dens(state) else: return self.neg_log_dens(state) + self.log_det_sqrt_gram(state) def dh1_dpos(self, state)-
Derivative of
h1Hamiltonian component with respect to position.Args
state:ChainState- State to compute value at.
Returns
array- Value of computed
h1derivative.
Expand source code Browse git
def dh1_dpos(self, state): if self.dens_wrt_hausdorff: return self.grad_neg_log_dens(state) else: return self.grad_neg_log_dens(state) + self.grad_log_det_sqrt_gram(state) def project_onto_cotangent_space(self, mom, state)-
Project a momentum on to the co-tangent space at a position.
Args
mom:array- Momentum (co-)vector as 1D array to project on to co-tangent space.
state:ChainState- State definining position on the manifold to project in to the co-tangent space of.
Returns
array- Projected momentum in the co-tangent space at
state.pos.
Expand source code Browse git
def project_onto_cotangent_space(self, mom, state): """Project a momentum on to the co-tangent space at a position. Args: mom (array): Momentum (co-)vector as 1D array to project on to co-tangent space. state (mici.states.ChainState): State definining position on the manifold to project in to the co-tangent space of. Returns: array: Projected momentum in the co-tangent space at `state.pos`. """ # Use parenthesis to force right-to-left evaluation to avoid # matrix-matrix products mom -= self.jacob_constr(state).T @ ( self.inv_gram(state) @ (self.jacob_constr(state) @ (self.metric.inv @ mom)) ) return mom def sample_momentum(self, state, rng)-
Sample a momentum from its conditional distribution given a position.
Args
state:ChainState- State defining position to condition on.
Returns
mom:array- Sampled momentum.
Expand source code Browse git
def sample_momentum(self, state, rng): mom = super().sample_momentum(state, rng) mom = self.project_onto_cotangent_space(mom, state) return mom def h2_flow(self, state, dt)-
Apply exact flow map corresponding to
h2Hamiltonian component.stateargument is modified in place.Args
state:ChainState- State to start flow at.
dt:float- Time interval to simulate flow for.
def dh2_flow_dmom(self, dt)-
Derivatives of
h2_flowflow map with respect to input momentum.Args
dt:float- Time interval flow simulated for.
Returns
dpos_dmom:Matrix- Matrix representing derivative
(Jacobian) of position output of
h2_flowwith respect to the value of the momentum component of the initial input state. dmom_dmom:Matrix- Matrix representing derivative
(Jacobian) of momentum output of
h2_flowwith respect to the value of the momentum component of the initial input state.
def neg_log_dens(self, state)-
Negative logarithm of unnormalized density of target distribution.
Args
state:ChainState- State to compute value at.
Returns
float- Value of computed negative log density.
def grad_neg_log_dens(self, state)-
Derivative of negative log density with respect to position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
neg_log_dens(state)derivative with respect tostate.pos.
def h1_flow(self, state, dt)-
Apply exact flow map corresponding to
h1Hamiltonian component.stateargument is modified in place.Args
state:ChainState- State to start flow at.
dt:float- Time interval to simulate flow for.
def h(self, state)-
Hamiltonian function for system.
Args
state:ChainState- State to compute value at.
Returns
float- Value of Hamiltonian.
def dh_dpos(self, state)-
Derivative of Hamiltonian with respect to position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
h(state)derivative with respect tostate.pos.
def dh_dmom(self, state)-
Derivative of Hamiltonian with respect to momentum.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
h(state)derivative with respect tostate.mom.
def h2(self, state)-
Hamiltonian component depending on momentum and optionally position.
Args
state:ChainState- State to compute value at.
Returns
float- Value of
h2Hamiltonian component.
def dh2_dmom(self, state)-
Derivative of
h2Hamiltonian component with respect to momentum.Args
state:ChainState- State to compute value at.
Returns
array- Value of
h2(state)derivative with respect tostate.pos.
class DenseConstrainedEuclideanMetricSystem (neg_log_dens, constr, metric=None, dens_wrt_hausdorff=True, grad_neg_log_dens=None, jacob_constr=None, mhp_constr=None)-
Euclidean Hamiltonian system subject to a dense set of constraints.
See
ConstrainedEuclideanMetricSystemfor more details about constrained systems.Args
neg_log_dens:Callable[[array],float]-
Function which given a position array returns the negative logarithm of an unnormalized probability density on the constrained position space with respect to the Hausdorff measure on the constraint manifold (if
dens_wrt_hausdorff == True) or alternatively the negative logarithm of an unnormalized probability density on the unconstrained (ambient) position space with respect to the Lebesgue measure. In the former case the target distribution it is wished to draw approximate samples from is assumed to be directly specified by the density function on the manifold. In the latter case the density function is instead taken to specify a prior distribution on the ambient space with the target distribution then corresponding to the posterior distribution when conditioning on the (zero Lebesgue measure) eventconstr(pos) == 0. This target posterior distribution has support on the differentiable manifold implicitly defined by the constraint equation, with density with respect to the Hausdorff measure on the manifold corresponding to the ratio of the prior density (specified byneg_log_dens) and the square-root of the determinant of the Gram matrix defined bygram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).Twhere
jacob_constris the Jacobian of the constraint functionconstrandmetricis the matrix representation of the metric on the ambient space. constr:Callable[[array],array]- Function which given a position array return as a 1D array the value of the (vector-valued) constraint function, the zero level-set of which implicitly defines the manifold the dynamic is simulated on.
metric:NoneorarrayorPositiveDefiniteMatrix- Matrix object
corresponding to matrix representation of metric on
unconstrained position space and covariance of Gaussian
marginal distribution on unconstrained momentum vector. If
Noneis passed (the default), the identity matrix will be used. If a 1D array is passed then this is assumed to specify a metric with positive diagonal matrix representation and the array the matrix diagonal. If a 2D array is passed then this is assumed to specify a metric with a dense positive definite matrix representation specified by the array. Otherwise if the value is aPositiveDefiniteMatrixsubclass it is assumed to directly specify the metric matrix representation. dens_wrt_hausdorff:bool- Whether the
neg_log_densfunction specifies the (negative logarithm) of the density of the target distribution with respect to the Hausdorff measure on the manifold directly (True) or alternatively the negative logarithm of a density of a prior distriubtion on the unconstrained (ambient) position space with respect to the Lebesgue measure, with the target distribution then corresponding to the posterior distribution when conditioning on the eventconst(pos) == 0(False). Note that in the former case the base Hausdorff measure on the manifold depends on the metric defined on the ambient space, with the Hausdorff measure being defined with respect to the metric induced on the manifold from this ambient metric. grad_neg_log_dens:NoneorCallable[[array],arrayorTuple[array,float]]-
Function which given a position array returns the derivative of
neg_log_denswith respect to the position array argument. Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the derivative and the second being the value of theneg_log_densevaluated at the passed position array. IfNoneis passed (the default) an automatic differentiation fallback will be used to attempt to construct a function to compute the derivative (and value) ofneg_log_densautomatically. jacob_constr:NoneorCallable[[array],arrayorTuple[array,array]]-
Function which given a position array computes the Jacobian (matrix / 2D array of partial derivatives) of the output of the constraint function
c = constr(q)with respect to the position array argumentq, returning the computed Jacobian as a 2D arrayjacobwithjacob[i, j] = ∂c[i] / ∂q[j]Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the Jacobian and the second being the value of
constrevaluated at the passed position array. IfNoneis passed (the default) an automatic differentiation fallback will be used to attempt to construct a function to compute the Jacobian (and value) ofneg_log_densautomatically. mhp_constr:NoneorCallable[[array],Callable[[array],array]] orCallable[[array],Tuple[Callable,array,array]]-
Function which given a position array returns another function which takes a 2D array as an argument and returns the matrix-Hessian-product (MHP) of the constraint function
constrwith respect to the position array argument. The MHP is here defined as a function of a(dim_constr, dim_pos)shaped 2D arraymmhp(m) = sum(m[:, :, None] * hess[:, :, :], axis=(0, 1))where
hessis the(dim_constr, dim_pos, dim_pos)shaped vector-Hessian ofc = constr(q)with respect toqi.e. the array of second-order partial derivatives of such thathess[i, j, k] = ∂²c[i] / (∂q[j] ∂q[k])Optionally the function may instead return a 3-tuple of values with the first a function to compute a MHP of
constr, the second a 2D array corresponding to the Jacobian ofconstr, and the third the value ofconstr, all evaluated at the passed position array. IfNoneis passed (the default) an automatic differentiation fallback will be used to attempt to construct a function which calculates the MHP (and Jacobian and value) ofconstrautomatically.
Expand source code Browse git
class DenseConstrainedEuclideanMetricSystem(ConstrainedEuclideanMetricSystem): r"""Euclidean Hamiltonian system subject to a dense set of constraints. See `ConstrainedEuclideanMetricSystem` for more details about constrained systems. """ def __init__( self, neg_log_dens, constr, metric=None, dens_wrt_hausdorff=True, grad_neg_log_dens=None, jacob_constr=None, mhp_constr=None, ): """ Args: neg_log_dens (Callable[[array], float]): Function which given a position array returns the negative logarithm of an unnormalized probability density on the constrained position space with respect to the Hausdorff measure on the constraint manifold (if `dens_wrt_hausdorff == True`) or alternatively the negative logarithm of an unnormalized probability density on the unconstrained (ambient) position space with respect to the Lebesgue measure. In the former case the target distribution it is wished to draw approximate samples from is assumed to be directly specified by the density function on the manifold. In the latter case the density function is instead taken to specify a prior distribution on the ambient space with the target distribution then corresponding to the posterior distribution when conditioning on the (zero Lebesgue measure) event `constr(pos) == 0`. This target posterior distribution has support on the differentiable manifold implicitly defined by the constraint equation, with density with respect to the Hausdorff measure on the manifold corresponding to the ratio of the prior density (specified by `neg_log_dens`) and the square-root of the determinant of the Gram matrix defined by gram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).T where `jacob_constr` is the Jacobian of the constraint function `constr` and `metric` is the matrix representation of the metric on the ambient space. constr (Callable[[array], array]): Function which given a position array return as a 1D array the value of the (vector-valued) constraint function, the zero level-set of which implicitly defines the manifold the dynamic is simulated on. metric (None or array or PositiveDefiniteMatrix): Matrix object corresponding to matrix representation of metric on *unconstrained* position space and covariance of Gaussian marginal distribution on *unconstrained* momentum vector. If `None` is passed (the default), the identity matrix will be used. If a 1D array is passed then this is assumed to specify a metric with positive diagonal matrix representation and the array the matrix diagonal. If a 2D array is passed then this is assumed to specify a metric with a dense positive definite matrix representation specified by the array. Otherwise if the value is a `mici.matrices.PositiveDefiniteMatrix` subclass it is assumed to directly specify the metric matrix representation. dens_wrt_hausdorff (bool): Whether the `neg_log_dens` function specifies the (negative logarithm) of the density of the target distribution with respect to the Hausdorff measure on the manifold directly (True) or alternatively the negative logarithm of a density of a prior distriubtion on the unconstrained (ambient) position space with respect to the Lebesgue measure, with the target distribution then corresponding to the posterior distribution when conditioning on the event `const(pos) == 0` (False). Note that in the former case the base Hausdorff measure on the manifold depends on the metric defined on the ambient space, with the Hausdorff measure being defined with respect to the metric induced on the manifold from this ambient metric. grad_neg_log_dens ( None or Callable[[array], array or Tuple[array, float]]): Function which given a position array returns the derivative of `neg_log_dens` with respect to the position array argument. Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the derivative and the second being the value of the `neg_log_dens` evaluated at the passed position array. If `None` is passed (the default) an automatic differentiation fallback will be used to attempt to construct a function to compute the derivative (and value) of `neg_log_dens` automatically. jacob_constr ( None or Callable[[array], array or Tuple[array, array]]): Function which given a position array computes the Jacobian (matrix / 2D array of partial derivatives) of the output of the constraint function `c = constr(q)` with respect to the position array argument `q`, returning the computed Jacobian as a 2D array `jacob` with jacob[i, j] = ∂c[i] / ∂q[j] Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the Jacobian and the second being the value of `constr` evaluated at the passed position array. If `None` is passed (the default) an automatic differentiation fallback will be used to attempt to construct a function to compute the Jacobian (and value) of `neg_log_dens` automatically. mhp_constr (None or Callable[[array], Callable[[array], array]] or Callable[[array], Tuple[Callable, array, array]]): Function which given a position array returns another function which takes a 2D array as an argument and returns the *matrix-Hessian-product* (MHP) of the constraint function `constr` with respect to the position array argument. The MHP is here defined as a function of a `(dim_constr, dim_pos)` shaped 2D array `m` mhp(m) = sum(m[:, :, None] * hess[:, :, :], axis=(0, 1)) where `hess` is the `(dim_constr, dim_pos, dim_pos)` shaped vector-Hessian of `c = constr(q)` with respect to `q` i.e. the array of second-order partial derivatives of such that hess[i, j, k] = ∂²c[i] / (∂q[j] ∂q[k]) Optionally the function may instead return a 3-tuple of values with the first a function to compute a MHP of `constr`, the second a 2D array corresponding to the Jacobian of `constr`, and the third the value of `constr`, all evaluated at the passed position array. If `None` is passed (the default) an automatic differentiation fallback will be used to attempt to construct a function which calculates the MHP (and Jacobian and value) of `constr` automatically. """ super().__init__( neg_log_dens=neg_log_dens, constr=constr, metric=metric, dens_wrt_hausdorff=dens_wrt_hausdorff, grad_neg_log_dens=grad_neg_log_dens, jacob_constr=jacob_constr, ) if not dens_wrt_hausdorff: self._mhp_constr = autodiff_fallback( mhp_constr, constr, "mhp_jacobian_and_value", "mhp_constr" ) @cache_in_state_with_aux("pos", ("jacob_constr", "constr")) def mhp_constr(self, state): return self._mhp_constr(state.pos) def jacob_constr_inner_product( self, jacob_constr_1, inner_product_matrix, jacob_constr_2=None ): if jacob_constr_2 is None or jacob_constr_2 is jacob_constr_1: return matrices.DensePositiveDefiniteMatrix( jacob_constr_1 @ (inner_product_matrix @ jacob_constr_1.T) ) else: return matrices.DenseSquareMatrix( jacob_constr_1 @ (inner_product_matrix @ jacob_constr_2.T) ) @cache_in_state("pos") def grad_log_det_sqrt_gram(self, state): # Evaluate MHP of constraint function before Jacobian as Jacobian value # will potentially be computed in 'forward' pass and cached mhp_constr = self.mhp_constr(state) return mhp_constr( self.inv_gram(state) @ self.jacob_constr(state) @ self.metric.inv )Ancestors
Subclasses
Methods
def mhp_constr(self, state)-
Expand source code Browse git
@cache_in_state_with_aux("pos", ("jacob_constr", "constr")) def mhp_constr(self, state): return self._mhp_constr(state.pos) def jacob_constr_inner_product(self, jacob_constr_1, inner_product_matrix, jacob_constr_2=None)-
Compute inner product of rows of constraint Jacobian matrices.
Computes
jacob_constr_1 @ inner_product_matrix @ jacob_constr_2.Tpotentially exploiting any structure / sparsity injacob_constr_1,jacob_constr_2andinner_product_matrix.Args
jacob_constr_1:Matrix- First constraint Jacobian in product.
inner_product_matrix:Matrix- Positive-definite matrix defining inner-product between rows of two constraint Jacobians.
jacob_constr_2:NoneorMatrix- Second constraint Jacobian in
product. Defaults to
jacob_constr_1if set toNone. Returns- Matrix: Object corresponding to computed inner products of the constraint Jacobian rows.
Expand source code Browse git
def jacob_constr_inner_product( self, jacob_constr_1, inner_product_matrix, jacob_constr_2=None ): if jacob_constr_2 is None or jacob_constr_2 is jacob_constr_1: return matrices.DensePositiveDefiniteMatrix( jacob_constr_1 @ (inner_product_matrix @ jacob_constr_1.T) ) else: return matrices.DenseSquareMatrix( jacob_constr_1 @ (inner_product_matrix @ jacob_constr_2.T) ) def grad_log_det_sqrt_gram(self, state)-
Derivative of (half of) log-determinant of Gram matrix wrt position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
log_det_sqrt_gram(state)derivative with respect tostate.pos.
Expand source code Browse git
@cache_in_state("pos") def grad_log_det_sqrt_gram(self, state): # Evaluate MHP of constraint function before Jacobian as Jacobian value # will potentially be computed in 'forward' pass and cached mhp_constr = self.mhp_constr(state) return mhp_constr( self.inv_gram(state) @ self.jacob_constr(state) @ self.metric.inv ) def constr(self, state)-
Constraint function at the current position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
constr(state.pos)as 1D array.
def jacob_constr(self, state)-
Jacobian of constraint function at the current position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of Jacobian of
constr(state.pos)as 2D array.
def gram(self, state)-
Gram matrix at current position.
The Gram matrix as a position
qis defined asgram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).Twhere
jacob_constris the Jacobian of the constraint functionconstrandmetricis the matrix representation of the metric on the ambient space.Args
state:ChainState- State to compute value at.
Returns
mici.matrices.PositiveDefiniteMatrix: Gram matrix as matrix object.
def inv_gram(self, state)-
Inverse of Gram matrix at current position.
Args
state:ChainState- State to compute value at.
Returns
PositiveDefiniteMatrix:InverseofGrammatrixas- matrix object.
def log_det_sqrt_gram(self, state)-
Value of (half of) log-determinant of Gram matrix.
def h1(self, state)-
Hamiltonian component depending only on position.
Args
state:ChainState- State to compute value at.
Returns
float- Value of
h1Hamiltonian component.
def dh1_dpos(self, state)-
Derivative of
h1Hamiltonian component with respect to position.Args
state:ChainState- State to compute value at.
Returns
array- Value of computed
h1derivative.
def project_onto_cotangent_space(self, mom, state)-
Project a momentum on to the co-tangent space at a position.
Args
mom:array- Momentum (co-)vector as 1D array to project on to co-tangent space.
state:ChainState- State definining position on the manifold to project in to the co-tangent space of.
Returns
array- Projected momentum in the co-tangent space at
state.pos.
def sample_momentum(self, state, rng)-
Sample a momentum from its conditional distribution given a position.
Args
state:ChainState- State defining position to condition on.
Returns
mom:array- Sampled momentum.
def h2_flow(self, state, dt)-
Apply exact flow map corresponding to
h2Hamiltonian component.stateargument is modified in place.Args
state:ChainState- State to start flow at.
dt:float- Time interval to simulate flow for.
def dh2_flow_dmom(self, dt)-
Derivatives of
h2_flowflow map with respect to input momentum.Args
dt:float- Time interval flow simulated for.
Returns
dpos_dmom:Matrix- Matrix representing derivative
(Jacobian) of position output of
h2_flowwith respect to the value of the momentum component of the initial input state. dmom_dmom:Matrix- Matrix representing derivative
(Jacobian) of momentum output of
h2_flowwith respect to the value of the momentum component of the initial input state.
def neg_log_dens(self, state)-
Negative logarithm of unnormalized density of target distribution.
Args
state:ChainState- State to compute value at.
Returns
float- Value of computed negative log density.
def grad_neg_log_dens(self, state)-
Derivative of negative log density with respect to position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
neg_log_dens(state)derivative with respect tostate.pos.
def h1_flow(self, state, dt)-
Apply exact flow map corresponding to
h1Hamiltonian component.stateargument is modified in place.Args
state:ChainState- State to start flow at.
dt:float- Time interval to simulate flow for.
def h(self, state)-
Hamiltonian function for system.
Args
state:ChainState- State to compute value at.
Returns
float- Value of Hamiltonian.
def dh_dpos(self, state)-
Derivative of Hamiltonian with respect to position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
h(state)derivative with respect tostate.pos.
def dh_dmom(self, state)-
Derivative of Hamiltonian with respect to momentum.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
h(state)derivative with respect tostate.mom.
def h2(self, state)-
Hamiltonian component depending on momentum and optionally position.
Args
state:ChainState- State to compute value at.
Returns
float- Value of
h2Hamiltonian component.
def dh2_dmom(self, state)-
Derivative of
h2Hamiltonian component with respect to momentum.Args
state:ChainState- State to compute value at.
Returns
array- Value of
h2(state)derivative with respect tostate.pos.
class GaussianDenseConstrainedEuclideanMetricSystem (neg_log_dens, constr, metric=None, grad_neg_log_dens=None, jacob_constr=None, mhp_constr=None)-
Gaussian Euclidean Hamiltonian system st. a dense set of constraints.
See
ConstrainedEuclideanMetricSystemfor more details about constrained systems andGaussianEuclideanMetricSystemfor Gaussian Euclidean metric systems.Args
neg_log_dens:Callable[[array],float]-
Function which given a position array returns the negative logarithm of an unnormalized probability density on the unconstrained (ambient) position space with respect to the standard Gaussian measure. The density function is taken to specify a prior distribution on the ambient space with the target distribution then corresponding to the posterior distribution when conditioning on the (zero Lebesgue measure) event
constr(pos) == 0. This target posterior distribution has support on the differentiable manifold implicitly defined by the constraint equation, with density with respect to the Hausdorff measure on the manifold corresponding to the ratio of the prior density (specified byneg_log_dens) and the square-root of the determinant of the Gram matrix defined bygram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).Twhere
jacob_constris the Jacobian of the constraint functionconstrandmetricis the matrix representation of the metric on the ambient space. constr:Callable[[array],array]- Function which given a position array return as a 1D array the value of the (vector-valued) constraint function, the zero level-set of which implicitly defines the manifold the dynamic is simulated on.
metric:NoneorarrayorPositiveDefiniteMatrix- Matrix object
corresponding to matrix representation of metric on
unconstrained position space and covariance of Gaussian
marginal distribution on unconstrained momentum vector. If
Noneis passed (the default), the identity matrix will be used. If a 1D array is passed then this is assumed to specify a metric with positive diagonal matrix representation and the array the matrix diagonal. If a 2D array is passed then this is assumed to specify a metric with a dense positive definite matrix representation specified by the array. Otherwise if a subclass ofPositiveDefiniteMatrixit is assumed to directly specify the metric matrix representation. grad_neg_log_dens:NoneorCallable[[array],arrayorTuple[array,float]]-
Function which given a position array returns the derivative of
neg_log_denswith respect to the position array argument. Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the derivative and the second being the value of theneg_log_densevaluated at the passed position array. IfNoneis passed (the default) an automatic differentiation fallback will be used to attempt to construct a function to compute the derivative (and value) ofneg_log_densautomatically. jacob_constr:NoneorCallable[[array],arrayorTuple[array,array]]-
Function which given a position array computes the Jacobian (matrix / 2D array of partial derivatives) of the output of the constraint function
c = constr(q)with respect to the position array argumentq, returning the computed Jacobian as a 2D arrayjacobwithjacob[i, j] = ∂c[i] / ∂q[j]Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the Jacobian and the second being the value of
constrevaluated at the passed position array. IfNoneis passed (the default) an automatic differentiation fallback will be used to attempt to construct a function to compute the Jacobian (and value) ofneg_log_densautomatically. mhp_constr:NoneorCallable[[array],Callable[[array],array]] orCallable[[array],Tuple[Callable,array,array]]-
Function which given a position array returns another function which takes a 2D array as an argument and returns the matrix-Hessian-product (MHP) of the constraint function
constrwith respect to the position array argument. The MHP is here defined as a function of a(dim_constr, dim_pos)shaped 2D arraymmhp(m) = sum(m[:, :, None] * hess[:, :, :], axis=(0, 1))where
hessis the(dim_constr, dim_pos, dim_pos)shaped vector-Hessian ofc = constr(q)with respect toqi.e. the array of second-order partial derivatives of such thathess[i, j, k] = ∂²c[i] / (∂q[j] ∂q[k])Optionally the function may instead return a 3-tuple of values with the first a function to compute a MHP of
constr, the second a 2D array corresponding to the Jacobian ofconstr, and the third the value ofconstr, all evaluated at the passed position array. IfNoneis passed (the default) an automatic differentiation fallback will be used to attempt to construct a function which calculates the MHP (and Jacobian and value) ofconstrautomatically.
Expand source code Browse git
class GaussianDenseConstrainedEuclideanMetricSystem( GaussianEuclideanMetricSystem, DenseConstrainedEuclideanMetricSystem ): r"""Gaussian Euclidean Hamiltonian system st. a dense set of constraints. See `ConstrainedEuclideanMetricSystem` for more details about constrained systems and `GaussianEuclideanMetricSystem` for Gaussian Euclidean metric systems. """ def __init__( self, neg_log_dens, constr, metric=None, grad_neg_log_dens=None, jacob_constr=None, mhp_constr=None, ): """ Args: neg_log_dens (Callable[[array], float]): Function which given a position array returns the negative logarithm of an unnormalized probability density on the unconstrained (ambient) position space with respect to the standard Gaussian measure. The density function is taken to specify a prior distribution on the ambient space with the target distribution then corresponding to the posterior distribution when conditioning on the (zero Lebesgue measure) event `constr(pos) == 0`. This target posterior distribution has support on the differentiable manifold implicitly defined by the constraint equation, with density with respect to the Hausdorff measure on the manifold corresponding to the ratio of the prior density (specified by `neg_log_dens`) and the square-root of the determinant of the Gram matrix defined by gram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).T where `jacob_constr` is the Jacobian of the constraint function `constr` and `metric` is the matrix representation of the metric on the ambient space. constr (Callable[[array], array]): Function which given a position array return as a 1D array the value of the (vector-valued) constraint function, the zero level-set of which implicitly defines the manifold the dynamic is simulated on. metric (None or array or PositiveDefiniteMatrix): Matrix object corresponding to matrix representation of metric on *unconstrained* position space and covariance of Gaussian marginal distribution on *unconstrained* momentum vector. If `None` is passed (the default), the identity matrix will be used. If a 1D array is passed then this is assumed to specify a metric with positive diagonal matrix representation and the array the matrix diagonal. If a 2D array is passed then this is assumed to specify a metric with a dense positive definite matrix representation specified by the array. Otherwise if a subclass of `mici.matrices.PositiveDefiniteMatrix` it is assumed to directly specify the metric matrix representation. grad_neg_log_dens ( None or Callable[[array], array or Tuple[array, float]]): Function which given a position array returns the derivative of `neg_log_dens` with respect to the position array argument. Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the derivative and the second being the value of the `neg_log_dens` evaluated at the passed position array. If `None` is passed (the default) an automatic differentiation fallback will be used to attempt to construct a function to compute the derivative (and value) of `neg_log_dens` automatically. jacob_constr ( None or Callable[[array], array or Tuple[array, array]]): Function which given a position array computes the Jacobian (matrix / 2D array of partial derivatives) of the output of the constraint function `c = constr(q)` with respect to the position array argument `q`, returning the computed Jacobian as a 2D array `jacob` with jacob[i, j] = ∂c[i] / ∂q[j] Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the Jacobian and the second being the value of `constr` evaluated at the passed position array. If `None` is passed (the default) an automatic differentiation fallback will be used to attempt to construct a function to compute the Jacobian (and value) of `neg_log_dens` automatically. mhp_constr (None or Callable[[array], Callable[[array], array]] or Callable[[array], Tuple[Callable, array, array]]): Function which given a position array returns another function which takes a 2D array as an argument and returns the *matrix-Hessian-product* (MHP) of the constraint function `constr` with respect to the position array argument. The MHP is here defined as a function of a `(dim_constr, dim_pos)` shaped 2D array `m` mhp(m) = sum(m[:, :, None] * hess[:, :, :], axis=(0, 1)) where `hess` is the `(dim_constr, dim_pos, dim_pos)` shaped vector-Hessian of `c = constr(q)` with respect to `q` i.e. the array of second-order partial derivatives of such that hess[i, j, k] = ∂²c[i] / (∂q[j] ∂q[k]) Optionally the function may instead return a 3-tuple of values with the first a function to compute a MHP of `constr`, the second a 2D array corresponding to the Jacobian of `constr`, and the third the value of `constr`, all evaluated at the passed position array. If `None` is passed (the default) an automatic differentiation fallback will be used to attempt to construct a function which calculates the MHP (and Jacobian and value) of `constr` automatically. """ DenseConstrainedEuclideanMetricSystem.__init__( self, neg_log_dens=neg_log_dens, constr=constr, metric=metric, dens_wrt_hausdorff=False, grad_neg_log_dens=grad_neg_log_dens, jacob_constr=jacob_constr, mhp_constr=mhp_constr, ) def jacob_constr_inner_product( self, jacob_constr_1, inner_product_matrix, jacob_constr_2=None ): if jacob_constr_2 is None or jacob_constr_2 is jacob_constr_1: return matrices.DenseSymmetricMatrix( jacob_constr_1 @ (inner_product_matrix @ jacob_constr_1.T) ) else: return matrices.DenseSquareMatrix( jacob_constr_1 @ (inner_product_matrix @ jacob_constr_2.T) )Ancestors
- GaussianEuclideanMetricSystem
- DenseConstrainedEuclideanMetricSystem
- ConstrainedEuclideanMetricSystem
- EuclideanMetricSystem
- System
- abc.ABC
Methods
def jacob_constr_inner_product(self, jacob_constr_1, inner_product_matrix, jacob_constr_2=None)-
Compute inner product of rows of constraint Jacobian matrices.
Computes
jacob_constr_1 @ inner_product_matrix @ jacob_constr_2.Tpotentially exploiting any structure / sparsity injacob_constr_1,jacob_constr_2andinner_product_matrix.Args
jacob_constr_1:Matrix- First constraint Jacobian in product.
inner_product_matrix:Matrix- Positive-definite matrix defining inner-product between rows of two constraint Jacobians.
jacob_constr_2:NoneorMatrix- Second constraint Jacobian in
product. Defaults to
jacob_constr_1if set toNone. Returns- Matrix: Object corresponding to computed inner products of the constraint Jacobian rows.
Expand source code Browse git
def jacob_constr_inner_product( self, jacob_constr_1, inner_product_matrix, jacob_constr_2=None ): if jacob_constr_2 is None or jacob_constr_2 is jacob_constr_1: return matrices.DenseSymmetricMatrix( jacob_constr_1 @ (inner_product_matrix @ jacob_constr_1.T) ) else: return matrices.DenseSquareMatrix( jacob_constr_1 @ (inner_product_matrix @ jacob_constr_2.T) ) def h2(self, state)-
Hamiltonian component depending on momentum and optionally position.
Args
state:ChainState- State to compute value at.
Returns
float- Value of
h2Hamiltonian component.
def h2_flow(self, state, dt)-
Apply exact flow map corresponding to
h2Hamiltonian component.stateargument is modified in place.Args
state:ChainState- State to start flow at.
dt:float- Time interval to simulate flow for.
def dh2_flow_dmom(self, dt)-
Derivatives of
h2_flowflow map with respect to input momentum.Args
dt:float- Time interval flow simulated for.
Returns
dpos_dmom:Matrix- Matrix representing derivative
(Jacobian) of position output of
h2_flowwith respect to the value of the momentum component of the initial input state. dmom_dmom:Matrix- Matrix representing derivative
(Jacobian) of momentum output of
h2_flowwith respect to the value of the momentum component of the initial input state.
def sample_momentum(self, state, rng)-
Sample a momentum from its conditional distribution given a position.
Args
state:ChainState- State defining position to condition on.
Returns
mom:array- Sampled momentum.
def neg_log_dens(self, state)-
Negative logarithm of unnormalized density of target distribution.
Args
state:ChainState- State to compute value at.
Returns
float- Value of computed negative log density.
def grad_neg_log_dens(self, state)-
Derivative of negative log density with respect to position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
neg_log_dens(state)derivative with respect tostate.pos.
def h1(self, state)-
Hamiltonian component depending only on position.
Args
state:ChainState- State to compute value at.
Returns
float- Value of
h1Hamiltonian component.
def dh1_dpos(self, state)-
Derivative of
h1Hamiltonian component with respect to position.Args
state:ChainState- State to compute value at.
Returns
array- Value of computed
h1derivative.
def h1_flow(self, state, dt)-
Apply exact flow map corresponding to
h1Hamiltonian component.stateargument is modified in place.Args
state:ChainState- State to start flow at.
dt:float- Time interval to simulate flow for.
def h(self, state)-
Hamiltonian function for system.
Args
state:ChainState- State to compute value at.
Returns
float- Value of Hamiltonian.
def dh_dpos(self, state)-
Derivative of Hamiltonian with respect to position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
h(state)derivative with respect tostate.pos.
def dh_dmom(self, state)-
Derivative of Hamiltonian with respect to momentum.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
h(state)derivative with respect tostate.mom.
def constr(self, state)-
Constraint function at the current position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
constr(state.pos)as 1D array.
def jacob_constr(self, state)-
Jacobian of constraint function at the current position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of Jacobian of
constr(state.pos)as 2D array.
def gram(self, state)-
Gram matrix at current position.
The Gram matrix as a position
qis defined asgram(q) = jacob_constr(q) @ inv(metric) @ jacob_constr(q).Twhere
jacob_constris the Jacobian of the constraint functionconstrandmetricis the matrix representation of the metric on the ambient space.Args
state:ChainState- State to compute value at.
Returns
mici.matrices.PositiveDefiniteMatrix: Gram matrix as matrix object.
def inv_gram(self, state)-
Inverse of Gram matrix at current position.
Args
state:ChainState- State to compute value at.
Returns
PositiveDefiniteMatrix:InverseofGrammatrixas- matrix object.
def log_det_sqrt_gram(self, state)-
Value of (half of) log-determinant of Gram matrix.
def project_onto_cotangent_space(self, mom, state)-
Project a momentum on to the co-tangent space at a position.
Args
mom:array- Momentum (co-)vector as 1D array to project on to co-tangent space.
state:ChainState- State definining position on the manifold to project in to the co-tangent space of.
Returns
array- Projected momentum in the co-tangent space at
state.pos.
def dh2_dmom(self, state)-
Derivative of
h2Hamiltonian component with respect to momentum.Args
state:ChainState- State to compute value at.
Returns
array- Value of
h2(state)derivative with respect tostate.pos.
def grad_log_det_sqrt_gram(self, state)-
Derivative of (half of) log-determinant of Gram matrix wrt position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
log_det_sqrt_gram(state)derivative with respect tostate.pos.
class RiemannianMetricSystem (neg_log_dens, metric_matrix_class, metric_func, vjp_metric_func=None, grad_neg_log_dens=None, metric_kwargs=None)-
Riemannian Hamiltonian system with a position-dependent metric.
This class allows for metric matrix representations of any generic type. In most cases a specialized subclass such as
DenseRiemannianMetricSystem,CholeskyFactoredRiemannianMetricSystem,DiagonalRiemannianMetricSystem,ScalarRiemannianMetricSystemorSoftAbsRiemannianMetricSystemwill provide a simpler method of constructng a system with a metric matrix representation of a specific type.The position space is assumed to be a Riemannian manifold with a metric with position-dependent positive definite matrix-representation M(q) where q is a position vector. The momentum p is then taken to have a zero-mean Gaussian conditional distribution given the position q, with covariance M(q), i.e. p \sim \mathcal{N}(0, M(q)) [1].
The h_1 Hamiltonian component is then
h_1(q) = \ell(q) + \frac{1}{2}\log\left|M(q)\right|
where \ell(q) is the negative log (unnormalized) density of the target distribution with respect to the Lebesgue measure at q. The h_2 Hamiltonian component is
h_2(q, p) = \frac{1}{2} p^T (M(q))^{-1} p.
Due to the coupling between the position and momentum variables in h_2, the Hamiltonian system is non-separable, requiring use of a numerical integrator with implicit steps when simulating the Hamiltonian dynamic associated with the system, e.g.
ImplicitLeapfrogIntegrator.References
- Girolami, M. and Calderhead, B., 2011. Riemann manifold Langevin and Hamiltonian Monte Varlo methods. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73(2), pp.123-214.
Args
neg_log_dens:Callable[[array],float]- Function which given a position array returns the negative logarithm of an unnormalized probability density on the position space with respect to the Lebesgue measure, with the corresponding distribution on the position space being the target distribution it is wished to draw approximate samples from.
metric_matrix_class:type[PositiveDefiniteMatrix]-
Class (or factory function returning an instance of the class) which defines type of matrix representation of metric. The class initializer should take a single positional argument which will be passed the array outputted by
metric_func, and which is assumed to be a parameter which fully defines the resulting matrix (e.g. the diagonal of aDiagonalMatrix). The class initializer may also optionally take one or more keyword arguments, with themetric_kwargsargument used to specify the value of these, if any. Together this means the metric matrix representation at a positionposis constructed asmetric = metric_matrix_class( metric_func(pos), **metric_kwargs)The
PositiveDefiniteMatrixsubclass should as a minimum defineinv,log_abs_det,grad_log_abs_det,grad_quadratic_form_inv,__matmul__and__rmatmul__methods / properties (see documentation ofPositiveDefiniteMatrixandDifferentiableMatrixfor definitions of the expected behaviour of these methods). metric_func:Callable[[array],array]- Function which given a
position array returns an array containing the parameter value
of the metric matrix representation passed as the single
positional argument to the
metric_matrix_classinitializer. vjp_metric_func:NoneorCallable[[array],Callable[[array],array]] orCallable[[array],Tuple[Callable[[array],array],array]]-
Function which given a position array returns another function which takes an array as an argument and returns the vector-Jacobian-product (VJP) of
metric_funcwith respect to the position array argument. The VJP is here defined as a function of an arrayv(of the same shape as the output ofmetric_func) corresponding tovjp(v) = sum(v[..., None] * jacob, tuple(range(v.ndim))where
jacobis the Jacobian ofm = metric_func(q)wrtqi.e. the array of partial derivatives of the function such thatjacob[..., i] = ∂m[...] / ∂q[i]Optionally the function may instead return a 2-tuple of values with the first a function to compute a VJP of
metric_funcand the second an array containing the value ofmetric_func, both evaluated at the passed position array. IfNoneis passed (the default) an automatic differentiation fallback will be used to attempt to construct a function which calculates the VJP (and value) ofmetric_funcautomatically. grad_neg_log_dens:NoneorCallable[[array],arrayorTuple[array,float]]-
Function which given a position array returns the derivative of
neg_log_denswith respect to the position array argument. Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the derivative and the second being the value of theneg_log_densevaluated at the passed position array. IfNoneis passed (the default) an automatic differentiation fallback will be used to attempt to construct the derivative ofneg_log_densautomatically. metric_kwargs:NoneorDict[str,object]- An optional dictionary
of any additional keyword arguments to the initializer of
metric_matrix_class.
Expand source code Browse git
class RiemannianMetricSystem(System): r"""Riemannian Hamiltonian system with a position-dependent metric. This class allows for metric matrix representations of any generic type. In most cases a specialized subclass such as `DenseRiemannianMetricSystem`, `CholeskyFactoredRiemannianMetricSystem`, `DiagonalRiemannianMetricSystem`, `ScalarRiemannianMetricSystem` or `SoftAbsRiemannianMetricSystem` will provide a simpler method of constructng a system with a metric matrix representation of a specific type. The position space is assumed to be a Riemannian manifold with a metric with position-dependent positive definite matrix-representation \(M(q)\) where \(q\) is a position vector. The momentum \(p\) is then taken to have a zero-mean Gaussian conditional distribution given the position \(q\), with covariance \(M(q)\), i.e. \(p \sim \mathcal{N}(0, M(q))\) [1]. The \(h_1\) Hamiltonian component is then \[ h_1(q) = \ell(q) + \frac{1}{2}\log\left|M(q)\right| \] where \(\ell(q)\) is the negative log (unnormalized) density of the target distribution with respect to the Lebesgue measure at \(q\). The \(h_2\) Hamiltonian component is \[ h_2(q, p) = \frac{1}{2} p^T (M(q))^{-1} p. \] Due to the coupling between the position and momentum variables in \(h_2\), the Hamiltonian system is non-separable, requiring use of a numerical integrator with implicit steps when simulating the Hamiltonian dynamic associated with the system, e.g. `mici.integrators.ImplicitLeapfrogIntegrator`. References: 1. Girolami, M. and Calderhead, B., 2011. Riemann manifold Langevin and Hamiltonian Monte Varlo methods. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73(2), pp.123-214. """ def __init__( self, neg_log_dens, metric_matrix_class, metric_func, vjp_metric_func=None, grad_neg_log_dens=None, metric_kwargs=None, ): """ Args: neg_log_dens (Callable[[array], float]): Function which given a position array returns the negative logarithm of an unnormalized probability density on the position space with respect to the Lebesgue measure, with the corresponding distribution on the position space being the target distribution it is wished to draw approximate samples from. metric_matrix_class (type[PositiveDefiniteMatrix]): Class (or factory function returning an instance of the class) which defines type of matrix representation of metric. The class initializer should take a single positional argument which will be passed the array outputted by `metric_func`, and which is assumed to be a parameter which fully defines the resulting matrix (e.g. the diagonal of a `mici.matrices.DiagonalMatrix`). The class initializer may also optionally take one or more keyword arguments, with the `metric_kwargs` argument used to specify the value of these, if any. Together this means the metric matrix representation at a position `pos` is constructed as metric = metric_matrix_class( metric_func(pos), **metric_kwargs) The `mici.matrices.PositiveDefiniteMatrix` subclass should as a minimum define `inv`, `log_abs_det`, `grad_log_abs_det`, `grad_quadratic_form_inv`, `__matmul__` and `__rmatmul__` methods / properties (see documentation of `mici.matrices.PositiveDefiniteMatrix` and `mici.matrices.DifferentiableMatrix` for definitions of the expected behaviour of these methods). metric_func (Callable[[array], array]): Function which given a position array returns an array containing the parameter value of the metric matrix representation passed as the single positional argument to the `metric_matrix_class` initializer. vjp_metric_func (None or Callable[[array], Callable[[array], array]] or Callable[[array], Tuple[Callable[[array], array], array]]): Function which given a position array returns another function which takes an array as an argument and returns the *vector-Jacobian-product* (VJP) of `metric_func` with respect to the position array argument. The VJP is here defined as a function of an array `v` (of the same shape as the output of `metric_func`) corresponding to vjp(v) = sum(v[..., None] * jacob, tuple(range(v.ndim)) where `jacob` is the Jacobian of `m = metric_func(q)` wrt `q` i.e. the array of partial derivatives of the function such that jacob[..., i] = ∂m[...] / ∂q[i] Optionally the function may instead return a 2-tuple of values with the first a function to compute a VJP of `metric_func` and the second an array containing the value of `metric_func`, both evaluated at the passed position array. If `None` is passed (the default) an automatic differentiation fallback will be used to attempt to construct a function which calculates the VJP (and value) of `metric_func` automatically. grad_neg_log_dens ( None or Callable[[array], array or Tuple[array, float]]): Function which given a position array returns the derivative of `neg_log_dens` with respect to the position array argument. Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the derivative and the second being the value of the `neg_log_dens` evaluated at the passed position array. If `None` is passed (the default) an automatic differentiation fallback will be used to attempt to construct the derivative of `neg_log_dens` automatically. metric_kwargs (None or Dict[str, object]): An optional dictionary of any additional keyword arguments to the initializer of `metric_matrix_class`. """ self._metric_matrix_class = metric_matrix_class self._metric_func = metric_func self._vjp_metric_func = autodiff_fallback( vjp_metric_func, metric_func, "vjp_and_value", "vjp_metric_func" ) self._metric_kwargs = {} if metric_kwargs is None else metric_kwargs super().__init__(neg_log_dens, grad_neg_log_dens) @cache_in_state("pos") def metric_func(self, state): """ Function computing the parameter of the metric matrix representation. Args: state (mici.states.ChainState): State to compute value at. Returns: array: Value of `metric_func(state.pos)`. """ return self._metric_func(state.pos) @cache_in_state_with_aux("pos", "metric_func") def vjp_metric_func(self, state): """ Function constructing a vector-Jacobian-product for `metric_func`. The vector-Jacobian-product is here defined as a function of an array `v` (of the same shape as the output of `metric_func`) corresponding to vjp(v) = sum(v[..., None] * jacob, axis=tuple(range(v.ndim)) where `jacob` is the Jacobian of `m = metric_func(q)` wrt `q` i.e. the array of partial derivatives of the function such that jacob[..., i] = ∂m[...] / ∂q[i] Args: state (mici.states.ChainState): State to compute VJP at. Returns: Callable[[array], array]: Vector-Jacobian-product function. """ return self._vjp_metric_func(state.pos) @cache_in_state("pos") def metric(self, state): """ Function computing the metric matrix representation. The returned type of this function is that specified by the `metric_matrix_class` argument to the initializer. Args: state (mici.states.ChainState): State to compute value at. Returns: mici.matrices.PositiveDefiniteMatrix: Metric matrix representation. """ return self._metric_matrix_class(self.metric_func(state), **self._metric_kwargs) def h(self, state): return self.h1(state) + self.h2(state) def h1(self, state): return self.neg_log_dens(state) + 0.5 * self.metric(state).log_abs_det def dh1_dpos(self, state): # Evaluate VJP of metric function before metric as metric value will # potentially be computed in forward pass and cached vjp_metric = self.vjp_metric_func(state) return self.grad_neg_log_dens(state) + 0.5 * vjp_metric( self.metric(state).grad_log_abs_det ) def h2(self, state): return 0.5 * state.mom @ self.metric(state).inv @ state.mom def dh2_dpos(self, state): # Evaluate VJP of metric function before metric as metric value will # potentially be computed in forward pass and cached vjp_metric = self.vjp_metric_func(state) return 0.5 * vjp_metric(self.metric(state).grad_quadratic_form_inv(state.mom)) def dh2_dmom(self, state): return self.metric(state).inv @ state.mom def sample_momentum(self, state, rng): return self.metric(state).sqrt @ rng.normal(size=state.pos.shape)Ancestors
- System
- abc.ABC
Subclasses
- ScalarRiemannianMetricSystem
- DiagonalRiemannianMetricSystem
- CholeskyFactoredRiemannianMetricSystem
- DenseRiemannianMetricSystem
- SoftAbsRiemannianMetricSystem
Methods
def metric_func(self, state)-
Function computing the parameter of the metric matrix representation.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
metric_func(state.pos).
Expand source code Browse git
@cache_in_state("pos") def metric_func(self, state): """ Function computing the parameter of the metric matrix representation. Args: state (mici.states.ChainState): State to compute value at. Returns: array: Value of `metric_func(state.pos)`. """ return self._metric_func(state.pos) def vjp_metric_func(self, state)-
Function constructing a vector-Jacobian-product for
metric_func.The vector-Jacobian-product is here defined as a function of an array
v(of the same shape as the output ofmetric_func) corresponding tovjp(v) = sum(v[..., None] * jacob, axis=tuple(range(v.ndim))where
jacobis the Jacobian ofm = metric_func(q)wrtqi.e. the array of partial derivatives of the function such thatjacob[..., i] = ∂m[...] / ∂q[i]Args
state:ChainState- State to compute VJP at.
Returns
Callable[[array], array]: Vector-Jacobian-product function.
Expand source code Browse git
@cache_in_state_with_aux("pos", "metric_func") def vjp_metric_func(self, state): """ Function constructing a vector-Jacobian-product for `metric_func`. The vector-Jacobian-product is here defined as a function of an array `v` (of the same shape as the output of `metric_func`) corresponding to vjp(v) = sum(v[..., None] * jacob, axis=tuple(range(v.ndim)) where `jacob` is the Jacobian of `m = metric_func(q)` wrt `q` i.e. the array of partial derivatives of the function such that jacob[..., i] = ∂m[...] / ∂q[i] Args: state (mici.states.ChainState): State to compute VJP at. Returns: Callable[[array], array]: Vector-Jacobian-product function. """ return self._vjp_metric_func(state.pos) def metric(self, state)-
Function computing the metric matrix representation.
The returned type of this function is that specified by the
metric_matrix_classargument to the initializer.Args
state:ChainState- State to compute value at.
Returns
mici.matrices.PositiveDefiniteMatrix: Metric matrix representation.
Expand source code Browse git
@cache_in_state("pos") def metric(self, state): """ Function computing the metric matrix representation. The returned type of this function is that specified by the `metric_matrix_class` argument to the initializer. Args: state (mici.states.ChainState): State to compute value at. Returns: mici.matrices.PositiveDefiniteMatrix: Metric matrix representation. """ return self._metric_matrix_class(self.metric_func(state), **self._metric_kwargs) def h(self, state)-
Hamiltonian function for system.
Args
state:ChainState- State to compute value at.
Returns
float- Value of Hamiltonian.
Expand source code Browse git
def h(self, state): return self.h1(state) + self.h2(state) def h1(self, state)-
Hamiltonian component depending only on position.
Args
state:ChainState- State to compute value at.
Returns
float- Value of
h1Hamiltonian component.
Expand source code Browse git
def h1(self, state): return self.neg_log_dens(state) + 0.5 * self.metric(state).log_abs_det def dh1_dpos(self, state)-
Derivative of
h1Hamiltonian component with respect to position.Args
state:ChainState- State to compute value at.
Returns
array- Value of computed
h1derivative.
Expand source code Browse git
def dh1_dpos(self, state): # Evaluate VJP of metric function before metric as metric value will # potentially be computed in forward pass and cached vjp_metric = self.vjp_metric_func(state) return self.grad_neg_log_dens(state) + 0.5 * vjp_metric( self.metric(state).grad_log_abs_det ) def h2(self, state)-
Hamiltonian component depending on momentum and optionally position.
Args
state:ChainState- State to compute value at.
Returns
float- Value of
h2Hamiltonian component.
Expand source code Browse git
def h2(self, state): return 0.5 * state.mom @ self.metric(state).inv @ state.mom def dh2_dpos(self, state)-
Expand source code Browse git
def dh2_dpos(self, state): # Evaluate VJP of metric function before metric as metric value will # potentially be computed in forward pass and cached vjp_metric = self.vjp_metric_func(state) return 0.5 * vjp_metric(self.metric(state).grad_quadratic_form_inv(state.mom)) def dh2_dmom(self, state)-
Derivative of
h2Hamiltonian component with respect to momentum.Args
state:ChainState- State to compute value at.
Returns
array- Value of
h2(state)derivative with respect tostate.pos.
Expand source code Browse git
def dh2_dmom(self, state): return self.metric(state).inv @ state.mom def sample_momentum(self, state, rng)-
Sample a momentum from its conditional distribution given a position.
Args
state:ChainState- State defining position to condition on.
Returns
mom:array- Sampled momentum.
Expand source code Browse git
def sample_momentum(self, state, rng): return self.metric(state).sqrt @ rng.normal(size=state.pos.shape) def neg_log_dens(self, state)-
Negative logarithm of unnormalized density of target distribution.
Args
state:ChainState- State to compute value at.
Returns
float- Value of computed negative log density.
def grad_neg_log_dens(self, state)-
Derivative of negative log density with respect to position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
neg_log_dens(state)derivative with respect tostate.pos.
def h1_flow(self, state, dt)-
Apply exact flow map corresponding to
h1Hamiltonian component.stateargument is modified in place.Args
state:ChainState- State to start flow at.
dt:float- Time interval to simulate flow for.
def dh_dpos(self, state)-
Derivative of Hamiltonian with respect to position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
h(state)derivative with respect tostate.pos.
def dh_dmom(self, state)-
Derivative of Hamiltonian with respect to momentum.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
h(state)derivative with respect tostate.mom.
class ScalarRiemannianMetricSystem (neg_log_dens, metric_scalar_func, vjp_metric_scalar_func=None, grad_neg_log_dens=None)-
Riemannian-metric system with scaled identity matrix representation.
Hamiltonian system with a position dependent scaled identity metric matrix representation which is specified by a scalar function
metric_scalar_functionof the positionqwhich outputs a strictly positive scalars = metric_scalar_func(q)with the the metric matrix representation then taken to bes * identity(q.shape[0]).See documentation of
RiemannianMetricSystemfor more general details about Riemannian-metric Hamiltonian systems.Args
neg_log_dens:Callable[[array],float]- Function which given a position array returns the negative logarithm of an unnormalized probability density on the position space with respect to the Lebesgue measure, with the corresponding distribution on the position space being the target distribution it is wished to draw approximate samples from.
metric_scalar_func:Callable[[array],float]- Function which given a position array returns a strictly positive scalar corresponding to the parameter value of the scaled identity metric matrix representation.
vjp_metric_scalar_func:NoneorCallable[[array],Callable[[array],float]] orCallable[[array],Tuple[Callable[[array,float]],float]]-
Function which given a position array returns another function which takes a scalar as an argument and returns the vector-Jacobian-product (VJP) of
metric_scalar_funcwith respect to the position array argument. The VJP is here defined as a function of a scalarvvjp(v) = v * gradwhere
gradis the(dim_pos,)shaped Jacobian (gradient) ofs = metric_scalar_func(q)with respect toqi.e. the array of partial derivatives of the function such thatgrad[i] = ∂s / ∂q[i]Optionally the function may instead return a 2-tuple of values with the first a function to compute a VJP of
metric_scalar_funcand the second a float containing the value ofmetric_scalar_func, both evaluated at the passed position array. IfNoneis passed (the default) an automatic differentiation fallback will be used to attempt to construct a function which calculates the VJP (and value) ofmetric_scalar_funcautomatically. grad_neg_log_dens:NoneorCallable[[array],arrayorTuple[array,float]]-
Function which given a position array returns the derivative of
neg_log_denswith respect to the position array argument. Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the derivative and the second being the value of theneg_log_densevaluated at the passed position array. IfNoneis passed (the default) an automatic differentiation fallback will be used to attempt to construct the derivative ofneg_log_densautomatically.
Expand source code Browse git
class ScalarRiemannianMetricSystem(RiemannianMetricSystem): """Riemannian-metric system with scaled identity matrix representation. Hamiltonian system with a position dependent scaled identity metric matrix representation which is specified by a scalar function `metric_scalar_function` of the position `q` which outputs a strictly positive scalar `s = metric_scalar_func(q)` with the the metric matrix representation then taken to be `s * identity(q.shape[0])`. See documentation of `RiemannianMetricSystem` for more general details about Riemannian-metric Hamiltonian systems. """ def __init__( self, neg_log_dens, metric_scalar_func, vjp_metric_scalar_func=None, grad_neg_log_dens=None, ): """ Args: neg_log_dens (Callable[[array], float]): Function which given a position array returns the negative logarithm of an unnormalized probability density on the position space with respect to the Lebesgue measure, with the corresponding distribution on the position space being the target distribution it is wished to draw approximate samples from. metric_scalar_func (Callable[[array], float]): Function which given a position array returns a strictly positive scalar corresponding to the parameter value of the scaled identity metric matrix representation. vjp_metric_scalar_func (None or Callable[[array], Callable[[array], float]] or Callable[[array], Tuple[Callable[[array, float]], float]]): Function which given a position array returns another function which takes a scalar as an argument and returns the *vector-Jacobian-product* (VJP) of `metric_scalar_func` with respect to the position array argument. The VJP is here defined as a function of a scalar `v` vjp(v) = v * grad where `grad` is the `(dim_pos,)` shaped Jacobian (gradient) of `s = metric_scalar_func(q)` with respect to `q` i.e. the array of partial derivatives of the function such that grad[i] = ∂s / ∂q[i] Optionally the function may instead return a 2-tuple of values with the first a function to compute a VJP of `metric_scalar_func` and the second a float containing the value of `metric_scalar_func`, both evaluated at the passed position array. If `None` is passed (the default) an automatic differentiation fallback will be used to attempt to construct a function which calculates the VJP (and value) of `metric_scalar_func` automatically. grad_neg_log_dens ( None or Callable[[array], array or Tuple[array, float]]): Function which given a position array returns the derivative of `neg_log_dens` with respect to the position array argument. Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the derivative and the second being the value of the `neg_log_dens` evaluated at the passed position array. If `None` is passed (the default) an automatic differentiation fallback will be used to attempt to construct the derivative of `neg_log_dens` automatically. """ super().__init__( neg_log_dens, matrices.PositiveScaledIdentityMatrix, metric_scalar_func, vjp_metric_scalar_func, grad_neg_log_dens, ) @cache_in_state("pos") def metric(self, state): return self._metric_matrix_class( self.metric_func(state), size=state.pos.shape[0] )Ancestors
- RiemannianMetricSystem
- System
- abc.ABC
Methods
def metric(self, state)-
Function computing the metric matrix representation.
The returned type of this function is that specified by the
metric_matrix_classargument to the initializer.Args
state:ChainState- State to compute value at.
Returns
mici.matrices.PositiveDefiniteMatrix: Metric matrix representation.
Expand source code Browse git
@cache_in_state("pos") def metric(self, state): return self._metric_matrix_class( self.metric_func(state), size=state.pos.shape[0] ) def metric_func(self, state)-
Function computing the parameter of the metric matrix representation.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
metric_func(state.pos).
def vjp_metric_func(self, state)-
Function constructing a vector-Jacobian-product for
metric_func.The vector-Jacobian-product is here defined as a function of an array
v(of the same shape as the output ofmetric_func) corresponding tovjp(v) = sum(v[..., None] * jacob, axis=tuple(range(v.ndim))where
jacobis the Jacobian ofm = metric_func(q)wrtqi.e. the array of partial derivatives of the function such thatjacob[..., i] = ∂m[...] / ∂q[i]Args
state:ChainState- State to compute VJP at.
Returns
Callable[[array], array]: Vector-Jacobian-product function.
def h(self, state)-
Hamiltonian function for system.
Args
state:ChainState- State to compute value at.
Returns
float- Value of Hamiltonian.
def h1(self, state)-
Hamiltonian component depending only on position.
Args
state:ChainState- State to compute value at.
Returns
float- Value of
h1Hamiltonian component.
def dh1_dpos(self, state)-
Derivative of
h1Hamiltonian component with respect to position.Args
state:ChainState- State to compute value at.
Returns
array- Value of computed
h1derivative.
def h2(self, state)-
Hamiltonian component depending on momentum and optionally position.
Args
state:ChainState- State to compute value at.
Returns
float- Value of
h2Hamiltonian component.
def dh2_dmom(self, state)-
Derivative of
h2Hamiltonian component with respect to momentum.Args
state:ChainState- State to compute value at.
Returns
array- Value of
h2(state)derivative with respect tostate.pos.
def sample_momentum(self, state, rng)-
Sample a momentum from its conditional distribution given a position.
Args
state:ChainState- State defining position to condition on.
Returns
mom:array- Sampled momentum.
def neg_log_dens(self, state)-
Negative logarithm of unnormalized density of target distribution.
Args
state:ChainState- State to compute value at.
Returns
float- Value of computed negative log density.
def grad_neg_log_dens(self, state)-
Derivative of negative log density with respect to position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
neg_log_dens(state)derivative with respect tostate.pos.
def h1_flow(self, state, dt)-
Apply exact flow map corresponding to
h1Hamiltonian component.stateargument is modified in place.Args
state:ChainState- State to start flow at.
dt:float- Time interval to simulate flow for.
def dh_dpos(self, state)-
Derivative of Hamiltonian with respect to position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
h(state)derivative with respect tostate.pos.
def dh_dmom(self, state)-
Derivative of Hamiltonian with respect to momentum.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
h(state)derivative with respect tostate.mom.
class DiagonalRiemannianMetricSystem (neg_log_dens, metric_diagonal_func, vjp_metric_diagonal_func=None, grad_neg_log_dens=None)-
Riemannian-metric system with diagonal matrix representation.
Hamiltonian system with a position dependent diagonal metric matrix representation which is specified by a vector-valued function
metric_diagonal_funcof the positionqwhich outputs a 1D array with strictly positive elementsd = metric_diagonal_func(q)with the metric matrix representation then taken to bediag(d).See documentation of
RiemannianMetricSystemfor more general details about Riemannian-metric Hamiltonian systems.Args
neg_log_dens:Callable[[array],float]- Function which given a position array returns the negative logarithm of an unnormalized probability density on the position space with respect to the Lebesgue measure, with the corresponding distribution on the position space being the target distribution it is wished to draw approximate samples from.
metric_diagonal_func:Callable[[array],array]- Function which given a position array returns a 1D array with strictly positive values corresponding to the diagonal values (left-to-right) of the diagonal metric matrix representation.
vjp_metric_diagonal_func:NoneorCallable[[array],Callable[[array],array]] orCallable[[array],Tuple[Callable[[array],array],array]]-
Function which given a position array returns another function which takes a 1D array as an argument and returns the vector-Jacobian-product (VJP) of
metric_diagonal_funcwith respect to the position array argument. The VJP is here defined as a function of a 1D arrayvvjp(v) = sum(v[:, None] * jacob[:, :], axis=0)where
jacobis the(dim_pos, dim_pos)shaped Jacobian ofd = metric_diagonal_func(q)with respect toqi.e. the array of partial derivatives of the function such thatjacob[i, j] = ∂d[i] / ∂q[j]Optionally the function may instead return a 2-tuple of values with the first a function to compute a VJP of
metric_diagonal_funcand the second a 1D array containing the value ofmetric_diagonal_func, both evaluated at the passed position array. IfNoneis passed (the default) an automatic differentiation fallback will be used to attempt to construct a function which calculates the VJP (and value) ofmetric_diagonal_funcautomatically. grad_neg_log_dens:NoneorCallable[[array],arrayorTuple[array,float]]-
Function which given a position array returns the derivative of
neg_log_denswith respect to the position array argument. Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the derivative and the second being the value of theneg_log_densevaluated at the passed position array. IfNoneis passed (the default) an automatic differentiation fallback will be used to attempt to construct the derivative ofneg_log_densautomatically.
Expand source code Browse git
class DiagonalRiemannianMetricSystem(RiemannianMetricSystem): """Riemannian-metric system with diagonal matrix representation. Hamiltonian system with a position dependent diagonal metric matrix representation which is specified by a vector-valued function `metric_diagonal_func` of the position `q` which outputs a 1D array with strictly positive elements `d = metric_diagonal_func(q)` with the metric matrix representation then taken to be `diag(d)`. See documentation of `RiemannianMetricSystem` for more general details about Riemannian-metric Hamiltonian systems. """ def __init__( self, neg_log_dens, metric_diagonal_func, vjp_metric_diagonal_func=None, grad_neg_log_dens=None, ): """ Args: neg_log_dens (Callable[[array], float]): Function which given a position array returns the negative logarithm of an unnormalized probability density on the position space with respect to the Lebesgue measure, with the corresponding distribution on the position space being the target distribution it is wished to draw approximate samples from. metric_diagonal_func (Callable[[array], array]): Function which given a position array returns a 1D array with strictly positive values corresponding to the diagonal values (left-to-right) of the diagonal metric matrix representation. vjp_metric_diagonal_func (None or Callable[[array], Callable[[array], array]] or Callable[[array], Tuple[Callable[[array], array], array]]): Function which given a position array returns another function which takes a 1D array as an argument and returns the *vector-Jacobian-product* (VJP) of `metric_diagonal_func` with respect to the position array argument. The VJP is here defined as a function of a 1D array `v` vjp(v) = sum(v[:, None] * jacob[:, :], axis=0) where `jacob` is the `(dim_pos, dim_pos)` shaped Jacobian of `d = metric_diagonal_func(q)` with respect to `q` i.e. the array of partial derivatives of the function such that jacob[i, j] = ∂d[i] / ∂q[j] Optionally the function may instead return a 2-tuple of values with the first a function to compute a VJP of `metric_diagonal_func` and the second a 1D array containing the value of `metric_diagonal_func`, both evaluated at the passed position array. If `None` is passed (the default) an automatic differentiation fallback will be used to attempt to construct a function which calculates the VJP (and value) of `metric_diagonal_func` automatically. grad_neg_log_dens ( None or Callable[[array], array or Tuple[array, float]]): Function which given a position array returns the derivative of `neg_log_dens` with respect to the position array argument. Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the derivative and the second being the value of the `neg_log_dens` evaluated at the passed position array. If `None` is passed (the default) an automatic differentiation fallback will be used to attempt to construct the derivative of `neg_log_dens` automatically. """ super().__init__( neg_log_dens, matrices.PositiveDiagonalMatrix, metric_diagonal_func, vjp_metric_diagonal_func, grad_neg_log_dens, )Ancestors
- RiemannianMetricSystem
- System
- abc.ABC
Methods
def metric_func(self, state)-
Function computing the parameter of the metric matrix representation.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
metric_func(state.pos).
def vjp_metric_func(self, state)-
Function constructing a vector-Jacobian-product for
metric_func.The vector-Jacobian-product is here defined as a function of an array
v(of the same shape as the output ofmetric_func) corresponding tovjp(v) = sum(v[..., None] * jacob, axis=tuple(range(v.ndim))where
jacobis the Jacobian ofm = metric_func(q)wrtqi.e. the array of partial derivatives of the function such thatjacob[..., i] = ∂m[...] / ∂q[i]Args
state:ChainState- State to compute VJP at.
Returns
Callable[[array], array]: Vector-Jacobian-product function.
def metric(self, state)-
Function computing the metric matrix representation.
The returned type of this function is that specified by the
metric_matrix_classargument to the initializer.Args
state:ChainState- State to compute value at.
Returns
mici.matrices.PositiveDefiniteMatrix: Metric matrix representation.
def h(self, state)-
Hamiltonian function for system.
Args
state:ChainState- State to compute value at.
Returns
float- Value of Hamiltonian.
def h1(self, state)-
Hamiltonian component depending only on position.
Args
state:ChainState- State to compute value at.
Returns
float- Value of
h1Hamiltonian component.
def dh1_dpos(self, state)-
Derivative of
h1Hamiltonian component with respect to position.Args
state:ChainState- State to compute value at.
Returns
array- Value of computed
h1derivative.
def h2(self, state)-
Hamiltonian component depending on momentum and optionally position.
Args
state:ChainState- State to compute value at.
Returns
float- Value of
h2Hamiltonian component.
def dh2_dmom(self, state)-
Derivative of
h2Hamiltonian component with respect to momentum.Args
state:ChainState- State to compute value at.
Returns
array- Value of
h2(state)derivative with respect tostate.pos.
def sample_momentum(self, state, rng)-
Sample a momentum from its conditional distribution given a position.
Args
state:ChainState- State defining position to condition on.
Returns
mom:array- Sampled momentum.
def neg_log_dens(self, state)-
Negative logarithm of unnormalized density of target distribution.
Args
state:ChainState- State to compute value at.
Returns
float- Value of computed negative log density.
def grad_neg_log_dens(self, state)-
Derivative of negative log density with respect to position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
neg_log_dens(state)derivative with respect tostate.pos.
def h1_flow(self, state, dt)-
Apply exact flow map corresponding to
h1Hamiltonian component.stateargument is modified in place.Args
state:ChainState- State to start flow at.
dt:float- Time interval to simulate flow for.
def dh_dpos(self, state)-
Derivative of Hamiltonian with respect to position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
h(state)derivative with respect tostate.pos.
def dh_dmom(self, state)-
Derivative of Hamiltonian with respect to momentum.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
h(state)derivative with respect tostate.mom.
class CholeskyFactoredRiemannianMetricSystem (neg_log_dens, metric_chol_func, vjp_metric_chol_func=None, grad_neg_log_dens=None)-
Riemannian-metric system with Cholesky-factored matrix representation.
Hamiltonian system with a position dependent metric matrix representation which is specified by its Cholesky factor by a matrix function
metric_chol_funcof the positionqwhich outputs a lower-triangular matrixL = metric_chol_func(q)with the metric matrix representation then taken to beL @ L.T.See documentation of
RiemannianMetricSystemfor more general details about Riemannian-metric Hamiltonian systems.Args
neg_log_dens:Callable[[array],float]- Function which given a position array returns the negative logarithm of an unnormalized probability density on the position space with respect to the Lebesgue measure, with the corresponding distribution on the position space being the target distribution it is wished to draw approximate samples from.
metric_chol_func:Callable[[array],array]- Function which given a position array returns a 2D array with zeros above the diagonal corresponding to the lower-triangular Cholesky-factor of the positive definite metric matrix representation.
vjp_metric_chol_func:NoneorCallable[[array],Callable[[array],array]] orCallable[[array],Tuple[Callable[[array],array],array]]-
Function which given a position array returns another function which takes a lower-triangular 2D array as an argument (any values in the array above the diagonal are ignored) and returns the vector-Jacobian-product (VJP) of
metric_chol_funcwith respect to the position array argument. The VJP is here defined as a function of a 2D arrayvvjp(v) = sum(v[:, :, None] * jacob[:, :, :], axis=(0, 1))where
jacobis the(dim_pos, dim_pos, dim_pos)shaped Jacobian ofL = metric_chol_func(q)with respect toqi.e. the array of partial derivatives of the function such thatjacob[i, j, k] = ∂L[i, j] / ∂q[k]Optionally the function may instead return a 2-tuple of values with the first a function to compute a VJP of
metric_chol_funcand the second a 2D array containing the value ofmetric_chol_func, both evaluated at the passed position array. IfNoneis passed (the default) an automatic differentiation fallback will be used to attempt to construct a function which calculates the VJP (and value) ofmetric_chol_funcautomatically. grad_neg_log_dens:NoneorCallable[[array],arrayorTuple[array,float]]-
Function which given a position array returns the derivative of
neg_log_denswith respect to the position array argument. Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the derivative and the second being the value of theneg_log_densevaluated at the passed position array. IfNoneis passed (the default) an automatic differentiation fallback will be used to attempt to construct the derivative ofneg_log_densautomatically.
Expand source code Browse git
class CholeskyFactoredRiemannianMetricSystem(RiemannianMetricSystem): """Riemannian-metric system with Cholesky-factored matrix representation. Hamiltonian system with a position dependent metric matrix representation which is specified by its Cholesky factor by a matrix function `metric_chol_func` of the position `q` which outputs a lower-triangular matrix `L = metric_chol_func(q)` with the metric matrix representation then taken to be `L @ L.T`. See documentation of `RiemannianMetricSystem` for more general details about Riemannian-metric Hamiltonian systems. """ def __init__( self, neg_log_dens, metric_chol_func, vjp_metric_chol_func=None, grad_neg_log_dens=None, ): """ Args: neg_log_dens (Callable[[array], float]): Function which given a position array returns the negative logarithm of an unnormalized probability density on the position space with respect to the Lebesgue measure, with the corresponding distribution on the position space being the target distribution it is wished to draw approximate samples from. metric_chol_func (Callable[[array], array]): Function which given a position array returns a 2D array with zeros above the diagonal corresponding to the lower-triangular Cholesky-factor of the positive definite metric matrix representation. vjp_metric_chol_func (None or Callable[[array], Callable[[array], array]] or Callable[[array], Tuple[Callable[[array], array], array]]): Function which given a position array returns another function which takes a lower-triangular 2D array as an argument (any values in the array above the diagonal are ignored) and returns the *vector-Jacobian-product* (VJP) of `metric_chol_func` with respect to the position array argument. The VJP is here defined as a function of a 2D array `v` vjp(v) = sum(v[:, :, None] * jacob[:, :, :], axis=(0, 1)) where `jacob` is the `(dim_pos, dim_pos, dim_pos)` shaped Jacobian of `L = metric_chol_func(q)` with respect to `q` i.e. the array of partial derivatives of the function such that jacob[i, j, k] = ∂L[i, j] / ∂q[k] Optionally the function may instead return a 2-tuple of values with the first a function to compute a VJP of `metric_chol_func` and the second a 2D array containing the value of `metric_chol_func`, both evaluated at the passed position array. If `None` is passed (the default) an automatic differentiation fallback will be used to attempt to construct a function which calculates the VJP (and value) of `metric_chol_func` automatically. grad_neg_log_dens ( None or Callable[[array], array or Tuple[array, float]]): Function which given a position array returns the derivative of `neg_log_dens` with respect to the position array argument. Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the derivative and the second being the value of the `neg_log_dens` evaluated at the passed position array. If `None` is passed (the default) an automatic differentiation fallback will be used to attempt to construct the derivative of `neg_log_dens` automatically. """ super().__init__( neg_log_dens, matrices.TriangularFactoredPositiveDefiniteMatrix, metric_chol_func, vjp_metric_chol_func, grad_neg_log_dens, metric_kwargs={"factor_is_lower": True}, )Ancestors
- RiemannianMetricSystem
- System
- abc.ABC
Methods
def metric_func(self, state)-
Function computing the parameter of the metric matrix representation.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
metric_func(state.pos).
def vjp_metric_func(self, state)-
Function constructing a vector-Jacobian-product for
metric_func.The vector-Jacobian-product is here defined as a function of an array
v(of the same shape as the output ofmetric_func) corresponding tovjp(v) = sum(v[..., None] * jacob, axis=tuple(range(v.ndim))where
jacobis the Jacobian ofm = metric_func(q)wrtqi.e. the array of partial derivatives of the function such thatjacob[..., i] = ∂m[...] / ∂q[i]Args
state:ChainState- State to compute VJP at.
Returns
Callable[[array], array]: Vector-Jacobian-product function.
def metric(self, state)-
Function computing the metric matrix representation.
The returned type of this function is that specified by the
metric_matrix_classargument to the initializer.Args
state:ChainState- State to compute value at.
Returns
mici.matrices.PositiveDefiniteMatrix: Metric matrix representation.
def h(self, state)-
Hamiltonian function for system.
Args
state:ChainState- State to compute value at.
Returns
float- Value of Hamiltonian.
def h1(self, state)-
Hamiltonian component depending only on position.
Args
state:ChainState- State to compute value at.
Returns
float- Value of
h1Hamiltonian component.
def dh1_dpos(self, state)-
Derivative of
h1Hamiltonian component with respect to position.Args
state:ChainState- State to compute value at.
Returns
array- Value of computed
h1derivative.
def h2(self, state)-
Hamiltonian component depending on momentum and optionally position.
Args
state:ChainState- State to compute value at.
Returns
float- Value of
h2Hamiltonian component.
def dh2_dmom(self, state)-
Derivative of
h2Hamiltonian component with respect to momentum.Args
state:ChainState- State to compute value at.
Returns
array- Value of
h2(state)derivative with respect tostate.pos.
def sample_momentum(self, state, rng)-
Sample a momentum from its conditional distribution given a position.
Args
state:ChainState- State defining position to condition on.
Returns
mom:array- Sampled momentum.
def neg_log_dens(self, state)-
Negative logarithm of unnormalized density of target distribution.
Args
state:ChainState- State to compute value at.
Returns
float- Value of computed negative log density.
def grad_neg_log_dens(self, state)-
Derivative of negative log density with respect to position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
neg_log_dens(state)derivative with respect tostate.pos.
def h1_flow(self, state, dt)-
Apply exact flow map corresponding to
h1Hamiltonian component.stateargument is modified in place.Args
state:ChainState- State to start flow at.
dt:float- Time interval to simulate flow for.
def dh_dpos(self, state)-
Derivative of Hamiltonian with respect to position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
h(state)derivative with respect tostate.pos.
def dh_dmom(self, state)-
Derivative of Hamiltonian with respect to momentum.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
h(state)derivative with respect tostate.mom.
class DenseRiemannianMetricSystem (neg_log_dens, metric_func, vjp_metric_func=None, grad_neg_log_dens=None)-
Riemannian-metric system with dense matrix representation.
Hamiltonian system with a position dependent metric matrix representation which is specified to be a dense matrix function
metric_funcof the positionqwhich is guaranteed to be positive definite almost-everywhere, withM = metric_func(q)then the metric matrix representation.See documentation of
RiemannianMetricSystemfor more general details about Riemannian-metric Hamiltonian systems.Args
neg_log_dens:Callable[[array],float]- Function which given a position array returns the negative logarithm of an unnormalized probability density on the position space with respect to the Lebesgue measure, with the corresponding distribution on the position space being the target distribution it is wished to draw approximate samples from.
metric_func:Callable[[array],array]- Function which given a
position array returns a 2D array corresponding to the positive
definite metric matrix representation. The returned matrices
(2D arrays) are assumed to be positive-definite for all input
positions and a
LinAlgErrorexception may be raised if this fails to be the case. vjp_metric_func:NoneorCallable[[array],Callable[[array],array]] orCallable[[array],Tuple[Callable[[array],array],array]]-
Function which given a position array returns another function which takes a 2D array as an argument and returns the vector-Jacobian-product (VJP) of
metric_funcwith respect to the position array argument. The VJP is here defined as a function of a 2D arrayvvjp(v) = sum(v[:, :, None] * jacob[:, :, :], axis=(0, 1))where
jacobis the(dim_pos, dim_pos, dim_pos)shaped Jacobian ofM = metric_func(q)with respect toqi.e. the array of partial derivatives of the function such thatjacob[i, j, k] = ∂M[i, j] / ∂q[k]Optionally the function may instead return a 2-tuple of values with the first a function to compute a VJP of
metric_funcand the second a 2D array containing the value ofmetric_func, both evaluated at the passed position array. IfNoneis passed (the default) an automatic differentiation fallback will be used to attempt to construct a function which calculates the VJP (and value) ofmetric_funcautomatically. grad_neg_log_dens:NoneorCallable[[array],arrayorTuple[array,float]]-
Function which given a position array returns the derivative of
neg_log_denswith respect to the position array argument. Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the derivative and the second being the value of theneg_log_densevaluated at the passed position array. IfNoneis passed (the default) an automatic differentiation fallback will be used to attempt to construct the derivative ofneg_log_densautomatically.
Expand source code Browse git
class DenseRiemannianMetricSystem(RiemannianMetricSystem): """Riemannian-metric system with dense matrix representation. Hamiltonian system with a position dependent metric matrix representation which is specified to be a dense matrix function `metric_func` of the position `q` which is guaranteed to be positive definite almost-everywhere, with `M = metric_func(q)` then the metric matrix representation. See documentation of `RiemannianMetricSystem` for more general details about Riemannian-metric Hamiltonian systems. """ def __init__( self, neg_log_dens, metric_func, vjp_metric_func=None, grad_neg_log_dens=None ): """ Args: neg_log_dens (Callable[[array], float]): Function which given a position array returns the negative logarithm of an unnormalized probability density on the position space with respect to the Lebesgue measure, with the corresponding distribution on the position space being the target distribution it is wished to draw approximate samples from. metric_func (Callable[[array], array]): Function which given a position array returns a 2D array corresponding to the positive definite metric matrix representation. The returned matrices (2D arrays) are assumed to be positive-definite for all input positions and a `LinAlgError` exception may be raised if this fails to be the case. vjp_metric_func (None or Callable[[array], Callable[[array], array]] or Callable[[array], Tuple[Callable[[array], array], array]]): Function which given a position array returns another function which takes a 2D array as an argument and returns the *vector-Jacobian-product* (VJP) of `metric_func` with respect to the position array argument. The VJP is here defined as a function of a 2D array `v` vjp(v) = sum(v[:, :, None] * jacob[:, :, :], axis=(0, 1)) where `jacob` is the `(dim_pos, dim_pos, dim_pos)` shaped Jacobian of `M = metric_func(q)` with respect to `q` i.e. the array of partial derivatives of the function such that jacob[i, j, k] = ∂M[i, j] / ∂q[k] Optionally the function may instead return a 2-tuple of values with the first a function to compute a VJP of `metric_func` and the second a 2D array containing the value of `metric_func`, both evaluated at the passed position array. If `None` is passed (the default) an automatic differentiation fallback will be used to attempt to construct a function which calculates the VJP (and value) of `metric_func` automatically. grad_neg_log_dens ( None or Callable[[array], array or Tuple[array, float]]): Function which given a position array returns the derivative of `neg_log_dens` with respect to the position array argument. Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the derivative and the second being the value of the `neg_log_dens` evaluated at the passed position array. If `None` is passed (the default) an automatic differentiation fallback will be used to attempt to construct the derivative of `neg_log_dens` automatically. """ super().__init__( neg_log_dens, matrices.DensePositiveDefiniteMatrix, metric_func, vjp_metric_func, grad_neg_log_dens, )Ancestors
- RiemannianMetricSystem
- System
- abc.ABC
Methods
def metric_func(self, state)-
Function computing the parameter of the metric matrix representation.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
metric_func(state.pos).
def vjp_metric_func(self, state)-
Function constructing a vector-Jacobian-product for
metric_func.The vector-Jacobian-product is here defined as a function of an array
v(of the same shape as the output ofmetric_func) corresponding tovjp(v) = sum(v[..., None] * jacob, axis=tuple(range(v.ndim))where
jacobis the Jacobian ofm = metric_func(q)wrtqi.e. the array of partial derivatives of the function such thatjacob[..., i] = ∂m[...] / ∂q[i]Args
state:ChainState- State to compute VJP at.
Returns
Callable[[array], array]: Vector-Jacobian-product function.
def metric(self, state)-
Function computing the metric matrix representation.
The returned type of this function is that specified by the
metric_matrix_classargument to the initializer.Args
state:ChainState- State to compute value at.
Returns
mici.matrices.PositiveDefiniteMatrix: Metric matrix representation.
def h(self, state)-
Hamiltonian function for system.
Args
state:ChainState- State to compute value at.
Returns
float- Value of Hamiltonian.
def h1(self, state)-
Hamiltonian component depending only on position.
Args
state:ChainState- State to compute value at.
Returns
float- Value of
h1Hamiltonian component.
def dh1_dpos(self, state)-
Derivative of
h1Hamiltonian component with respect to position.Args
state:ChainState- State to compute value at.
Returns
array- Value of computed
h1derivative.
def h2(self, state)-
Hamiltonian component depending on momentum and optionally position.
Args
state:ChainState- State to compute value at.
Returns
float- Value of
h2Hamiltonian component.
def dh2_dmom(self, state)-
Derivative of
h2Hamiltonian component with respect to momentum.Args
state:ChainState- State to compute value at.
Returns
array- Value of
h2(state)derivative with respect tostate.pos.
def sample_momentum(self, state, rng)-
Sample a momentum from its conditional distribution given a position.
Args
state:ChainState- State defining position to condition on.
Returns
mom:array- Sampled momentum.
def neg_log_dens(self, state)-
Negative logarithm of unnormalized density of target distribution.
Args
state:ChainState- State to compute value at.
Returns
float- Value of computed negative log density.
def grad_neg_log_dens(self, state)-
Derivative of negative log density with respect to position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
neg_log_dens(state)derivative with respect tostate.pos.
def h1_flow(self, state, dt)-
Apply exact flow map corresponding to
h1Hamiltonian component.stateargument is modified in place.Args
state:ChainState- State to start flow at.
dt:float- Time interval to simulate flow for.
def dh_dpos(self, state)-
Derivative of Hamiltonian with respect to position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
h(state)derivative with respect tostate.pos.
def dh_dmom(self, state)-
Derivative of Hamiltonian with respect to momentum.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
h(state)derivative with respect tostate.mom.
class SoftAbsRiemannianMetricSystem (neg_log_dens, grad_neg_log_dens=None, hess_neg_log_dens=None, mtp_neg_log_dens=None, softabs_coeff=1.0)-
SoftAbs Riemmanian metric Hamiltonian system.
Hamiltonian system with a position dependent metric matrix representation which is specified to be a dense matrix function
metric_funcof the positionqwhich is guaranteed to be positive definite almost-everywhere, withM = metric_func(q)then the metric matrix representation.Hamiltonian system with a position dependent metric matrix representation which is specified to be an eigenvalue-regularized transformation of the Hessian of the negative log density function (the symmetric matrix of second derivatives the negative log density function with respect to the position array components. Specifically if
hess_neg_log_densis a symmetric 2D square array valued function of the positionq, withH = hess_neg_log_dens(q)then if an eigenvalue decomposition ofHis computed, i.e.eigval, eigvec = eigh(H), witheigvala 1D array of real eigenvalues, andeigvecthe corresponding 2D array (orthogonal matrix) with eigenvectors as columns, then the resulting positive-definite metric matrix representationMis computed asM = eigvec @ diag(softabs(eigval, softabs_coeff)) @ eigvec.Twith
softabs(x, softabs_coeff) = x / tanh(x * softabs_coeff)an elementwise function which acts as a smooth approximation to the absolute function (ensuring all the eigenvalues ofMare strictly positive) with the additional scalar parametersoftabs_coeffcontrolling the smoothness of the approximation, withsoftabstending to the piecewise linearabsfunction assoftabs_coefftends to infinity, and becoming increasingly smooth assoftabs_coefftends to zero.See documentation of
RiemannianMetricSystemfor more general details about Riemannian-metric Hamiltonian systems.References
- Betancourt, M., 2013. A general metric for Riemannian manifold Hamiltonian Monte Carlo. In Geometric science of information (pp. 327-334).
Args
neg_log_dens:Callable[[array],float]- Function which given a position array returns the negative logarithm of an unnormalized probability density on the position space with respect to the Lebesgue measure, with the corresponding distribution on the position space being the target distribution it is wished to draw approximate samples from.
grad_neg_log_dens:NoneorCallable[[array],arrayorTuple[array,float]]-
Function which given a position array returns the derivative of
neg_log_denswith respect to the position array argument. Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the derivative and the second being the value of theneg_log_densevaluated at the passed position array. IfNoneis passed (the default) an automatic differentiation fallback will be used to attempt to construct the derivative ofneg_log_densautomatically. hess_neg_log_dens:NoneorCallable[[array],arrayorTuple[array,array,float]]-
Function which given a position array returns the Hessian of
neg_log_denswith respect to the position array argument as a 2D array. Optionally the function may instead return a 3-tuple of values with the first a 2D array containting the Hessian ofneg_log_dens, the second a 1D array containing the gradient ofneg_log_densand the third the value ofneg_log_dens, all evaluated at the passed position array. IfNoneis passed (the default) an automatic differentiation fallback will be used to attempt to construct a function which calculates the Hessian (and gradient and value) ofneg_log_densautomatically. mtp_neg_log_dens:NoneorCallable[[array],Callable[[array],array]] orCallable[[array],Tuple[Callable,array,array,float]]-
Function which given a position array returns another function which takes a 2D array (matrix) as an argument and returns the matrix-Tressian-product (MTP) of
neg_log_denswith respect to the position array argument. The MTP is here defined as a function of a matrixmcorresponding tomtp(m) = sum(m[:, :, None] * tress[:, :, :], axis=(0, 1))where
tressis the 'Tressian' off = neg_log_dens(q)wrtqi.e. the 3D array of third-order partial derivatives of the scalar-valued function such thattress[i, j, k] = ∂³f / (∂q[i] ∂q[j] ∂q[k])Optionally the function may instead return a 4-tuple of values with the first a function to compute a MTP of
neg_log_dens, the second a 2D array containing the Hessian ofneg_log_dens, the third a 1D array containing the gradient ofneg_log_densand the fourth the value ofneg_log_dens, all evaluated at the passed position array. IfNoneis passed (the default) an automatic differentiation fallback will be used to attempt to construct a function which calculates the MTP (and Hesisan and gradient and value) ofneg_log_densautomatically. softabs_coeff:float- Positive regularisation coefficient for smooth approximation to absolute value used to regularize Hessian eigenvalues in metric matrix representation. As the value tends to infinity the approximation becomes increasingly close to the absolute function.
Expand source code Browse git
class SoftAbsRiemannianMetricSystem(RiemannianMetricSystem): """SoftAbs Riemmanian metric Hamiltonian system. Hamiltonian system with a position dependent metric matrix representation which is specified to be a dense matrix function `metric_func` of the position `q` which is guaranteed to be positive definite almost-everywhere, with `M = metric_func(q)` then the metric matrix representation. Hamiltonian system with a position dependent metric matrix representation which is specified to be an eigenvalue-regularized transformation of the Hessian of the negative log density function (the symmetric matrix of second derivatives the negative log density function with respect to the position array components. Specifically if `hess_neg_log_dens` is a symmetric 2D square array valued function of the position `q`, with `H = hess_neg_log_dens(q)` then if an eigenvalue decomposition of `H` is computed, i.e. `eigval, eigvec = eigh(H)`, with `eigval` a 1D array of real eigenvalues, and `eigvec` the corresponding 2D array (orthogonal matrix) with eigenvectors as columns, then the resulting positive-definite metric matrix representation `M` is computed as M = eigvec @ diag(softabs(eigval, softabs_coeff)) @ eigvec.T with `softabs(x, softabs_coeff) = x / tanh(x * softabs_coeff)` an elementwise function which acts as a smooth approximation to the absolute function (ensuring all the eigenvalues of `M` are strictly positive) with the additional scalar parameter `softabs_coeff` controlling the smoothness of the approximation, with `softabs` tending to the piecewise linear `abs` function as `softabs_coeff` tends to infinity, and becoming increasingly smooth as `softabs_coeff` tends to zero. See documentation of `RiemannianMetricSystem` for more general details about Riemannian-metric Hamiltonian systems. References: 1. Betancourt, M., 2013. A general metric for Riemannian manifold Hamiltonian Monte Carlo. In Geometric science of information (pp. 327-334). """ def __init__( self, neg_log_dens, grad_neg_log_dens=None, hess_neg_log_dens=None, mtp_neg_log_dens=None, softabs_coeff=1.0, ): """ Args: neg_log_dens (Callable[[array], float]): Function which given a position array returns the negative logarithm of an unnormalized probability density on the position space with respect to the Lebesgue measure, with the corresponding distribution on the position space being the target distribution it is wished to draw approximate samples from. grad_neg_log_dens ( None or Callable[[array], array or Tuple[array, float]]): Function which given a position array returns the derivative of `neg_log_dens` with respect to the position array argument. Optionally the function may instead return a 2-tuple of values with the first being the array corresponding to the derivative and the second being the value of the `neg_log_dens` evaluated at the passed position array. If `None` is passed (the default) an automatic differentiation fallback will be used to attempt to construct the derivative of `neg_log_dens` automatically. hess_neg_log_dens (None or Callable[[array], array or Tuple[array, array, float]]): Function which given a position array returns the Hessian of `neg_log_dens` with respect to the position array argument as a 2D array. Optionally the function may instead return a 3-tuple of values with the first a 2D array containting the Hessian of `neg_log_dens`, the second a 1D array containing the gradient of `neg_log_dens` and the third the value of `neg_log_dens`, all evaluated at the passed position array. If `None` is passed (the default) an automatic differentiation fallback will be used to attempt to construct a function which calculates the Hessian (and gradient and value) of `neg_log_dens` automatically. mtp_neg_log_dens (None or Callable[[array], Callable[[array], array]] or Callable[[array], Tuple[Callable, array, array, float]]): Function which given a position array returns another function which takes a 2D array (matrix) as an argument and returns the *matrix-Tressian-product* (MTP) of `neg_log_dens` with respect to the position array argument. The MTP is here defined as a function of a matrix `m` corresponding to mtp(m) = sum(m[:, :, None] * tress[:, :, :], axis=(0, 1)) where `tress` is the 'Tressian' of `f = neg_log_dens(q)` wrt `q` i.e. the 3D array of third-order partial derivatives of the scalar-valued function such that tress[i, j, k] = ∂³f / (∂q[i] ∂q[j] ∂q[k]) Optionally the function may instead return a 4-tuple of values with the first a function to compute a MTP of `neg_log_dens`, the second a 2D array containing the Hessian of `neg_log_dens`, the third a 1D array containing the gradient of `neg_log_dens` and the fourth the value of `neg_log_dens`, all evaluated at the passed position array. If `None` is passed (the default) an automatic differentiation fallback will be used to attempt to construct a function which calculates the MTP (and Hesisan and gradient and value) of `neg_log_dens` automatically. softabs_coeff (float): Positive regularisation coefficient for smooth approximation to absolute value used to regularize Hessian eigenvalues in metric matrix representation. As the value tends to infinity the approximation becomes increasingly close to the absolute function. """ self._hess_neg_log_dens = autodiff_fallback( hess_neg_log_dens, neg_log_dens, "hessian_grad_and_value", "neg_log_dens" ) self._mtp_neg_log_dens = autodiff_fallback( mtp_neg_log_dens, neg_log_dens, "mtp_hessian_grad_and_value", "mtp_neg_log_dens", ) super().__init__( neg_log_dens, matrices.SoftAbsRegularizedPositiveDefiniteMatrix, self._hess_neg_log_dens, self._mtp_neg_log_dens, grad_neg_log_dens, metric_kwargs={"softabs_coeff": softabs_coeff}, ) def metric_func(self, state): return self.hess_neg_log_dens(state) def vjp_metric_func(self, state): return self.mtp_neg_log_dens(state) @cache_in_state_with_aux("pos", ("grad_neg_log_dens", "neg_log_dens")) def hess_neg_log_dens(self, state): """Hessian of negative log density with respect to position. Args: state (mici.states.ChainState): State to compute value at. Returns: hessian (array): 2D array of `neg_log_dens(state)` second derivatives with respect to `state.pos`, with `hessian[i, j]` the second derivative of `neg_log_dens(state)` with respect to `state.pos[i]` and `state.pos[j]`. """ return self._hess_neg_log_dens(state.pos) @cache_in_state_with_aux( "pos", ("hess_neg_log_dens", "grad_neg_log_dens", "neg_log_dens") ) def mtp_neg_log_dens(self, state): """Generate MTP of negative log density with respect to position. The matrix-Tressian-product (MTP) is here defined as a function of a matrix `m` corresponding to mtp(m) = sum(m[:, :, None] * tress[:, :, :], axis=(0, 1)) where `tress` is the 'Tressian' of `f = neg_log_dens(q)` with respect to `q = state.pos` i.e. the 3D array of third-order partial derivatives of the scalar-valued function such that tress[i, j, k] = ∂³f / (∂q[i] ∂q[j] ∂q[k]) Args: state (mici.states.ChainState): State to compute value at. Returns: mtp (Callable[[array], array]): Function which accepts a 2D array of shape `(state.pos.shape[0], state.pos.shape[0])` as an argument and returns an array of shape `state.pos.shape` containing the computed MTP value. """ return self._mtp_neg_log_dens(state.pos)Ancestors
- RiemannianMetricSystem
- System
- abc.ABC
Methods
def metric_func(self, state)-
Function computing the parameter of the metric matrix representation.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
metric_func(state.pos).
Expand source code Browse git
def metric_func(self, state): return self.hess_neg_log_dens(state) def vjp_metric_func(self, state)-
Function constructing a vector-Jacobian-product for
metric_func.The vector-Jacobian-product is here defined as a function of an array
v(of the same shape as the output ofmetric_func) corresponding tovjp(v) = sum(v[..., None] * jacob, axis=tuple(range(v.ndim))where
jacobis the Jacobian ofm = metric_func(q)wrtqi.e. the array of partial derivatives of the function such thatjacob[..., i] = ∂m[...] / ∂q[i]Args
state:ChainState- State to compute VJP at.
Returns
Callable[[array], array]: Vector-Jacobian-product function.
Expand source code Browse git
def vjp_metric_func(self, state): return self.mtp_neg_log_dens(state) def hess_neg_log_dens(self, state)-
Hessian of negative log density with respect to position.
Args
state:ChainState- State to compute value at.
Returns
hessian:array- 2D array of
neg_log_dens(state)second derivatives with respect tostate.pos, withhessian[i, j]the second derivative ofneg_log_dens(state)with respect tostate.pos[i]andstate.pos[j].
Expand source code Browse git
@cache_in_state_with_aux("pos", ("grad_neg_log_dens", "neg_log_dens")) def hess_neg_log_dens(self, state): """Hessian of negative log density with respect to position. Args: state (mici.states.ChainState): State to compute value at. Returns: hessian (array): 2D array of `neg_log_dens(state)` second derivatives with respect to `state.pos`, with `hessian[i, j]` the second derivative of `neg_log_dens(state)` with respect to `state.pos[i]` and `state.pos[j]`. """ return self._hess_neg_log_dens(state.pos) def mtp_neg_log_dens(self, state)-
Generate MTP of negative log density with respect to position.
The matrix-Tressian-product (MTP) is here defined as a function of a matrix
mcorresponding tomtp(m) = sum(m[:, :, None] * tress[:, :, :], axis=(0, 1))where
tressis the 'Tressian' off = neg_log_dens(q)with respect toq = state.posi.e. the 3D array of third-order partial derivatives of the scalar-valued function such thattress[i, j, k] = ∂³f / (∂q[i] ∂q[j] ∂q[k])Args
state:ChainState- State to compute value at.
Returns
mtp:Callable[[array],array]- Function which accepts a 2D array
of shape
(state.pos.shape[0], state.pos.shape[0])as an argument and returns an array of shapestate.pos.shapecontaining the computed MTP value.
Expand source code Browse git
@cache_in_state_with_aux( "pos", ("hess_neg_log_dens", "grad_neg_log_dens", "neg_log_dens") ) def mtp_neg_log_dens(self, state): """Generate MTP of negative log density with respect to position. The matrix-Tressian-product (MTP) is here defined as a function of a matrix `m` corresponding to mtp(m) = sum(m[:, :, None] * tress[:, :, :], axis=(0, 1)) where `tress` is the 'Tressian' of `f = neg_log_dens(q)` with respect to `q = state.pos` i.e. the 3D array of third-order partial derivatives of the scalar-valued function such that tress[i, j, k] = ∂³f / (∂q[i] ∂q[j] ∂q[k]) Args: state (mici.states.ChainState): State to compute value at. Returns: mtp (Callable[[array], array]): Function which accepts a 2D array of shape `(state.pos.shape[0], state.pos.shape[0])` as an argument and returns an array of shape `state.pos.shape` containing the computed MTP value. """ return self._mtp_neg_log_dens(state.pos) def metric(self, state)-
Function computing the metric matrix representation.
The returned type of this function is that specified by the
metric_matrix_classargument to the initializer.Args
state:ChainState- State to compute value at.
Returns
mici.matrices.PositiveDefiniteMatrix: Metric matrix representation.
def h(self, state)-
Hamiltonian function for system.
Args
state:ChainState- State to compute value at.
Returns
float- Value of Hamiltonian.
def h1(self, state)-
Hamiltonian component depending only on position.
Args
state:ChainState- State to compute value at.
Returns
float- Value of
h1Hamiltonian component.
def dh1_dpos(self, state)-
Derivative of
h1Hamiltonian component with respect to position.Args
state:ChainState- State to compute value at.
Returns
array- Value of computed
h1derivative.
def h2(self, state)-
Hamiltonian component depending on momentum and optionally position.
Args
state:ChainState- State to compute value at.
Returns
float- Value of
h2Hamiltonian component.
def dh2_dmom(self, state)-
Derivative of
h2Hamiltonian component with respect to momentum.Args
state:ChainState- State to compute value at.
Returns
array- Value of
h2(state)derivative with respect tostate.pos.
def sample_momentum(self, state, rng)-
Sample a momentum from its conditional distribution given a position.
Args
state:ChainState- State defining position to condition on.
Returns
mom:array- Sampled momentum.
def neg_log_dens(self, state)-
Negative logarithm of unnormalized density of target distribution.
Args
state:ChainState- State to compute value at.
Returns
float- Value of computed negative log density.
def grad_neg_log_dens(self, state)-
Derivative of negative log density with respect to position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
neg_log_dens(state)derivative with respect tostate.pos.
def h1_flow(self, state, dt)-
Apply exact flow map corresponding to
h1Hamiltonian component.stateargument is modified in place.Args
state:ChainState- State to start flow at.
dt:float- Time interval to simulate flow for.
def dh_dpos(self, state)-
Derivative of Hamiltonian with respect to position.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
h(state)derivative with respect tostate.pos.
def dh_dmom(self, state)-
Derivative of Hamiltonian with respect to momentum.
Args
state:ChainState- State to compute value at.
Returns
array- Value of
h(state)derivative with respect tostate.mom.