Statistics – Methodology
Scientific paper
2009-11-24
Statistical Science 2011, Vol. 26, No. 3, 332-351
Statistics
Methodology
Published in at http://dx.doi.org/10.1214/11-STS365 the Statistical Science (http://www.imstat.org/sts/) by the Institute of M
Scientific paper
10.1214/11-STS365
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 Fr\"{u}hwirth-Schnatter [J. Amer. Statist. Assoc. 96 (2001) 194--209] that is based on random label switching.
Hobert James P.
Robert Christian P.
Roy Vivekananda
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