Some contributions to particle Markov chain Monte Carlo algorithms

Hidden Markov models (HMMs) (Cappe et al., 2005) and discrete time stopped Markov processes (Del Moral, 2004, Section 2.2.3) are used to model phenomena in a wide range of fields. However, as practitioners develop more intricate models, analytical Bayesian inference becomes very difficult. In light of this issue, this work focuses on sampling from the posteriors of HMMs and stopped Markov processes using sequential Monte Carlo (SMC) (Doucet et al. 2008, Doucet et al. 2001, Gordon et al. 1993) and, more importantly, particle Markov chain Monte Carlo (PMCMC) (Andrieu et al., 2010). The thesis consists of three major contributions, which enhance the performance of PMCMC. The first work focuses on HMMs, and it begins by introducing a new SMC smoothing (Briers et al. 2010, Fearnhead et al. 2010) estimate of the HMM's normalising constant; we prove the estimate's unbiasedness and a central limit theorem. We use this estimate to develop new PMCMC algorithms that, under certain algorithmic settings, require less computational time than the algorithms of Andrieu et al. (2010). Our new estimate also leads to the discovery of an optimal setting for the smoothers of Briers et al. (2010) and Fearnhead et al. (2010). As this setting is not available for the general class of HMMs, we develop three algorithms for approximating it. The second major work builds from Jasra et al. (2013) and Whiteley et al. (2012) to develop new SMC and PMCMC algorithms that draw from HMMs whose observations have intractable density functions. While these types of algorithms have appeared before (see Jasra et al. 2013, Jasra et al. 2012, and Martin et al. 2012), this work uses twisted proposals as in Whiteley et al. (2012) to reduce the variance of SMC estimates of the normalising constant to improve the convergence of PMCMC in some scenarios. Finally, the third project is concerned with inferring the unknown parameters of stopped Markov processes that are only observed upon reaching their terminal sets. Bayesian inference has not been attempted on this class of problems before. The parameters are inferred through two new adaptive and non-adaptive PMCMC algorithms.Open Acces