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

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 Frühwirth-Schnatter [J. Amer. Statist. Assoc. 96 (2001) 194--209] that is based on random label switching.

Published in at http://dx.doi.org/10.1214/11-STS365 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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Cited by in corpus (7)

• A spectral analytic comparison of trace-class data augmentation algorithms and their sandwich variants
• Improving the Convergence Properties of the Data Augmentation Algorithm with an Application to Bayesian Mixture Modeling
• BAMBI: An R package for Fitting Bivariate Angular Mixture Models
• Consistent estimation of the spectrum of trace class data augmentation algorithms
• Trace-class Monte Carlo Markov Chains for Bayesian Multivariate Linear Regression with Non-Gaussian Errors
• A Bayesian Approach for Analyzing Data on the Stiefel Manifold
• Fast Markov chain Monte Carlo for high dimensional Bayesian regression models with shrinkage priors