Work

Hastings Ratio of the LOCAL Proposal Used in Bayesian Phylogenetics

Year: 2005

Type: article

Abstract: As part of another study, we estimated the marginal likelihoods of trees using different proposal algorithms and discovered repeatable discrepancies that implied that the published Hastings ratio for a proposal mechanism used in many Bayesian phylogenetic analyses is incorrect. In this article, we derive the correct Hastings ratio for the (Larget and Simon, 1999) “LOCAL move without a molecular clock.” The derivation illustrates how a recently described method for determining the acceptance probabilities for proposals in Markov chain Monte Carlo (Green, 2003) provides an intuitive method for calculating Hastings ratios. Although the use of the previously reported Hastings ratio could result in a bias toward shorter branch lengths, the effect is very minor and is overwhelmed by the information contained within even small data sets. Markov chain Monte Carlo (MCMC) methods are widely used to explore posterior probability densities by simulating a walk through tree/model space (Simon and Larget, 2001; Huelsenbeck and Ronquist, 2001). Many MCMC simulations employ the Metropolis-Hastings algorithm, which uses a stochastic function to propose a new state, x′, for the chain based upon the current state, x. The “state of the chain” refers to the values of all of the parameters in the model (including branch lengths for the tree). Let q(x, dx′) denote the probability density of proposing a move from x→x. The Metropolis algorithm (Metropolis et al., 1953) is limited to simulation schemes in which q(x, dx′) = q(x′, dx). Hastings (1970) significantly eased the task of implementating MCMC methods by modifying the Metropolis algorithm to allow for the use of asymmetric proposal densities. If one is sampling the posterior density (which is proportional to the product of the likelihood, , and the prior probability density, p), then the probability of accepting a proposal α(x, x′) in the Metropolis-Hastings algorithm is: ...

Source: Systematic Biology

DOI: 10.1080/10635150500354670

Authors: Mark T. Holder, Paul O. Lewis, David L. Swofford, Bret Larget

Institutions: Florida State University, University of Connecticut, University of Wisconsin–Madison

Language: English

Topic: Markov Chains and Monte Carlo Methods

Subfield: Statistics and Probability

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Institutes of Health, National Institute of General Medical Sciences

Awards: G2817901981, Improving Bayesian Phylogeny, G5185559849, Extending Bayesian Phylogenetic Analysis