| 要約: | This thesis provides a set of novel Monte Carlo methods to perform Bayesian inference, with an emphasis on a state-space modelling framework. The thesis comprises three self-contained research papers. The first paper works at the interface of modelling and inference, extending existing work to develop a new model for medium and high-dimensional data possessing step-changes changes in the correlation structure. A bespoke inference routine is developed, with experimental results demonstrating the speed-up of convergence to equilibrium under a novel alteration of a particle Gibbs sampler for change-point detection. The second paper is concerned with the problem of obtaining unbiased estimators for the smoothing problem in state-space models. New methodology is developed leveraging recent developments in unbiased Markov chain Monte Carlo (MCMC) in conjunction with a particle independent Metropolis–Hastings Markov kernel. We show that the resulting coupling time can be precisely characterised. As a result, it is possible to study the large sample limit of the coupling time to help guide the choice in an algorithmic tuning parameter (the number of particles in this case). The third and final paper develops this methodology further, to obtain unbiased estimators for MCMC algorithms in settings where the likelihood may be unavailable to evaluate pointwise. The motivation for this work is to find a simple means by which to parallelise exact Bayesian inference in state-space models. The focus is primarily on pseudo-marginal algorithms and particle MCMC, in which the mixing behaviour of the algorithm is typically polynomially ergodic. In this setting we propose new methodology to allow the obtention of unbiased estimators of expectations of stationary distributions of these Markov chains. In addition, we extend existing results to both prove the validity and characterise the computational efficiency of these estimators, in terms of algorithmic tuning parameters. We evaluate the resulting methodology on otherwise intractable state-space and latent variable models, including a multivariate probit model popular in econometrics and a nonlinear state-space model arising in neuroscience. Additional algorithmic extensions are provided and subsequently applied to a parameter inference problem in a high-dimensional Ising model.
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