  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 Poincare and log-Sobolev constants of Markov chains in terms of
similar decompositions [JSTV04, MR02, MR06, MY09]. Our proofs are simple,
relying largely on recent results relating hitting and mixing times of
reversible Markov chains [PS13, Oli12]. We describe situations in which our
results give substantially better bounds than those obtained by applying
existing decomposition results and provide examples for illustration.
