Geometric ergodicity of a Metropolis-Hastings algorithm for Bayesian inference of phylogenetic branch lengths
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Geometric ergodicity of a Metropolis-Hastings algorithm for Bayesian inference of phylogenetic branch lengths David A. Spade1 Received: 11 June 2019 / Accepted: 15 February 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This manuscript extends the work of Spade et al. (Math Biosci 268:9–21, 2015) to an examination of a fully-updating version of a Metropolis-Hastings algorithm for inference of phylogenetic branch lengths. This approach serves as an intermediary between theoretical assessment of Markov chain convergence, which in phylogenetic settings is typically difficult to do analytically, and output-based convergence diagnostics, which suffer from several of their own limitations. In this manuscript, we will also examine the performance of the convergence assessment techniques for this Markov chain and the convergence behavior of this type of Markov chain compared to the one-at-a-time updating scheme investigated in Spade et al. (Math Biosci 268:9–21, 2015). We will also vary the choices of the drift function in order to obtain a sense of how the choice of the drift function affects the estimated bound on the chain’s mixing time. Keywords Statistical Phylogenetics · Mixing time · Markov chain Monte Carlo · Bayesian methods
1 Introduction In phylogenetics, a common goal is to make inferences about the evolutionary pattern among a group of individuals, such as genes or species, given a set of genetic data pertaining to these individuals. These individuals are known as taxa. Genetic material may come in many forms, including DNA sequences, amino acid sequences, or protein sequences, but this work assumes that the genetic material that is available on each taxon is in the form of a DNA sequence. A common way that evolutionary biologists represent the evolutionary structure among a group of taxa is through a phylogenetic
This work has been supported in part by National Science Foundation Grant NSF-DMS-1228244.
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David A. Spade [email protected] University of Wisconsin–Milwaukee, 3200 North Cramer Street, Milwaukee, WI 53201, USA
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tree. A phylogenetic tree is a branching diagram that may be viewed as an acyclic graph with n external vertices, known as leaves or tips, where each leaf represents one taxon. The tree may be rooted or unrooted, and if the tree is rooted, then it has one internal node of degree 2 that represents the most recent common ancestor among the taxa. A rooted tree may be assumed to satisfy a molecular clock, where the evolutionary time that elapses between each leaf and the root is equal. This work focuses on rooted trees for which a molecular clock is not assumed. A phylogenetic tree consists of two parts, both of which must be estimated. The first of these is the tree topology, which contains information about the branching structure of the evolutionary history among the taxa. The branch lengths contain information about the expected number of substitutions per unit time at stationarity. In the literature, many approaches to phylogenetic tree esti
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