Geometric ergodicity of a hybrid sampler for Bayesian inference of phylogenetic branch lengths

One of the fundamental goals in phylogenetics is to make inferences about the evolutionary pattern among a group of individuals, such as genes or species, using present-day genetic material. This pattern is represented by a phylogenetic tree, and as computational methods have caught up to the statistical theory, Bayesian methods of making inferences about phylogenetic trees have become increasingly popular. Bayesian inference of phylogenetic trees requires sampling from intractable probability distributions. Common methods of sampling from these distributions include Markov chain Monte Carlo (MCMC) and Sequential Monte Carlo (SMC) methods, and one way that both of these methods can proceed is by first simulating a tree topology and then taking a sample from the posterior distribution of the branch lengths given the tree topology and the data set. In many MCMC methods, it is difficult to verify that the underlying Markov chain is geometrically ergodic, and thus, it is necessary to rely on output-based convergence diagnostics in order to assess convergence on an ad hoc basis. These diagnostics suffer from several important limitations, so in an effort to circumvent these limitations, this work establishes geometric convergence for a particular Markov chain that is used to sample branch lengths under a fairly general class of nucleotide substitution models and provides a numerical method for estimating the time this Markov chain takes to converge.