Improving the Convergence Properties of the Data Augmentation Algorithm with an Application to Bayesian Mixture Modeling

TL;DR

This paper analyzes the spectral properties of data augmentation algorithms, demonstrating that under certain conditions their convergence can be characterized and improved, with a focus on Bayesian mixture models.

Contribution

It proves that the spectra of DA and sandwich algorithms are finite and ordered, providing insights into their convergence properties, especially for finite augmentation spaces.

Findings

Spectra of DA and sandwich operators are finite and lie in [0,1).

Sandwich operator spectrum dominates DA spectrum, indicating faster convergence.

Application to Bayesian mixture models shows practical implications for algorithm efficiency.

Abstract

The reversible Markov chains that drive the data augmentation (DA) and sandwich algorithms define self-adjoint operators whose spectra encode the convergence properties of the algorithms. When the target distribution has uncountable support, as is nearly always the case in practice, it is generally quite difficult to get a handle on these spectra. We show that, if the augmentation space is finite, then (under regularity conditions) the operators defined by the DA and sandwich chains are compact, and the spectra are finite subsets of $[0,1)$. Moreover, we prove that the spectrum of the sandwich operator dominates the spectrum of the DA operator in the sense that the ordered elements of the former are all less than or equal to the corresponding elements of the latter. As a concrete example, we study a widely used DA algorithm for the exploration of posterior densities associated with Bayesian mixture models [J. Roy. Statist. Soc. Ser. B 56 (1994) 363--375]. In particular, we compare this mixture DA algorithm with an alternative algorithm proposed by Fr\"{u}hwirth-Schnatter [J. Amer. Statist. Assoc. 96 (2001) 194--209] that is based on random label switching.