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Licensed Unlicensed Requires Authentication Published by De Gruyter March 26, 2013

Monte Carlo estimation of total variation distance of Markov chains on large spaces, with application to phylogenetics

Radu Herbei and Laura Kubatko EMAIL logo


Markov chains are widely used for modeling in many areas of molecular biology and genetics. As the complexity of such models advances, it becomes increasingly important to assess the rate at which a Markov chain converges to its stationary distribution in order to carry out accurate inference. A common measure of convergence to the stationary distribution is the total variation distance, but this measure can be difficult to compute when the state space of the chain is large. We propose a Monte Carlo method to estimate the total variation distance that can be applied in this situation, and we demonstrate how the method can be efficiently implemented by taking advantage of GPU computing techniques. We apply the method to two Markov chains on the space of phylogenetic trees, and discuss the implications of our findings for the development of algorithms for phylogenetic inference.

Corresponding author: Laura Kubatko, The Ohio State University – Statistics, 404 Cockins Hall, 1958, Neil Avenue, Columbus, OH 43210, USA, Phone: +1-614-247-8846, Fax: +1-614-292-2096

We acknowledge computing support from the Ohio Supercomputer Center (

Conflict of interest statement

Funding: The first author is supported in part by the National Science Foundation award DMS-1209142. The second author is supported in part by the National Science Foundation award DMS-1106706.


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Published Online: 2013-03-26

©2013 by Walter de Gruyter Berlin Boston

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