Uniformization for sampling realizations of Markov processes: applications to Bayesian implementations of codon substitution models

Abstract

We describe a general method, based on a uniformization technique, which can be utilized to generate realizations of a Markovian substitution process conditional on an alignment of character states and a given tree topology. The method is applicable under a wide range of evolutionary models, and to illustrate its usefulness in practice, we embed it within a data augmentation-based Markov chain Monte Carlo sampler, for approximating posterior distributions under previously proposed codon substitution models. The sampler is found to be more efficient than the conventional pruning-based sampler with the decorrelation times between draws from the posterior reduced by a factor of 20 or more.