Bayesian Approaches to the Design of Markov Chain Monte Carlo Samplers

Abstract

In the decades since Markov chain Monte Carlo methods were first introduced, they have revolutionised Bayesian approaches to statistical inference. Each new advance in MCMC methodology produces near immediate benefits for Bayesian practitioners, expanding the range of problems they can feasibly solve. In this paper, we explore ways in which Bayesian approaches can return something of the debt owed to MCMC, by using explicitly Bayesian concepts to aid in the design of MCMC samplers. The art of efficient MCMC sampling lies in designing a Markov process that (a) has the required limiting distribution, (b) has good convergence and mixing properties and (c) can be implemented in a computationally efficient manner. In this paper, we explore the idea that the selection of an appropriate process, and in particular the tuning of the parameters of the process to achieve the above goals, can be regarded as a problem of estimation. As such, it is amenable to a conventional Bayesian approach, in which a prior distribution for optimal parameters of the sampler is specified, data relevant to sampler performance is obtained and a posterior distribution for optimal parameters is formed. Sampling from this posterior distribution can then be incorporated into the MCMC sampler to produce an adaptive method. We present a new MCMC algorithm for Bayesian adaptive Metropolis-Hasting sampling (BAMS), using an explicitly Bayesian inference to update the proposal distribution. We show that author Keith’s earlier Bayesian adaptive independence sampler (BAIS) and a new Bayesian adaptive random walk sampler (BARS) emerge as instances. More important than either of these instances, BAMS provides a general framework within which to explore adaptive schemes that are guaranteed to converge to the required limiting distribution.