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

作     者：Hobert, James P. Roy, Vivekananda Robert, Christian P. 

作者机构：Univ Florida Dept Stat Gainesville FL 32611 USA Iowa State Univ Dept Stat Ames IA 50011 USA Univ Paris 09 CEREMADE F-75775 Paris 16 France CREST F-92245 Malakoff France 

出 版 物：《STATISTICAL SCIENCE》 (统计科学)

年 卷 期：2011年第26卷第3期

页      面：332-351页

核心收录：

学科分类：0202[经济学-应用经济学] 02[经济学] 020208[经济学-统计学] 07[理学] 0714[理学-统计学（可授理学、经济学学位）] 

基　　金：NSF [DMS-08-05860] Agence Nationale de la Recherche (ANR) [212, rue de Bercy 75012 Paris, ANR-08-BLAN-0218 Big'MC] Universite Paris Dauphine Direct For Mathematical & Physical Scien Division Of Mathematical Sciences Funding Source: National Science Foundation 

主　　题：Compact operator convergence rate eigenvalue label switching Markov operator Monte Carlo operator norm positive operator reversible Markov chain sandwich algorithm spectrum 

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