Title: Confident Bayesian phylogenetic inference.
Speaker: Dr. Luke Kelly, School of Mathematical Sciences, U.C.C.
Title: Confident Bayesian phylogenetic inference.
Phyogenetic inference aims to reconstruct the phylogeny of species descended from a common ancestor. The phylogeny is typically a graph depicting evolutionary relationships, typically a binary tree with the observed taxa at the tips and edge lengths representing elapsed time. Although an intractable statistical problem on a general state space with a likelihood that may only be evaluated numerically, we can take a Bayesian approach and construct a Markov chain to explore the posterior distribution on the space of tree topologies, branch lengths and components of the evolutionary model. In performing Bayesian phylogenetic inference, we would like to ensure that our Markov chains are mixing over the entire posterior distribution, and that our models are consistent as the amount of data increases.
In this talk, I will discuss two recent strands of work. The first describes lagged couplings of Markov chains exploring phylogenetic posteriors. We demonstrate how to successfully couple standard Markov chain Monte Carlo transition kernels for phylogenetic trees and models so that coupled chains meet exactly after a random finite number of iterations. By coupling chains at a lag, we are able to estimate a bound on the total variation distance to stationarity which decays to 0 and be confident that our Markov chains are mixing. In the second part of the talk, I will discuss results which establish posterior consistency as the number of taxa and length of data sequences increase under two widely used prior distributions. The convergence rates match known frequentist rates obtained under stronger assumptions.