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An adjacent-swap Markov chain on coalescent trees

Part of: Markov processes

Published online by Cambridge University Press:  02 September 2022

Mackenzie Simper Open the ORCID record for Mackenzie Simper [Opens in a new window]  and
Julia A. Palacios
Mackenzie Simper*
Affiliation:
Stanford University
Julia A. Palacios*
Affiliation:
Stanford University
*
*Postal address: Department of Mathematics, Building 380, 450 Jane Stanford Way, Stanford, CA 94305, USA. Email address: msimper@stanford.edu
**Postal address: Department of Statistics and Department of Biomedical Data Science, Sequoia Hall, 390 Jane Stanford Way, Stanford, CA 94305, USA. Email address: juliapr@stanford.edu
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Abstract

The standard coalescent is widely used in evolutionary biology and population genetics to model the ancestral history of a sample of molecular sequences as a rooted and ranked binary tree. In this paper we present a representation of the space of ranked trees as a space of constrained ordered matched pairs. We use this representation to define ergodic Markov chains on labeled and unlabeled ranked tree shapes analogously to transposition chains on the space of permutations. We show that an adjacent-swap chain on labeled and unlabeled ranked tree shapes has a mixing time at least of order $n^3$ , and at most of order $n^{4}$ . Bayesian inference methods rely on Markov chain Monte Carlo methods on the space of trees. Thus it is important to define good Markov chains which are easy to simulate and for which rates of convergence can be studied.

Keywords

Random transpositionslumped Markov chainTajima distribution

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 4 , December 2022 , pp. 1243 - 1260
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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