Marginal Markov Chain Monte Carlo Methods

Marginal Data Augmentation and Parameter-Expanded Data Augmentation are related methods for improving the the convergence properties of the two-step Gibbs sampler know as the Data Augmentation sampler. These methods expand the parameter space with a so-callled working parameter that is unidentifiable given the observed data but is identifiable given the so-called augmented data. Although these methods can result in enormous computational gains, their use has been somewhat limited do the constrained framework they are constructed under and the necessary identification of a working parameter. This article proposes a new prescriptive framework that greatly expands the class of problems that can benefit from the key idea underlying these methods. In particular, we show how working parameters can automatically be introduced into any Gibbs sampler. Since these samplers are typically used in a Bayesian framework, the working parameter requires a prior distribution and the convergence properties of the Markov chain depend on the choice of this choice distribution. Under certain conditions the optimal choice is improper and results in a joint Markov chain on the expanded parameter space that is not positive recurrent. This leads to unexplored technical difficulties when one attempts to exploit the computational advantage in multi-step mcmc samplers, the very chains that might benefit most from this technology. In this article we develop strategies and theory that allow optimal marginal methods to be used in multistep samplers. We illustrate the potential to dramatically improve the convergence properties of mcmc samplers by applying the marginal Gibbs sampler to a logistic mixed model. 1 Expanding State Spaces in MCMC Constructing a Markov chain on an expanded state space in the context of Monte Carlo sampling can greatly simplify the required component draws or lead to chains with better mixing properties. Professor van Dyk’s research was supported in part by NSF grants DMS-04-06085 and SES-05-50980.