Perturbation bounds for Monte Carlo within Metropolis via restricted approximations

The Monte Carlo within Metropolis (MCwM) algorithm, interpreted as a perturbed Metropolis-Hastings (MH) algorithm, provides an approach for approximate sampling when the target distribution is intractable. Assuming the unperturbed Markov chain is geometrically ergodic, we show explicit estimates of the difference between the n th step distributions of the perturbed MCwM and the unperturbed MH chains. These bounds are based on novel perturbation results for Markov chains which are of interest beyond the MCwM setting. To apply the bounds, we need to control the difference between the transition probabilities of the two chains and to verify stability of the perturbed chain.

Sequential Monte Carlo with transformations

This paper examines methodology for performing Bayesian inference sequentially on a sequence of posteriors on spaces of different dimensions. For this, we use sequential Monte Carlo samplers, introducing the innovation of using deterministic transformations to move particles effectively between target distributions with different dimensions. This approach, combined with adaptive methods, yields an extremely flexible and general algorithm for Bayesian model comparison that is suitable for use in applications where the acceptance rate in reversible jump Markov chain Monte Carlo is low. We use this approach on model comparison for mixture models, and for inferring coalescent trees sequentially, as data arrives.