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This paper studies a Markov chain for phylogenetic reconstruction which usesa popular transition between tree topologies known as subtreepruning-and-regrafting (SPR). We analyze the Markov chain in the simplersetting that the generating tree consists of very short edge lengths, shortenough so that each sample from the generating tree (or character inphylogenetic terminology) is likely to have only one mutation, and that thereenough samples so that the data looks like the generating distribution. Weprove in this setting that the Markov chain is rapidly mixing, i.e., it quicklyconverges to its stationary distribution, which is the posterior distributionover tree topologies. Our proofs use that the leading term of the maximumlikelihood function of a tree T is the maximum parsimony score, which is thesize of the minimum cut in T needed to realize single edge cuts of thegenerating tree. Our main contribution is a combinatorial proof that in oursimplified setting, SPR moves are guaranteed to converge quickly to the maximumparsimony tree. Our results are in contrast to recent works showing exampleswith heterogeneous data (namely, the data is generated from a mixturedistribution) where many natural Markov chains are exponentially slow toconverge to the stationary distribution.

Fast Convergence of MCMC Algorithms for Phylogenetic Reconstruction with Homogeneous Data on Closely Related Species