Metropolis-Hastings Within Partially Collapsed Gibbs Samplers

David A. Van Dyk,Xiyun Jiao

doi:10.1080/10618600.2014.930041

Abstract

The partially collapsed Gibbs (PCG) sampler offers a new strategy for improving the convergence of a Gibbs sampler. PCG achieves faster convergence by reducing the conditioning in some of the draws of its parent Gibbs sampler. Although this can significantly improve convergence, care must be taken to ensure that the stationary distribution is preserved. The conditional distributions sampled in a PCG sampler may be incompatible and permuting their order may upset the stationary distribution of the chain. Extra care must be taken when Metropolis-Hastings (MH) updates are used in some or all of the updates. Reducing the conditioning in an MH within Gibbs sampler can change the stationary distribution, even when the PCG sampler would work perfectly if MH were not used. In fact, a number of samplers of this sort that have been advocated in the literature do not actually have the target stationary distributions. In this article, we illustrate the challenges that may arise when using MH within a PCG sampler and develop a general strategy for using such updates while maintaining the desired stationary distribution. Theoretical arguments provide guidance when choosing between different MH within PCG sampling schemes. Finally, we illustrate the MH within PCG sampler and its computational advantage using several examples from our applied work.

Highlights

The popularity of the Gibbs sampler stems from its simplicity and power to effectively generate samples from a high-dimensional probability distribution
The Partially Collapsed Gibbs (PCG) sampler offers a strategy for improving the convergence characteristics of a Gibbs sampler
Because reducing the conditioning can significantly improve the rate of convergence of the sampler, while permutation typically has a minor effect, and trimming has no effect on the rate of convergence, we generally expect the PCG sampler to exhibit better and often much better convergence properties than its parent Gibbs sampler

Summary

Introduction

The popularity of the Gibbs sampler stems from its simplicity and power to effectively generate samples from a high-dimensional probability distribution. Unlike the ordinary Gibbs sampler, the conditional distributions sampled in a PCG sampler may be incompatible, meaning there is no joint distribution of which they are simultaneously the conditional distributions In this case, permuting the order of the updates can change the stationary distribution of the chain. Woodard et al (2012), for example, points out this problem in certain samplers described in the literature for regression with functional predictors They do not use the framework of PCG, these samplers are simple special cases of improper MH within PCG samplers. We illustrate difficulties that may arise when using MH updates within a PCG sampler and develop a general strategy for using such updates while maintaining the target stationary distribution.

Notation

Spectral analysis in X-ray astronomy

A common error in the simplest PCG sampler

Convergence of the Partially Collapsed Gibbs sampler

Identifying the stationary distributions

Using MH following a reduced step

To block or not to block

Comparing the iterated and joint MH strategies

Comparing the samplers with and without blocking

The simplest MH within PCG sampler

Spectral analysis with narrow lines in high-energy astrophysics

Relating ECME with Newton-type updates to MH within PCG samplers