19 Dubious Ways to Compute the Marginal Likelihood of a Phylogenetic Tree Topology

Mathieu Fourment,Chris Whidden,Arman Bilge,Frederick A Matsen,Andrew F Magee,Vladimir N Minin,David Posada

doi:10.1093/sysbio/syz046

Abstract

The marginal likelihood of a model is a key quantity for assessing the evidence provided by the data in support of a model. The marginal likelihood is the normalizing constant for the posterior density, obtained by integrating the product of the likelihood and the prior with respect to model parameters. Thus, the computational burden of computing the marginal likelihood scales with the dimension of the parameter space. In phylogenetics, where we work with tree topologies that are high-dimensional models, standard approaches to computing marginal likelihoods are very slow. Here, we study methods to quickly compute the marginal likelihood of a single fixed tree topology. We benchmark the speed and accuracy of 19 different methods to compute the marginal likelihood of phylogenetic topologies on a suite of real data sets under the JC69 model. These methods include several new ones that we develop explicitly to solve this problem, as well as existing algorithms that we apply to phylogenetic models for the first time. Altogether, our results show that the accuracy of these methods varies widely, and that accuracy does not necessarily correlate with computational burden. Our newly developed methods are orders of magnitude faster than standard approaches, and in some cases, their accuracy rivals the best established estimators.

Highlights

In phylogenetic inference, the tree topology forms a key object of inference
Markov chain Monte Carlo (MCMC) over topologies is computationally expensive [Lakner et al, 2008, Hohna et al, 2008]. These MCMC algorithms spend a nontrivial amount of time marginalizing over branch lengths and substitution models parameters and discarding them so that the estimated posterior probability of a tree topology is the proportion of MCMC iterations in which it appears
We review existing methods and develop new ones to compute the posterior probabilities of tree topologies by quickly marginalizing out branch lengths to compute the marginal likelihood of a given topology

Summary

Introduction

In Bayesian phylogenetics, this translates to approximating the posterior distribution of tree topologies. A joint posterior distribution of tree topologies and continuous parameters, including branch lengths and substitution model parameters, is approximated directly via Markov chain Monte Carlo (MCMC), as done in the popular Bayesian phylogenetics software MrBayes [Ronquist et al, 2012]. MCMC over topologies is computationally expensive [Lakner et al, 2008, Hohna et al, 2008] These MCMC algorithms spend a nontrivial amount of time marginalizing over branch lengths and substitution models parameters and discarding them so that the estimated posterior probability of a tree topology is the proportion of MCMC iterations in which it appears. We compare speed and accuracy of 19 methods and examine whether there is a speed-accuracy trade off

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Discussion

Conclusion