Local degeneracy of Markov chain Monte Carlo methods

TL;DR

This paper investigates the asymptotic behavior of Monte Carlo methods, focusing on local degeneracy, and demonstrates how marginal augmentation can improve local properties in Gibbs samplers, especially for models with fewer categories.

Contribution

It introduces the concept of local degeneracy in Monte Carlo methods, provides equivalent conditions for it, and analyzes the impact of marginal augmentation on Gibbs samplers for cumulative logit models.

Findings

Natural Gibbs sampler often fails to be locally consistent.

Marginal augmentation improves asymptotic properties for small category counts.

Both methods lack local consistency when the number of categories is large.

Abstract

We study asymptotic behavior of Monte Carlo method. Local consistency is one of an ideal property of Monte Carlo method. However, it may fail to hold local consistency for several reason. In fact, in practice, it is more important to study such a non-ideal behavior. We call local degeneracy for one of a non-ideal behavior of Monte Carlo methods. We show some equivalent conditions for local degeneracy. As an application we study a Gibbs sampler (data augmentation) for cumulative logit model with or without marginal augmentation. It is well known that natural Gibbs sampler does not work well for this model. In a sense of local consistency and degeneracy, marginal augmentation is shown to improve the asymptotic property. However, when the number of categories is large, both methods are not locally consistent.