Elementary bounds on mixing times for decomposable Markov chains

Many finite-state reversible Markov chains can be naturally decomposed into “projection” and “restriction” chains. In this paper we provide bounds on the total variation mixing times of the original chain in terms of the mixing properties of these related chains. This paper is in the tradition of existing bounds on Poincaré and log-Sobolev constants of Markov chains in terms of similar decompositions (Jerrum et al., 2004; Madras and Randall, 2002; Martin and Randall, 2006; Madras and Yuen, 2009). Our proofs are simple, relying largely on recent results relating hitting and mixing times of reversible Markov chains (Peres and Sousi, 2013; Oliveira, 2012). We describe situations in which our results give substantially better bounds than those obtained by applying existing decomposition results and provide examples for illustration.