A SPECTRAL ANALYTIC COMPARISON OF TRACE-CLASS DATA AUGMENTATION ALGORITHMS AND THEIR SANDWICH VARIANTS

• 成果类型: Article
• 署名作者: Khare, Kshitij; Hobert, James P.
• 署名单位: State University System of Florida; University of Florida
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/11-AOS916
• 发表日期: 2011
• 页码: 2585-2606
• 关键词: interweaving strategy asis; boosting mcmc efficiency; GIBBS SAMPLER; marginal augmentation; covariance structure; monte-carlo; schemes

摘要

The data augmentation (DA) algorithm is a widely used Markov chain Monte Carlo algorithm that is easy to implement but often suffers from slow convergence. The sandwich algorithm is an alternative that can converge much faster while requiring roughly the same computational effort per iteration. Theoretically, the sandwich algorithm always converges at least as fast as the corresponding DA algorithm in the sense that parallel to K*parallel to <= parallel to K parallel to, where K and K* are the Markov operators associated with the DA and sandwich algorithms, respectively, and parallel to . parallel to denotes operator norm. In this paper, a substantial refinement of this operator norm inequality is developed. In particular, under regularity conditions implying that K is a trace-class operator, it is shown that K* is also a positive, trace-class operator, and that the spectrum of K* dominates that of K in the sense that the ordered elements of the former are all less than or equal to the corresponding elements of the latter. Furthermore, if the sandwich algorithm is constructed using a group action, as described by Liu and Wu [J. Amer Statist. Assoc. 94 (1999) 1264-1274] and Hobert and Marchev [Ann. Statist. 36 (2008) 532-554], then there is strict inequality between at least one pair of eigenvalues. These results are applied to a new DA algorithm for Bayesian quantile regression introduced by Kozumi and Kobayashi [J. Stat. Comput. Simul. 81 (2011) 1565-1578].