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Several computational methods have recently been proposed for delimiting species using multilocus sequence data. Among them, the Bayesian method of Yang and Rannala uses the multispecies coalescent model in the likelihood framework to calculate the posterior probabilities for the different species-delimitation models. It has a sound statistical basis and is found to have nice statistical properties in simulation studies, such as low error rates of undersplitting and oversplitting. However, the method suffers from poor mixing of the reversible-jump Markov chain Monte Carlo (rjMCMC) algorithms. Here, we describe several modifications to the algorithms. We propose a flexible prior that allows the user to specify the probability that each node on the guide tree represents a true speciation event. We also introduce modifications to the rjMCMC algorithms that remove the constraint on the new species divergence time when splitting and alter the gene trees to remove incompatibilities. The new algorithms are found to improve mixing of the Markov chain for both simulated and empirical data sets.

In the absence of recent admixture between species, bipartitions of individuals in gene trees that are shared across loci can potentially be used to infer the presence of two or more species. This approach to species delimitation via molecular sequence data has been constrained by the fact that genealogies for individual loci are often poorly resolved and that ancestral lineage sorting, hybridization, and other population genetic processes can lead to discordant gene trees. Here we use a Bayesian modeling approach to generate the posterior probabilities of species assignments taking account of uncertainties due to unknown gene trees and the ancestral coalescent process. For tractability, we rely on a user-specified guide tree to avoid integrating over all possible species delimitations. The statistical performance of the method is examined using simulations, and the method is illustrated by analyzing sequence data from rotifers, fence lizards, and human populations.

We extend our recently developed Markov chain Monte Carlo algorithm for Bayesian estimation of species divergence times to allow variable evolutionary rates among lineages. The method can use heterogeneous data from multiple gene loci and accommodate multiple fossil calibrations. Uncertainties in fossil calibrations are described using flexible statistical distributions. The prior for divergence times for nodes lacking fossil calibrations is specified by use of a birth-death process with species sampling. The prior for lineage-specific substitution rates is specified using either a model with autocorrelated rates among adjacent lineages (based on a geometric Brownian motion model of rate drift) or a model with independent rates among lineages specified by a log-normal probability distribution. We develop an infinite-sites theory, which predicts that when the amount of sequence data approaches infinity, the width of the posterior credibility interval and the posterior mean of divergence times form a perfect linear relationship, with the slope indicating uncertainties in time estimates that cannot be reduced by sequence data alone. Simulations are used to study the influence of among-lineage rate variation and the number of loci sampled on the uncertainty of divergence time estimates. The analysis suggests that posterior time estimates typically involve considerable uncertainties even with an infinite amount of sequence data, and that the reliability and precision of fossil calibrations are critically important to divergence time estimation. We apply our new algorithms to two empirical data sets and compare the results with those obtained in previous Bayesian and likelihood analyses. The results demonstrate the utility of our new algorithms.

Bayesian analysis of macroevolutionary mixtures (BAMM) has recently taken the study of lineage diversification by storm. BAMM estimates the diversification-rate parameters (speciation and extinction) for every branch of a study phylogeny and infers the number and location of diversification-rate shifts across branches of a tree. Our evaluation of BAMM reveals two major theoretical errors: (i) the likelihood function (which estimates the model parameters from the data) is incorrect, and (ii) the compound Poisson process prior model (which describes the prior distribution of diversification-rate shifts across branches) is incoherent. Using simulation, we demonstrate that these theoretical issues cause statistical pathologies; posterior estimates of the number of diversification-rate shifts are strongly influenced by the assumed prior, and estimates of diversification-rate parameters are unreliable. Moreover, the inability to correctly compute the likelihood or to correctly specify the prior for rate-variable trees precludes the use of Bayesian approaches for testing hypotheses regarding the number and location of diversification-rate shifts using BAMM.

Gene flow among populations or species and incomplete lineage sorting (ILS) are two evolutionary processes responsible for generating gene tree discordance and therefore hindering species tree estimation. Numerous studies have evaluated the impacts of ILS on species tree inference, yet the ramifications of gene flow on species trees remain less studied. Here, we simulate and analyse multilocus sequence data generated with ILS and gene flow to quantify their impacts on species tree inference. We characterize species tree estimation errors under various models of gene flow, such as the isolation-migration model, the n-island model, and gene flow between non-sister species or involving ancestral species, and species boundaries crossed by a single gene copy (allelic introgression) or by a single migrant individual. These patterns of gene flow are explored on species trees of different sizes (4 vs. 10 species), at different time scales (shallow vs. deep), and with different migration rates. Species trees are estimated with the multispecies coalescent model using Bayesian methods (BEST and *BEAST) and with a summary statistic approach (MPEST) that facilitates phylogenomic-scale analysis. Even in cases where the topology of the species tree is estimated with high accuracy, we find that gene flow can result in overestimates of population sizes (species tree dilation) and underestimates of species divergence times (species tree compression). Signatures of migration events remain present in the distribution of coalescent times for gene trees, and with sufficient data it is possible to identify those loci that have crossed species boundaries. These results highlight the need for careful sampling design in phylogeographic and species delimitation studies as gene flow, introgression, or incorrect sample assignments can bias the estimation of the species tree topology and of parameter estimates such as population sizes and divergence times.

Population phylogenomics uses sampled genomes to jointly infer population genetic processes (ancestral and contemporary population sizes, historical gene flow) and a phylogenetic tree relating species or populations including species split times. This challenging problem has been tackled most successfully in the Bayesian framework under the multispecies coalescent (MSC) model via Markov chain Monte Carlo (MCMC) computational algorithms. However, MCMC methods suffer from two serious problems: (i) mixing difficulties due to the high-dimensional state space with complex constraints, and (ii) the intrinsically serial nature of MCMC algorithms that defies parallelisation. To deal with both issues, we develop a new method, called Virtual Dimension Reduction allowing Parallelisation (VDRoP), that achieves the same MCMC mixing efficiency as dimension reduction through analytical integration of parameters, but without sacrificing parallel computation and without the restriction to conjugate priors. We implement the new method in the Bayesian program BPP and apply it to genomic datasets from Adansonia baobab trees, Anopheles mosquitoes, and Heliconius butterflies. The new algorithms reduce the run-time of MCMC analyses by 3 to 8 fold and improve the mixing efficiency by up to 50 fold for representative empirical datasets. Bayesian inference of gene flow under the multispecies coalescent involves heavy computation. This study presents VDRoP, a parallel virtual dimension-reduction framework, which preserves the mixing efficiency of reduced dimension but samples all parameters such as population sizes and rates of gene flow.

What does the posterior probability of a phylogenetic tree mean? This simulation study shows that Bayesian posterior probabilities have the meaning that is typically ascribed to them; the posterior probability of a tree is the probability that the tree is correct, assuming that the model is correct. At the same time, the Bayesian method can be sensitive to model misspecification, and the sensitivity of the Bayesian method appears to be greater than the sensitivity of the nonparametric bootstrap method (using maximum likelihood to estimate trees). Although the estimates of phylogeny obtained by use of the method of maximum likelihood or the Bayesian method are likely to be similar, the assessment of the uncertainty of inferred trees via either bootstrapping (for maximum likelihood estimates) or posterior probabilities (for Bayesian estimates) is not likely to be the same. We suggest that the Bayesian method be implemented with the most complex models of those currently available, as this should reduce the chance that the method will concentrate too much probability on too few trees.

We develop a Bayesian method for inferring the species phylogeny under the multispecies coalescent (MSC) model. To improve the mixing properties of the Markov chain Monte Carlo (MCMC) algorithm that traverses the space of species trees, we implement two efficient MCMC proposals: the first is based on the Subtree Pruning and Regrafting (SPR) algorithm and the second is based on a node-slider algorithm. Like the Nearest-Neighbor Interchange (NNI) algorithm we implemented previously, both new algorithms propose changes to the species tree, while simultaneously altering the gene trees at multiple genetic loci to automatically avoid conflicts with the newly proposed species tree. The method integrates over gene trees, naturally taking account of the uncertainty of gene tree topology and branch lengths given the sequence data. A simulation study was performed to examine the statistical properties of the new method. The method was found to show excellent statistical performance, inferring the correct species tree with near certainty when 10 loci were included in the dataset. The prior on species trees has some impact, particularly for small numbers of loci. We analyzed several previously published datasets (both real and simulated) for rattlesnakes and Philippine shrews, in comparison with alternative methods. The results suggest that the Bayesian coalescent-based method is statistically more efficient than heuristic methods based on summary statistics, and that our implementation is computationally more efficient than alternative full-likelihood methods under the MSC. Parameter estimates for the rattlesnake data suggest drastically different evolutionary dynamics between the nuclear and mitochondrial loci, even though they support largely consistent species trees. We discuss the different challenges facing the marginal likelihood calculation and transmodel MCMC as alternative strategies for estimating posterior probabilities for species trees.

The Bayesian method for estimating species phylogenies from molecular sequence data provides an attractive alternative to maximum likelihood with nonparametric bootstrap due to the easy interpretation of posterior probabilities for trees and to availability of efficient computational algorithms. However, for many data sets it produces extremely high posterior probabilities, sometimes for apparently incorrect clades. Here we use both computer simulation and empirical data analysis to examine the effect of the prior model for internal branch lengths. We found that posterior probabilities for trees and clades are sensitive to the prior for internal branch lengths, and priors assuming long internal branches cause high posterior probabilities for trees. In particular, uniform priors with high upper bounds bias Bayesian clade probabilities in favor of extreme values. We discuss possible remedies to the problem, including empirical and full Bayesian methods and subjective procedures suggested in Bayesian hypothesis testing. Our results also suggest that the bootstrap proportion and Bayesian posterior probability are different measures of accuracy, and that the bootstrap proportion, if interpreted as the probability that the clade is true, can be either too liberal or too conservative.

Bayesian inference of species divergence times is an unusual statistical problem, because the divergence time parameters are not identifiable unless both fossil calibrations and sequence data are available. Commonly used marginal priors on divergence times derived from fossil calibrations may conflict with node order on the phylogenetic tree causing a change in the prior on divergence times for a particular topology. Care should be taken to avoid confusing this effect with changes due to informative sequence data. This effect is illustrated with examples. A topology-consistent prior that preserves the marginal priors is defined and examples are constructed. Conflicts between fossil calibrations and relative branch lengths (based on sequence data) can cause estimates of divergence times that are grossly incorrect, yet have a narrow posterior distribution. An example of this effect is given; it is recommended that overly narrow posterior distributions of divergence times should be carefully scrutinized. This article is part of the themed issue ‘Dating species divergences using rocks and clocks’.