Investigating Convergence of Markov Chain Monte Carlo Methods for Bayesian Phylogenetic Inference

In biology, it is commonly of interest to investigate the evolutionary pattern that gave rise to an existing group of individuals, such as species or genes. This pattern is most often represented pictorially by a phylogenetic tree. Many methods of inferring evolutionary patterns have been proposed, but as advances in computational capabilities have made Bayesian inference more approachable, it has become an increasingly popular technique for phylogenetic inference. In Bayesian inference, it is often the case that the posterior density cannot be written out in its entirety due to the intractability of the normalizing constant. One way of working around this is to use a Markov chain Monte Carlo (MCMC) method. The idea is that after several (possibly many) iterations, the chain has approximately converged to its stationary distribution, namely, the posterior distribution. After these initial iterations, subsequent steps of the chain represent an approximate sample from the posterior distribution, thus enabling Bayesian inference. The biggest question one faces when using MCMC methods is the question of how long the chain should be run before sampling can begin, i.e., the mixing time of the chain. Many methods exist that aim to answer this question by