A spectral analytic comparison of trace-class data augmentation algorithms and their sandwich variants

Abstract

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 $\Vert {K^*}\Vert \le \Vert {K}\Vert$, where $K$ and $K^*$ are the Markov operators associated with the DA and sandwich algorithms, respectively, and $\Vert\cdot\Vert$ 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].