Bayesian estimation of concordance among gene trees

Multigene sequence data have great potential for elucidating important and interesting evolutionary processes, but statistical methods for extracting information from such data remain limited. Although various biological processes may cause different genes to have different genealogical histories (and hence different tree topologies), we also may expect that the number of distinct topologies among a set of genes is relatively small compared with the number of possible topologies. Therefore evidence about the tree topology for one gene should influence our inferences of the tree topology on a different gene, but to what extent? In this paper, we present a new approach for modeling and estimating concordance among a set of gene trees given aligned molecular sequence data. Our approach introduces a one-parameter probability distribution to describe the prior distribution of concordance among gene trees. We describe a novel 2-stage Markov chain Monte Carlo (MCMC) method that first obtains independent Bayesian posterior probability distributions for individual genes using standard methods. These posterior distributions are then used as input for a second MCMC procedure that estimates a posterior distribution of gene-to-tree maps (GTMs). The posterior distribution of GTMs can then be summarized to provide revised posterior probability distributions for each gene (taking account of concordance) and to allow estimation of the proportion of the sampled genes for which any given clade is true (the sample-wide concordance factor). Further, under the assumption that the sampled genes are drawn randomly from a genome of known size, we show how one can obtain an estimate, with credibility intervals, on the proportion of the entire genome for which a clade is true (the genome-wide concordance factor). We demonstrate the method on a set of 106 genes from 8 yeast species.     

Fundamental differences between the methods of maximum likelihood and maximum posterior probability in phylogenetics

Using a four-taxon example under a simple model of evolution, we show that the methods of maximum likelihood and maximum posterior probability (which is a Bayesian method of inference) may not arrive at the same optimal tree topology. Some patterns that are separately uninformative under the maximum likelihood method are separately informative under the Bayesian method. We also show that this difference has impact on the bootstrap frequencies and the posterior probabilities of topologies, which therefore are not necessarily approximately equal. Efron et al. (Proc. Natl. Acad. Sci. USA 93:13429-13434, 1996) stated that bootstrap frequencies can, under certain circumstances, be interpreted as posterior probabilities. This is true only if one includes a non-informative prior distribution of the possible data patterns, and most often the prior distributions are instead specified in terms of topology and branch lengths. [Bayesian inference; maximum likelihood method; Phylogeny; support.].     

Is there a star tree paradox?

Concerns have been raised that posterior probabilities on phylogenetic trees can be unreliable when the true tree is unresolved or has very short internal branches, because existing methods for Bayesian phylogenetic analysis do not explicitly evaluate unresolved trees. Two recent papers have proposed that evaluating only resolved trees results in a "star tree paradox": when the true tree is unresolved or close to it, posterior probabilities were predicted to become increasingly unpredictable as sequence length grows, resulting in inflated confidence in one resolved tree or another and an increasing risk of false-positive inferences. Here we show that this is not the case; existing Bayesian methods do not lead to an inflation of statistical confidence, provided the evolutionary model is correct and uninformative priors are assumed. Posterior probabilities do not become increasingly unpredictable with increasing sequence length, and they exhibit conservative type I error rates, leading to a low rate of false-positive inferences. With infinite data, posterior probabilities give equal support for all resolved trees, and the rate of false inferences falls to zero. We conclude that there is no star tree paradox caused by not sampling unresolved trees.     

Strange bayes indeed: uniform topological priors imply non-uniform clade priors

While Bayesian analysis has become common in phylogenetics, the effects of topological prior probabilities on tree inference have not been investigated. In Bayesian analyses, the prior probability of topologies is almost always considered equal for all possible trees, and clade support is calculated from the majority rule consensus of the approximated posterior distribution of topologies. These uniform priors on tree topologies imply non-uniform prior probabilities of clades, which are dependent on the number of taxa in a clade as well as the number of taxa in the analysis. As such, uniform topological priors do not model ignorance with respect to clades. Here, we demonstrate that Bayesian clade support, bootstrap support, and jackknife support from 17 empirical studies are significantly and positively correlated with non-uniform clade priors resulting from uniform topological priors. Further, we demonstrate that this effect disappears for bootstrap and jackknife when data sets are free from character conflict, but remains pronounced for Bayesian clade supports, regardless of tree shape. Finally, we propose the use of a Bayes factor to account for the fact that uniform topological priors do not model ignorance with respect to clade probability.     

Guided tree topology proposals for Bayesian phylogenetic inference

Increasingly, large data sets pose a challenge for computationally intensive phylogenetic methods such as Bayesian Markov chain Monte Carlo (MCMC). Here, we investigate the performance of common MCMC proposal distributions in terms of median and variance of run time to convergence on 11 data sets. We introduce two new Metropolized Gibbs Samplers for moving through "tree space." MCMC simulation using these new proposals shows faster average run time and dramatically improved predictability in performance, with a 20-fold reduction in the variance of the time to estimate the posterior distribution to a given accuracy. We also introduce conditional clade probabilities and demonstrate that they provide a superior means of approximating tree topology posterior probabilities from samples recorded during MCMC.     

Effects of branch length uncertainty on Bayesian posterior probabilities for phylogenetic hypotheses

In Bayesian phylogenetics, confidence in evolutionary relationships is expressed as posterior probability--the probability that a tree or clade is true given the data, evolutionary model, and prior assumptions about model parameters. Model parameters, such as branch lengths, are never known in advance; Bayesian methods incorporate this uncertainty by integrating over a range of plausible values given an assumed prior probability distribution for each parameter. Little is known about the effects of integrating over branch length uncertainty on posterior probabilities when different priors are assumed. Here, we show that integrating over uncertainty using a wide range of typical prior assumptions strongly affects posterior probabilities, causing them to deviate from those that would be inferred if branch lengths were known in advance; only when there is no uncertainty to integrate over does the average posterior probability of a group of trees accurately predict the proportion of correct trees in the group. The pattern of branch lengths on the true tree determines whether integrating over uncertainty pushes posterior probabilities upward or downward. The magnitude of the effect depends on the specific prior distributions used and the length of the sequences analyzed. Under realistic conditions, however, even extraordinarily long sequences are not enough to prevent frequent inference of incorrect clades with strong support. We found that across a range of conditions, diffuse priors--either flat or exponential distributions with moderate to large means--provide more reliable inferences than small-mean exponential priors. An empirical Bayes approach that fixes branch lengths at their maximum likelihood estimates yields posterior probabilities that more closely match those that would be inferred if the true branch lengths were known in advance and reduces the rate of strongly supported false inferences compared with fully Bayesian integration.     

Bayesian species delimitation in West African forest geckos (Hemidactylus fasciatus)

Genealogical data are an important source of evidence for delimiting species, yet few statistical methods are available for calculating the probabilities associated with different species delimitations. Bayesian species delimitation uses reversible-jump Markov chain Monte Carlo (rjMCMC) in conjunction with a user-specified guide tree to estimate the posterior distribution for species delimitation models containing different numbers of species. We apply Bayesian species delimitation to investigate the speciation history of forest geckos (Hemidactylus fasciatus) from tropical West Africa using five nuclear loci (and mtDNA) for 51 specimens representing 10 populations. We find that species diversity in H. fasciatus is currently underestimated, and describe three new species to reflect the most conservative estimate for the number of species in this complex. We examine the impact of the guide tree, and the prior distributions on ancestral population sizes (θ) and root age (τ₀), on the posterior probabilities for species delimitation. Mis-specification of the guide tree or the prior distribution for θ can result in strong support for models containing more species. We describe a new statistic for summarizing the posterior distribution of species delimitation models, called speciation probabilities, which summarize the posterior support for each speciation event on the starting guide tree.     

Constraining prior probabilities of phylogenetic trees

Although Bayesian methods are widely used in phylogenetic systematics today, the foundations of this methodology are still debated among both biologists and philosophers. The Bayesian approach to phylogenetic inference requires the assignment of prior probabilities to phylogenetic trees. As in other applications of Bayesian epistemology, the question of whether there is an objective way to assign these prior probabilities is a contested issue. This paper discusses the strategy of constraining the prior probabilities of phylogenetic trees by means of the Principal Principle. In particular, I discuss a proposal due to Velasco (Biol Philos 23:455–473, 2008) of assigning prior probabilities to tree topologies based on the Yule process. By invoking the Principal Principle I argue that prior probabilities of tree topologies should rather be assigned a weighted mixture of probability distributions based on Pinelis’ (P Roy Soc Lond B Bio 270:1425–1431, 2003) multi-rate branching process including both the Yule distribution and the uniform distribution. However, I argue that this solves the problem of the priors of phylogenetic trees only in a weak form.     

Bayesian phylogenetic inference using DNA sequences: a Markov Chain Monte Carlo Method

An improved Bayesian method is presented for estimating phylogenetic trees using DNA sequence data. The birth-death process with species sampling is used to specify the prior distribution of phylogenies and ancestral speciation times, and the posterior probabilities of phylogenies are used to estimate the maximum posterior probability (MAP) tree. Monte Carlo integration is used to integrate over the ancestral speciation times for particular trees. A Markov Chain Monte Carlo method is used to generate the set of trees with the highest posterior probabilities. Methods are described for an empirical Bayesian analysis, in which estimates of the speciation and extinction rates are used in calculating the posterior probabilities, and a hierarchical Bayesian analysis, in which these parameters are removed from the model by an additional integration. The Markov Chain Monte Carlo method avoids the requirement of our earlier method for calculating MAP trees to sum over all possible topologies (which limited the number of taxa in an analysis to about five). The methods are applied to analyze DNA sequences for nine species of primates, and the MAP tree, which is identical to a maximum-likelihood estimate of topology, has a probability of approximately 95%.      

Model‐based approach to test hard polytomies in the Eulaemus clade of the most diverse South American lizard genus Liolaemus (Liolaemini,Squamata)

Lack of resolution in a phylogenetic tree is usually represented as a polytomy, and often adding more data (loci and taxa) resolves the species tree. These are the ‘soft’ polytomies, but in other cases additional data fail to resolve relationships; these are the ‘hard’ polytomies. This latter case is often interpreted as a simultaneous radiation of lineages in the history of a clade. Although hard polytomies are difficult to address, model‐based approaches provide new tools to test these hypotheses. Here, we used a clade of 144 species of the South American lizard clade Eulaemus to estimate phylogenies using a traditional concatenated matrix and three species tree methods: *BEAST, BEST, and minimizing deep coalescences (MDC). The different species tree methods recovered largely discordant results, but all resolved the same polytomy (e.g. very short internodes amongst lineages and low nodal support in Bayesian methods). We simulated data sets under eight explicit evolutionary models (including hard polytomies), tested these against empirical data (a total of 14 loci), and found support for two polytomies as the most plausible hypothesis for diversification of this clade. We discuss the performance of these methods and their limitations under the challenging scenario of hard polytomies. © 2015 The Linnean Society of London     

Bayesian estimation of species divergence times under a molecular clock using multiple fossil calibrations with soft bounds

We implement a Bayesian Markov chain Monte Carlo algorithm for estimating species divergence times that uses heterogeneous data from multiple gene loci and accommodates multiple fossil calibration nodes. A birth-death process with species sampling is used to specify a prior for divergence times, which allows easy assessment of the effects of that prior on posterior time estimates. We propose a new approach for specifying calibration points on the phylogeny, which allows the use of arbitrary and flexible statistical distributions to describe uncertainties in fossil dates. In particular, we use soft bounds, so that the probability that the true divergence time is outside the bounds is small but nonzero. A strict molecular clock is assumed in the current implementation, although this assumption may be relaxed. We apply our new algorithm to two data sets concerning divergences of several primate species, to examine the effects of the substitution model and of the prior for divergence times on Bayesian time estimation. We also conduct computer simulation to examine the differences between soft and hard bounds. We demonstrate that divergence time estimation is intrinsically hampered by uncertainties in fossil calibrations, and the error in Bayesian time estimates will not go to zero with increased amounts of sequence data. Our analyses of both real and simulated data demonstrate potentially large differences between divergence time estimates obtained using soft versus hard bounds and a general superiority of soft bounds. Our main findings are as follows. (1) When the fossils are consistent with each other and with the molecular data, and the posterior time estimates are well within the prior bounds, soft and hard bounds produce similar results. (2) When the fossils are in conflict with each other or with the molecules, soft and hard bounds behave very differently; soft bounds allow sequence data to correct poor calibrations, while poor hard bounds are impossible to overcome by any amount of data. (3) Soft bounds eliminate the need for "safe" but unrealistically high upper bounds, which may bias posterior time estimates. (4) Soft bounds allow more reliable assessment of estimation errors, while hard bounds generate misleadingly high precisions when fossils and molecules are in conflict.     

Adaptive Bayesian sum of trees model for covariate-dependent spectral analysis

This paper introduces a flexible and adaptive nonparametric method for estimating the association between multiple covariates and power spectra of multiple time series. The proposed approach uses a Bayesian sum of trees model to capture complex dependencies and interactions between covariates and the power spectrum, which are often observed in studies of biomedical time series. Local power spectra corresponding to terminal nodes within trees are estimated nonparametrically using Bayesian penalized linear splines. The trees are considered to be random and fit using a Bayesian backfitting Markov chain Monte Carlo (MCMC) algorithm that sequentially considers tree modifications via reversible-jump MCMC techniques. For high-dimensional covariates, a sparsity-inducing Dirichlet hyperprior on tree splitting proportions is considered, which provides sparse estimation of covariate effects and efficient variable selection. By averaging over the posterior distribution of trees, the proposed method can recover both smooth and abrupt changes in the power spectrum across multiple covariates. Empirical performance is evaluated via simulations to demonstrate the proposed method's ability to accurately recover complex relationships and interactions. The proposed methodology is used to study gait maturation in young children by evaluating age-related changes in power spectra of stride interval time series in the presence of other covariates.     

Reanalysis of Murphy et al.'s data gives various mammalian phylogenies and suggests overcredibility of Bayesian trees

Murphy and colleagues reported that the mammalian phylogeny was resolved by Bayesian phylogenetics. However, the DNA sequences they used had many alignment gaps and undetermined nucleotide sites. We therefore reanalyzed their data by minimizing unshared nucleotide sites and retaining as many species as possible (13 species). In constructing phylogenetic trees, we used the Bayesian, maximum likelihood (ML), maximum parsimony (MP), and neighbor-joining (NJ) methods with different substitution models. These trees were constructed by using both protein and DNA sequences. The results showed that the posterior probabilities for Bayesian trees were generally much higher than the bootstrap values for ML, MP, and NJ trees. Two different Bayesian topologies for the same set of species were sometimes supported by high posterior probabilities, implying that two different topologies can be judged to be correct by Bayesian phylogenetics. This suggests that the posterior probability in Bayesian analysis can be excessively high as an indication of statistical confidence and therefore Murphy et al.'s tree, which largely depends on Bayesian posterior probability, may not be correct.     

An examination of the monophyly of morning glory taxa using Bayesian phylogenetic inference

The objective of this study was to obtain a quantitative assessment of the monophyly of morning glory taxa, specifically the genus Ipomoea and the tribe Argyreieae. Previous systematic studies of morning glories intimated the paraphyly of Ipomoea by suggesting that the genera within the tribe Argyreieae are derived from within Ipomoea; however, no quantitative estimates of statistical support were developed to address these questions. We applied a Bayesian analysis to provide quantitative estimates of monophyly in an investigation of morning glory relationships using DNA sequence data. We also explored various approaches for examining convergence of the Markov chain Monte Carlo (MCMC) simulation of the Bayesian analysis by running 18 separate analyses varying in length. We found convergence of the important components of the phylogenetic model (the tree with the maximum posterior probability, branch lengths, the parameter values from the DNA substitution model, and the posterior probabilities for clade support) for these data after one million generations of the MCMC simulations. In the process, we identified a run where the parameter values obtained were often outside the range of values obtained from the other runs, suggesting an aberrant result. In addition, we compared the Bayesian method of phylogenetic analysis to maximum likelihood and maximum parsimony. The results from the Bayesian analysis and the maximum likelihood analysis were similar for topology, branch lengths, and parameters of the DNA substitution model. Topologies also were similar in the comparison between the Bayesian analysis and maximum parsimony, although the posterior probabilities and the bootstrap proportions exhibited some striking differences. In a Bayesian analysis of three data sets (ITS sequences, waxy sequences, and ITS + waxy sequences) no supoort for the monophyly of the genus Ipomoea, or for the tribe Argyreieae, was observed, with the estimate of the probability of the monophyly of these taxa being less than 3.4 x 10(-7).     

Fair-balance paradox, star-tree paradox, and Bayesian phylogenetics

The star-tree paradox refers to the conjecture that the posterior probabilities for the three unrooted trees for four species (or the three rooted trees for three species if the molecular clock is assumed) do not approach 1/3 when the data are generated using the star tree and when the amount of data approaches infinity. It reflects the more general phenomenon of high and presumably spurious posterior probabilities for trees or clades produced by the Bayesian method of phylogenetic reconstruction, and it is perceived to be a manifestation of the deeper problem of the extreme sensitivity of Bayesian model selection to the prior on parameters. Analysis of the star-tree paradox has been hampered by the intractability of the integrals involved. In this article, I use Laplacian expansion to approximate the posterior probabilities for the three rooted trees for three species using binary characters evolving at a constant rate. The approximation enables calculation of posterior tree probabilities for arbitrarily large data sets. Both theoretical analysis of the analogous fair-coin and fair-balance problems and computer simulation for the tree problem confirmed the existence of the star-tree paradox. When the data size n --> infinity, the posterior tree probabilities do not converge to 1/3 each, but they vary among data sets according to a statistical distribution. This distribution is characterized. Two strategies for resolving the star-tree paradox are explored: (1) a nonzero prior probability for the degenerate star tree and (2) an increasingly informative prior forcing the internal branch length toward zero. Both appear to be effective in resolving the paradox, but the latter is simpler to implement. The posterior tree probabilities are found to be very sensitive to the prior.     

Measuring gametic disequilibrium from multilocus data

We describe a Bayesian approach to analyzing multilocus genotype or haplotype data to assess departures from gametic (linkage) equilibrium. Our approach employs a Markov chain Monte Carlo (MCMC) algorithm to approximate the posterior probability distributions of disequilibrium parameters. The distributions are computed exactly in some simple settings. Among other advantages, posterior distributions can be presented visually, which allows the uncertainties in parameter estimates to be readily assessed. In addition, background knowledge can be incorporated, where available, to improve the precision of inferences. The method is illustrated by application to previously published datasets; implications for multilocus forensic match probabilities and for simple association-based gene mapping are also discussed.     

A Variational Bayes Approach to the Analysis of Occupancy Models

Detection-nondetection data are often used to investigate species range dynamics using Bayesian occupancy models which rely on the use of Markov chain Monte Carlo (MCMC) methods to sample from the posterior distribution of the parameters of the model. In this article we develop two Variational Bayes (VB) approximations to the posterior distribution of the parameters of a single-season site occupancy model which uses logistic link functions to model the probability of species occurrence at sites and of species detection probabilities. This task is accomplished through the development of iterative algorithms that do not use MCMC methods. Simulations and small practical examples demonstrate the effectiveness of the proposed technique. We specifically show that (under certain circumstances) the variational distributions can provide accurate approximations to the true posterior distributions of the parameters of the model when the number of visits per site (K) are as low as three and that the accuracy of the approximations improves as K increases. We also show that the methodology can be used to obtain the posterior distribution of the predictive distribution of the proportion of sites occupied (PAO).     

Survival of bowhead whales,Balaena mysticetus,estimated from 1981-1998 photoidentification data

Annual survival probability of bowhead whales, Balaena mysticetus, was estimated using both Bayesian and maximum likelihood implementations of Cormack and Jolly-Seber (JS) models for capture-recapture estimation in open populations and reduced-parameter generalizations of these models. Aerial photographs of naturally marked bowheads collected between 1981 and 1998 provided the data. The marked whales first photographed in a particular year provided the initial 'capture' and 'release' of those marked whales and photographs in subsequent years the 'recaptures'. The Cormack model, often called the Cormack-Jolly-Seber (CJS) model, and the program MARK were used to identify the model with a single survival and time-varying capture probabilities as the most appropriate for these data. When survival was constrained to be one or less, the maximum likelihood estimate computed by MARK was one, invalidating confidence interval computations based on the asymptotic standard error or profile likelihood. A Bayesian Markov chain Monte Carlo (MCMC) implementation of the model was used to produce a posterior distribution for annual survival. The corresponding reduced-parameter JS model was also fit via MCMC because it is the more appropriate of the two models for these photoidentification data. Because the CJS model ignores much of the information on capture probabilities provided by the data, its results are less precise and more sensitive to the prior distributions used than results from the JS model. With priors for annual survival and capture probabilities uniform from 0 to 1, the posterior mean for bowhead survival rate from the JS model is 0.984, and 95% of the posterior probability lies between 0.948 and 1. This high estimated survival rate is consistent with other bowhead life history data.     

Model parameterization, prior distributions, and the general time-reversible model in Bayesian phylogenetics

Bayesian phylogenetic methods require the selection of prior probability distributions for all parameters of the model of evolution. These distributions allow one to incorporate prior information into a Bayesian analysis, but even in the absence of meaningful prior information, a prior distribution must be chosen. In such situations, researchers typically seek to choose a prior that will have little effect on the posterior estimates produced by an analysis, allowing the data to dominate. Sometimes a prior that is uniform (assigning equal prior probability density to all points within some range) is chosen for this purpose. In reality, the appropriate prior depends on the parameterization chosen for the model of evolution, a choice that is largely arbitrary. There is an extensive Bayesian literature on appropriate prior choice, and it has long been appreciated that there are parameterizations for which uniform priors can have a strong influence on posterior estimates. We here discuss the relationship between model parameterization and prior specification, using the general time-reversible model of nucleotide evolution as an example. We present Bayesian analyses of 10 simulated data sets obtained using a variety of prior distributions and parameterizations of the general time-reversible model. Uniform priors can produce biased parameter estimates under realistic conditions, and a variety of alternative priors avoid this bias.     

Searching for convergence in phylogenetic Markov chain Monte Carlo

Markov chain Monte Carlo (MCMC) is a methodology that is gaining widespread use in the phylogenetics community and is central to phylogenetic software packages such as MrBayes. An important issue for users of MCMC methods is how to select appropriate values for adjustable parameters such as the length of the Markov chain or chains, the sampling density, the proposal mechanism, and, if Metropolis-coupled MCMC is being used, the number of heated chains and their temperatures. Although some parameter settings have been examined in detail in the literature, others are frequently chosen with more regard to computational time or personal experience with other data sets. Such choices may lead to inadequate sampling of tree space or an inefficient use of computational resources. We performed a detailed study of convergence and mixing for 70 randomly selected, putatively orthologous protein sets with different sizes and taxonomic compositions. Replicated runs from multiple random starting points permit a more rigorous assessment of convergence, and we developed two novel statistics, delta and epsilon, for this purpose. Although likelihood values invariably stabilized quickly, adequate sampling of the posterior distribution of tree topologies took considerably longer. Our results suggest that multimodality is common for data sets with 30 or more taxa and that this results in slow convergence and mixing. However, we also found that the pragmatic approach of combining data from several short, replicated runs into a "metachain" to estimate bipartition posterior probabilities provided good approximations, and that such estimates were no worse in approximating a reference posterior distribution than those obtained using a single long run of the same length as the metachain. Precision appears to be best when heated Markov chains have low temperatures, whereas chains with high temperatures appear to sample trees with high posterior probabilities only rarely.