A Comparison Theorem for Data Augmentation Algorithms with Applications

The data augmentation (DA) algorithm is considered a useful Markov chain Monte Carlo algorithm that sometimes suffers from slow convergence. It is often possible to convert a DA algorithm into a sandwich algorithm that is computationally equivalent to the DA algorithm, but converges much faster. Theoretically, the reversible Markov chain that drives the sandwich algorithm is at least as good as the corresponding DA chain in terms of performance in the central limit theorem and in the operator norm sense. In this paper, we use the sandwich machinery to compare two DA algorithms. In particular, we provide conditions under which one DA chain can be represented as a sandwich version of the other. Our results are used to extend Hobert and Marchev’s (2008) results on the Haar PX-DA algorithm and to improve the collapsing theorem of Liu, Wong and Kong (1994) and Liu (1994).