Efficient Monte Carlo methods for Bayesian state-space model inference

Abstract

State-space models are widely used to model time series data where the observations depend on a latent process. The latent process consists of a sequence of latent states that evolve according to a specified system process. The observed time series data are then modelled as a function of the latent states via an observation process. Estimating the parameters of a state-space model within a Bayesian framework can be challenging. In this thesis, we consider these challenges and develop efficient Monte Carlo algorithms to aid inference. We propose two classes of method that, as a central theme, leverage approximate hidden Markov models for efficient inference.

In the first part of this thesis, we propose that approximate hidden Markov models can be used to design efficient Markov chain Monte Carlo proposal distributions, defined such that the usual theoretical guarantees apply. We discuss how the hidden Markov models are constructed under the proposed approach, the associated generality arising from the tuning parameters, and how these tuning parameters can be chosen efficiently in practice. We demonstrate that this proposed algorithm provides an efficient and robust alternative method for fitting state-space models, even for those that exhibit near-chaotic behaviour.

In the second part of this thesis, we develop an approximate hidden Markov model approach to designing efficient particle Markov chain Monte Carlo algorithms. In particular, we propose an approach to particle Gibbs with ancestor sampling that leverages approximate hidden Markov models to combat impoverishment within the sequential Monte Carlo steps of the original algorithm. We additionally propose that fixed approximations to the hidden Markov model can be used to substantially reduce the computational cost of the hidden Markov model and particle Gibbs with ancestor sampling algorithm. We demonstrate the efficiency of this proposed approach in several traditionally challenging examples, focusing on state-space models with regime switching.