Properties of Markov Chain Monte Carlo Performance across Many Empirical Alignments.

Nearly all current Bayesian phylogenetic applications rely on Markov chain Monte Carlo (MCMC) methods to approximate the posterior distribution for trees and other parameters of the model. These approximations are only reliable if Markov chains adequately converge and sample from the joint posterior distribution. Although several studies of phylogenetic MCMC convergence exist, these have focused on simulated data sets or select empirical examples. Therefore, much that is considered common knowledge about MCMC in empirical systems derives from a relatively small family of analyses under ideal conditions. To address this, we present an overview of commonly applied phylogenetic MCMC diagnostics and an assessment of patterns of these diagnostics across more than 18,000 empirical analyses. Many analyses appeared to perform well and failures in convergence were most likely to be detected using the average standard deviation of split frequencies, a diagnostic that compares topologies among independent chains. Different diagnostics yielded different information about failed convergence, demonstrating that multiple diagnostics must be employed to reliably detect problems. The number of taxa and average branch lengths in analyses have clear impacts on MCMC performance, with more taxa and shorter branches leading to more difficult convergence. We show that the usage of models that include both Γ-distributed among-site rate variation and a proportion of invariable sites is not broadly problematic for MCMC convergence but is also unnecessary. Changes to heating and the usage of model-averaged substitution models can both offer improved convergence in some cases, but neither are a panacea.

The Behavior of Metropolis-Coupled Markov Chains When Sampling Rugged Phylogenetic Distributions.

Bayesian phylogenetic inference relies on the use of Markov chain Monte Carlo (MCMC) to provide numerical approximations of high-dimensional integrals and estimate posterior probabilities. However, MCMC performs poorly when posteriors are very rugged (i.e., regions of high posterior density are separated by regions of low posterior density). One technique that has become popular for improving numerical estimates from MCMC when distributions are rugged is Metropolis coupling (MC$^3$). In MC$^3$, additional chains are employed to sample flattened transformations of the posterior and improve mixing. Here, we highlight several underappreciated behaviors of MC3. Notably, estimated posterior probabilities may be incorrect but appear to converge, when individual chains do not mix well, despite different chains sampling trees from all relevant areas in tree space. Counterintuitively, such behavior can be more difficult to diagnose with increased numbers of chains. We illustrate these surprising behaviors of MC$^3$ using a simple, non-phylogenetic example and phylogenetic examples involving both constrained and unconstrained analyses. To detect and mitigate the effects of these behaviors, we recommend increasing the number of independent analyses and varying the temperature of the hottest chain in current versions of Bayesian phylogenetic software. Convergence diagnostics based on the behavior of the hottest chain may also help detect these behaviors and could form a useful addition to future software releases.

Bayes Factors Unmask Highly Variable Information Content, Bias, and Extreme Influence in Phylogenomic Analyses.

As the application of genomic data in phylogenetics has become routine, a number of cases have arisen where alternative data sets strongly support conflicting conclusions. This sensitivity to analytical decisions has prevented firm resolution of some of the most recalcitrant nodes in the tree of life. To better understand the causes and nature of this sensitivity, we analyzed several phylogenomic data sets using an alternative measure of topological support (the Bayes factor) that both demonstrates and averts several limitations of more frequently employed support measures (such as Markov chain Monte Carlo estimates of posterior probabilities). Bayes factors reveal important, previously hidden, differences across six "phylogenomic" data sets collected to resolve the phylogenetic placement of turtles within Amniota. These data sets vary substantially in their support for well-established amniote relationships, particularly in the proportion of genes that contain extreme amounts of information as well as the proportion that strongly reject these uncontroversial relationships. All six data sets contain little information to resolve the phylogenetic placement of turtles relative to other amniotes. Bayes factors also reveal that a very small number of extremely influential genes (less than 1% of genes in a data set) can fundamentally change significant phylogenetic conclusions. In one example, these genes are shown to contain previously unrecognized paralogs. This study demonstrates both that the resolution of difficult phylogenomic problems remains sensitive to seemingly minor analysis details and that Bayes factors are a valuable tool for identifying and solving these challenges.