Sumeetpal S. Singh

Nikolas Kantas, Sumeetpal S. Singh, Arnaud Doucet

We show that the sensor self-localization problem can be cast as a static parameter estimation problem for Hidden Markov Models and we implement fully decentralized versions of the Recursive Maximum Likelihood and on-line Expectation-Maximization algorithms to localize the sensor network simultaneously with target tracking. For linear Gaussian models, our algorithms can be implemented exactly using a distributed version of the Kalman filter and a novel message passing algorithm. The latter allows each node to compute the local derivatives of the likelihood or the sufficient statistics needed for Expectation-Maximization. In the non-linear case, a solution based on local linearization in the spirit of the Extended Kalman Filter is proposed. In numerical examples we demonstrate that the developed algorithms are able to learn the localization parameters.

Chon Wai Ho, Sumeetpal S. Singh, Jiaqi Guo

Sequential decision-making problems are often modelled as a Markov decision process (MDP). We focus on the stochastic shortest path (SSP) problem, which is an infinite-horizon undiscounted MDP with absorbing terminal states. We develop a Bayesian framework to learn the optimal decision strategy through interactions with the decision-making task. Specifically, we learn the optimal action-value function $Q^*$, but unlike many existing Bayesian approaches, we do not rely on unrealistic modelling assumptions and ad-hoc approximations. Our approach is to directly construct the posterior beliefs for $Q^*$ through Bellman's optimality equations. For deterministic rewards, we characterise the posterior as a distribution with a manifold density. To facilitate simpler inference, we relax the likelihood so that a Lebesgue density exists. The flip side is to create unidentifiability issues. Specifically, the relaxed posterior can have significant mass on improper decision rules, while the exact posterior will not. We also calculate the exact posterior probabilities for optimal action selections for the tabular parametrisation of $Q^*$, a Gaussian likelihood relaxation and a Gaussian prior, which is useful in benchmarking studies. Numerical studies on variants of the Deep Sea benchmark verify our findings. We demonstrate that our framework faithfully quantifies uncertainty and, compared to other temporal-difference-based Bayesian methodologies, is more data efficient. We conclude with recommendations for future work.

Jiaqi Guo, Chon Wai Ho, Sumeetpal S. Singh

The Markov Decision Process (MDP) is a popular framework for sequential decision-making problems, and uncertainty quantification is an essential component of it to learn optimal decision-making strategies. In particular, a Bayesian framework is used to maintain beliefs about the optimal decisions and the unknown ingredients of the model, which are also to be learned from the data, such as the rewards and state dynamics. However, many existing Bayesian approaches for learning the optimal decision-making strategy are based on unrealistic modelling assumptions and utilise approximate inference techniques. This raises doubts whether the benefits of Bayesian uncertainty quantification are fully realised or can be relied upon. We focus on infinite-horizon and undiscounted MDPs, with finite state and action spaces, and a terminal state. We provide a full Bayesian framework, from modelling to inference to decision-making. For modelling, we introduce a likelihood function with minimal assumptions for learning the optimal action-value function based on Bellman's optimality equations, analyse its properties, and clarify connections to existing works. For deterministic rewards, the likelihood is degenerate and we introduce artificial observation noise to relax it, in a controlled manner, to facilitate more efficient Monte Carlo-based inference. For inference, we propose an adaptive sequential Monte Carlo algorithm to both sample from and adjust the sequence of relaxed posterior distributions. For decision-making, we choose actions using samples from the posterior distribution over the optimal strategies. While commonly done, we provide new insight that clearly shows that it is a generalisation of Thompson sampling from multi-arm bandit problems. Finally, we evaluate our framework on the Deep Sea benchmark problem and demonstrate the exploration benefits of posterior sampling in MDPs.

Torben Sell, Sumeetpal S. Singh

This paper introduces a new neural network based prior for real valued functions on $\mathbb R^d$ which, by construction, is more easily and cheaply scaled up in the domain dimension $d$ compared to the usual Karhunen-Loève function space prior. The new prior is a Gaussian neural network prior, where each weight and bias has an independent Gaussian prior, but with the key difference that the variances decrease in the width of the network in such a way that the resulting function is \emph{almost surely} well defined in the limit of an infinite width network. We show that in a Bayesian treatment of inferring unknown functions, the induced posterior over functions is amenable to Monte Carlo sampling using Hilbert space Markov chain Monte Carlo (MCMC) methods. This type of MCMC is popular, e.g. in the Bayesian Inverse Problems literature, because it is stable under \emph{mesh refinement}, i.e. the acceptance probability does not shrink to $0$ as more parameters of the function's prior are introduced, even \emph{ad infinitum}. In numerical examples we demonstrate these stated competitive advantages over other function space priors. We also implement examples in Bayesian Reinforcement Learning to automate tasks from data and demonstrate, for the first time, stability of MCMC to mesh refinement for these type of problems.

Lan Jiang, Sumeetpal S. Singh

A new Bayesian state and parameter learning algorithm for multiple target tracking (MTT) models with image observations is proposed. Specifically, a Markov chain Monte Carlo algorithm is designed to sample from the posterior distribution of the unknown number of targets, their birth and death times, states and model parameters, which constitutes the complete solution to the tracking problem. The conventional approach is to pre-process the images to extract point observations and then perform tracking. We model the image generation process directly to avoid potential loss of information when extracting point observations. Numerical examples show that our algorithm has improved tracking performance over commonly used techniques, for both synthetic examples and real florescent microscopy data, especially in the case of dim targets with overlapping illuminated regions.

Lan Jiang, Sumeetpal S. Singh, Sinan Yıldırım

We propose a new Bayesian tracking and parameter learning algorithm for non-linear non-Gaussian multiple target tracking (MTT) models. We design a Markov chain Monte Carlo (MCMC) algorithm to sample from the posterior distribution of the target states, birth and death times, and association of observations to targets, which constitutes the solution to the tracking problem, as well as the model parameters. In the numerical section, we present performance comparisons with several competing techniques and demonstrate significant performance improvements in all cases.

Sinan Yildirim, A. Taylan Cemgil, Sumeetpal S. Singh

In this paper we formulate the nonnegative matrix factorisation (NMF) problem as a maximum likelihood estimation problem for hidden Markov models and propose online expectation-maximisation (EM) algorithms to estimate the NMF and the other unknown static parameters. We also propose a sequential Monte Carlo approximation of our online EM algorithm. We show the performance of the proposed method with two numerical examples.