Metropolis-Hastings within Partially Collapsed Gibbs Samplers

TL;DR

This paper discusses the integration of Metropolis-Hastings updates within Partially Collapsed Gibbs samplers, highlighting challenges and proposing strategies to preserve the correct stationary distribution for improved convergence.

Contribution

It provides a general strategy for correctly incorporating MH updates into PCG samplers, ensuring the stationary distribution is maintained, along with theoretical guidance and practical examples.

Findings

Identifies issues with stationary distribution when using MH in PCG

Develops a strategy to preserve the target distribution with MH updates

Demonstrates computational advantages through examples

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.