Introduction

I am a research fellow in the Department of Statistics and Applied Probability at the National University of Singapore, where I am part of Alex Thiery's research group. My research is focused on developing methods for performing efficient approximate inference in complex probabilistic models.

Research interests: Markov chain Monte Carlo methods, Hamiltonian Monte Carlo, approximate Bayesian computation, data assimilation, numerical simulation.

Contact

Address
Office 05-97, Block S16, 6 Science Drive 2, National University of Singapore, Singapore, 117546

Publications

Journal articles

  • 2017/12 Asymptotically exact inference in differentiable generative models

    Matthew M. Graham and Amos J. Storkey

    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 move between inputs exactly consistent with observations. We validate the method by performing inference experiments in a diverse set of models.

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.

Theses and dissertations

  • 2018/07 Auxiliary variable Markov chain Monte Carlo methods

    Matthew M. Graham

    PhD thesis, University of Edinburgh

    Markov chain Monte Carlo (MCMC) methods are a widely applicable class of algorithms for estimating integrals in statistical inference problems. A common approach in MCMC methods is to introduce additional auxiliary variables into the Markov chain state and perform transitions in the joint space of target and auxiliary variables. In this thesis we consider novel methods for using auxiliary variables within MCMC methods to allow approximate inference in otherwise intractable models and to improve sampling performance in models exhibiting challenging properties such as multimodality. We first consider the pseudo-marginal framework. This extends the Metropolis–Hastings algorithm to cases where we only have access to an unbiased estimator of the density of target distribution. The resulting chains can sometimes show ‘sticking’ behaviour where long series of proposed updates are rejected. Further the algorithms can be difficult to tune and it is not immediately clear how to generalise the approach to alternative transition operators. We show that if the auxiliary variables used in the density estimator are included in the chain state it is possible to use new transition operators such as those based on slice-sampling algorithms within a pseudo-marginal setting. This auxiliary pseudo-marginal approach leads to easier to tune methods and is often able to improve sampling efficiency over existing approaches. As a second contribution we consider inference in probabilistic models defined via a generative process with the probability density of the outputs of this process only implicitly defined. The approximate Bayesian computation (ABC) framework allows inference in such models when conditioning on the values of observed model variables by making the approximation that generated observed variables are ‘close’ rather than exactly equal to observed data. Although making the inference problem more tractable, the approximation error introduced in ABC methods can be difficult to quantify and standard algorithms tend to perform poorly when conditioning on high dimensional observations. This often requires further approximation by reducing the observations to lower dimensional summary statistics. We show how including all of the random variables used in generating model outputs as auxiliary variables in a Markov chain state can allow the use of more efficient and robust MCMC methods such as slice sampling and Hamiltonian Monte Carlo (HMC) within an ABC framework. In some cases this can allow inference when conditioning on the full set of observed values when standard ABC methods require reduction to lower dimensional summaries for tractability. Further we introduce a novel constrained HMC method for performing inference in a restricted class of differentiable generative models which allows conditioning the generated observed variables to be arbitrarily close to observed data while maintaining computational tractability. As a final topicwe consider the use of an auxiliary temperature variable in MCMC methods to improve exploration of multimodal target densities and allow estimation of normalising constants. Existing approaches such as simulated tempering and annealed importance sampling use temperature variables which take on only a discrete set of values. The performance of these methods can be sensitive to the number and spacing of the temperature values used, and the discrete nature of the temperature variable prevents the use of gradient-based methods such as HMC to update the temperature alongside the target variables. We introduce new MCMC methods which instead use a continuous temperature variable. This both removes the need to tune the choice of discrete temperature values and allows the temperature variable to be updated jointly with the target variables within a HMC method.
  • 2013/07 Insect olfactory landmark navigation

    Matthew M. Graham

    MSc by Research dissertation, University of Edinburgh

    The natural world is full of chemical signals - organisms of all scales and taxonomic classifications transmit and receive chemical signals to guide the full gamut of life’s processes: from helping forming mother-infant bonds, to identifying potential mates and even signalling their own deaths. Insects are particularly reliant on chemical cues to guide their behaviour and understanding how insects respond to and use chemical cues in their environment is a high active research area. In a series of recent studies Steck et al. produced evidence that foragers of the Saharan desert ant species Cataglyphis fortis are able to learn an association between an array of odour sources arranged around the entrance to their nest and the relative location of the nest entrance and later use the information they receive from the odour sources to help them navigate to the visually inconspicious nest entrance. This ability to use odour sources as olfactory landmarks had not been previously seen experimentally in insects, and is a remarkable behaviour given the extremely complex and highly dynamic nature of the olfactory signals received by the ants from the turbulent odour plumes the chemicals travel in from the sources. After an introductory chapter covering some relevant background theory to the work in this project, the second chapter of this dissertation will detail a field study conducted with the European desert ant species Cataglyphis velox. As in the studies of Steck et al. the ants were constrained to moving a linear channel and so the navigation task limited to being one-dimensional, the aim of this study was to see if there was any evidence supporting the hypothesis that Cataglyphis velox ants are able to use olfactory landmarks to navigate in a more realistic open environment. The results of the study were inconclusive, due to the low sample sizes that were collected and small effect size in the study design used, however it is proposed that the study could be considered usefully as pilot for a full study at a later date, and an adjusted study design is proposed that might overcome a lot of the issues encountered in the current study. In the third and final chapter of this dissertation, a modelling study of what information is available in the olfactory signal received from a turbulent odour plume about the location of the source of that plume is presented, with this work aiming to explore the information which may being used by Cataglyphis desert ants when using olfactory landmarks to navigate. The details of the plume and olfactory sensor models used are described and the results of an analysis of the estimated mutual information between the modelled olfactory signals and the location of odour source presented. It is found that the locational informational content of individual signal segment statistics seems to be low, though combining multiple statistics does potentially allow more useful reductions in uncertainty.

Talks

Projects

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

My PhD project was focussed on developments to Markov Chain Monte Carlo (MCMC) methods. I specifically considered 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.

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

University of Edinburgh, School of Informatics

I was the teaching assistant for the coursework-based Machine Learning Practical in the 2016-2017 academic year. The Jupyter notebooks and Python framework I helped co-develop for the course lab sessions are available at the course Github repository. I have also previously tutored for Machine Learning and Pattern Recognition, 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 during my undergraduate studies. 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. During my PhD in Edinburgh I was a member of Edinburgh University Hillwalking Club and also began climbing in and outdoors more regularly (though still sadly not all that often!).