  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.

Methodology;Computation