Improving the convergence properties of the data augmentation algorithm with an application to Bayesian mixture modeling. (English) Zbl 1246.60095

Hobert, James P.; Roy, Vivekananda; Robert, Christian P.

Stat. Sci. 26, No. 3, 332-351 (2011).

Summary: 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. Diebolt and C. P. Robert [J. R. Stat. Soc., Ser. B, Stat. Methodol. 56, No. 2, 363–375 (1994; Zbl 0796.62028)]. In particular, we compare this mixture DA algorithm with an alternative algorithm proposed by S. Frühwirth-Schnatter [J. Am. Stat. Assoc. 96, No. 453, 194–209 (2001; Zbl 1015.62022)] that is based on random label switching.

MSC:

60J22	Computational methods in Markov chains
62F15	Bayesian inference
65C05	Monte Carlo methods

Keywords:

compact operator; convergence rate; eigenvalue; label switching; Markov operator; Monte Carlo; operator norm; positive operator; reversible Markov chain; sandwich algorithm; spectrum

Citations:

Zbl 0796.62028; Zbl 1015.62022

References:

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