  Phylogenetic inference is an intractable statistical problem on a complex
space. Markov chain Monte Carlo methods are the primary tool for Bayesian
phylogenetic inference but it is challenging to construct efficient schemes to
explore the associated posterior distribution or assess their performance.
Existing approaches are unable to diagnose mixing or convergence of Markov
schemes jointly across all components of a phylogenetic model. Lagged couplings
of Markov chain Monte Carlo algorithms have recently been developed on simpler
spaces to diagnose convergence and construct unbiased estimators. We describe a
contractive coupling of Markov chains targeting a posterior distribution over a
space of phylogenetic trees with branch lengths, scalar parameters and latent
variables. We use these couplings to assess mixing and convergence of Markov
chains jointly across all components of the phylogenetic model on trees with up
to 200 leaves. Samples from our coupled chains may also be used to construct
unbiased estimators.
