Introduction

I am a PhD student in the Doctoral Training Centre in Neuroinformatics, part of the School of Informatics at University of Edinburgh. My supervisor is Amos Storkey and I am part of his machine learning research group.

My research is focused on developing methods for performing efficient approximate inference in complex probabilistic models.

Contact

Address
Room 5.08, Informatics Forum, University of Edinburgh, 10 Crichton Street, Edinburgh, EH8 9AB

Research

PhD project: Auxiliary variable MCMC methods (Supervisor: Amos Storkey)

My PhD project has been primarily concerned with developments to Markov Chain Monte Carlo (MCMC) methods. These are a group of techniques for performing approximate inference with probabilistic models.

The core idea of MCMC methods is to simulate a stochastic dynamical system where the probability distribution over the state of the system converges to the distribution of interest. The samples of the system state can then be used to estimate expectations (averages under a probability distribution) of the model. How well the dynamic is able to explore the state-space determines how quickly these estimates converge to the correct values.

I am specifically considering methods which augment the system state space with additional auxiliary variables. In some cases this allows the robustness or efficiency of sampling methods to be improved, for example by making it easier for the sampler to coherently explore the target distribution or to increase movement between modes in multimodal target distributions. In other settings redefining the state space of the problem can allow us to perform inference in settings where we do not have an explicit form for the distribution on the variables of interest.

Pre-prints

Journal articles

  • Asymptotically exact inference in differentiable generative models

    Matthew M. Graham and Amos J. Storkey

    To appear in: Electronic Journal of Statistics

    Many generative models can be expressed as a differentiable function applied to input variables sampled from a known probability distribution. This framework includes both the generative component of learned parametric models such as variational autoencoders and generative adversarial networks, and also procedurally defined simulator models which involve only differentiable operations. Though the distribution on the input variables to such models is known, often the distribution on the output variables is only implicitly defined. We present a method for performing efficient Markov chain Monte Carlo inference in such models when conditioning on observations of the model output. For some models this offers an asymptotically exact inference method where approximate Bayesian computation might otherwise be employed. We use the intuition that computing conditional expectations is equivalent to integrating over a density defined on the manifold corresponding to the set of inputs consistent with the observed outputs. This motivates the use of a constrained variant of Hamiltonian Monte Carlo which leverages the smooth geometry of the manifold to coherently move between inputs exactly consistent with observations. We validate the method by performing inference experiments in a diverse set of models.

Publications

Conference proceedings

  • 08/2017 Continuously tempered Hamiltonian Monte Carlo

    Matthew M. Graham and Amos J. Storkey

    Proceedings of the 33rd Conference on Uncertainty in Artificial Intelligence

    Hamiltonian Monte Carlo (HMC) is a powerful Markov chain Monte Carlo (MCMC) method for performing approximate inference in complex probabilistic models of continuous variables. In common with many MCMC methods, however, the standard HMC approach performs poorly in distributions with multiple isolated modes. We present a method for augmenting the Hamiltonian system with an extra continuous temperature control variable which allows the dynamic to bridge between sampling a complex target distribution and a simpler unimodal base distribution. This augmentation both helps improve mixing in multimodal targets and allows the normalisation constant of the target distribution to be estimated. The method is simple to implement within existing HMC code, requiring only a standard leapfrog integrator. We demonstrate experimentally that the method is competitive with annealed importance sampling and simulating tempering methods at sampling from challenging multimodal distributions and estimating their normalising constants.
  • 04/2017 Asymptotically exact inference in differentiable generative models

    Matthew M. Graham and Amos J. Storkey

    Proceedings of the 20th International Conference on Artificial Intelligence and Statistics

    Many generative models can be expressed as a differentiable function of random inputs drawn from some simple probability density. This framework includes both deep generative architectures such as Variational Autoencoders and a large class of procedurally defined simulator models. We present a method for performing efficient MCMC inference in such models when conditioning on observations of the model output. For some models this offers an asymptotically exact inference method where Approximate Bayesian Computation might otherwise be employed. We use the intuition that inference corresponds to integrating a density across the manifold corresponding to the set of inputs consistent with the observed outputs. This motivates the use of a constrained variant of Hamiltonian Monte Carlo which leverages the smooth geometry of the manifold to coherently move between inputs exactly consistent with observations. We validate the method by performing inference tasks in a diverse set of models.
  • 05/2016 Pseudo-Marginal Slice Sampling

    Iain Murray and Matthew M. Graham

    Proceedings of the 19th International Conference on Artificial Intelligence and Statistics

    Markov chain Monte Carlo (MCMC) methods asymptotically sample from complex probability distributions. The pseudo-marginal MCMC framework only requires an unbiased estimator of the unnormalized probability distribution function to construct a Markov chain. However, the resulting chains are harder to tune to a target distribution than conventional MCMC, and the types of updates available are limited. We describe a general way to clamp and update the random numbers used in a pseudo-marginal method's unbiased estimator. In this framework we can use slice sampling and other adaptive methods. We obtain more robust Markov chains, which often mix more quickly.

Workshop papers

  • 08/2017 Inference in differentiable generative models

    Matthew M. Graham and Amos J. Storkey

    ICML 2017 workshop: Implicit generative models

    Many generative models can be expressed as a differentiable function of random inputs drawn from a known probability distribution. This framework includes both learnt parametric generative models and a large class of procedurally defined simulator models. We present a method for performing efficient Markov chain Monte Carlo (MCMC) inference in such models when conditioning on observations of the model output. For some models this offers an asymptotically exact inference method where Approximate Bayesian Computation might otherwise be employed. We use the intuition that inference corresponds to integrating a density across the manifold corresponding to the set of inputs consistent with the observed outputs. This motivates the use of a constrained variant of Hamiltonian Monte Carlo which leverages the smooth geometry of the manifold to move between inputs exactly consistent with observations.
  • 12/2016 Continuously tempered Hamiltonian Monte Carlo

    Matthew M. Graham and Amos J. Storkey

    NIPS 2016 workshop: Advances in Approximate Bayesian Inference

    Hamiltonian Monte Carlo (HMC) is a powerful Markov chain Monte Carlo (MCMC) method for performing approximate inference in complex probabilistic models of continuous variables. In common with many MCMC methods however the standard HMC approach performs poorly in distributions with multiple isolated modes. Based on an approach proposed in the statistical physics literature, we present a method for augmenting the Hamiltonian system with an extra continuous temperature control variable which allows the dynamic to bridge between sampling a complex target distribution and a simpler uni-modal base distribution. This augmentation both helps increase mode-hopping in multi-modal targets and allows the normalisation constant of the target distribution to be estimated. The method is simple to implement within existing HMC code, requiring only a standard leapfrog integrator. It produces MCMC samples from the target distribution which can be used to directly estimate expectations without any importance re-weighting.

Talks

Previous projects

MSc by Research project: Insect olfactory landmark navigation (Supervisor: Barbara Webb)

This project was motivated by the work of Kathrin Steck and colleagues who discovered that Cataglyphis fortis, a Saharan desert ant species, are able to use odour sources in their environment as 'landmarks' to help when navigating back to their nest. A field study was conducted attempting to see if the previous results could be observed when ants were subjected to a more complex navigation task and an information-theoretic analysis used to try to establish how much positional information is available from local olfactory sensation of remote odour sources.

MEng project: Measuring tissue stiffness with ultrasound (Supervisor: Graham Treece)

This project was based around the technique of ultrasound elastography. In particular I was trying to develop a technique for estimating absolute stiffness at a small set of points in an ultrasound image plane by tracking the propagation of shear waves produced by a surface tap using standard ultrasound imaging hardware.

Code

Most of my code can be found on my Github profile. Below are a selection of the possibly more generally useful repositories.

Teaching

I am currently the teaching assistant for the coursework-based Machine Learning Practical. The Jupyter notebooks and Python framework we are developing for the course lab sessions are being distributed on the course Github repository. I am also a tutor for Machine Learning and Pattern Recognition. I have also previously tutored Information Theory and Probabilistic Modelling and Reasoning (PMR). There a couple of Jupyter notebooks with notes on PMR topics I made when tutoring on Github here.

I did a short tutorial on Hamiltonian Monte Carlo for the Institute for Adaptive and Neural Computation PIGlets discussion group. The slides are available here and an associated Jupyter notebook going through an example implementation is available on Github.

Non-work interests

Dawn sky over the Cairngorm plateau

In my spare time I'm a keen hillwalker and mountaineer. I walked extensively with the Cambridge University Hillwalking Club while doing my undergraduate there. In the summer of 2012 I helped organise and took part in an expedition to the Tien Shan range in Kyrgyzstan with seven other CUHWC members. Since coming to Edinburgh I've joined the Edinburgh University Hillwalking Club and have started climbing regularly at the wall in the University sports centre.