Improving the Convergence Properties of the Data Augmentation Algorithm with an Application to Bayesian Mixture Modelling

Every reversible Markov chain defines an operator whose spectrum encodes the convergenceproperties of the chain. When the state space is finite, the spectrum is just the set ofeigenvalues of the corresponding Markov transition matrix. However, when the state space isinfinite, the spectrum may be uncountable, and is nearly always impossible to calculate. In mostapplications of the data augmentation (DA) algorithm, the state space of the DA Markov chainis infinite. However, we show that, under regularity conditions that include the finiteness of theaugmented space, the operators defined by the DA chain and Hobert and Marchev’s (2008) alternativechain are both compact, and the corresponding spectra are both finite subsets of [0; 1).Moreover, we prove that the spectrum of Hobert and Marchev’s (2008) chain dominates thespectrum of the DA chain in the sense that the ordered elements of the former are all less thanor equal to the corresponding elements of the latter. As a concrete example, we study a widelyused DA algorithm for the exploration of posterior densities associated with Bayesian mixturemodels (Diebolt and Robert, 1994). In particular, we compare this mixture DA algorithm withan alternative algorithm proposed by Fr¨uhwirth-Schnatter (2001) that is based on random labelswitching.