Frequentist properties of Bayesian posterior probabilities of phylogenetic trees under simple and complex substitution models

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.     

Comparison of Bayesian and maximum likelihood bootstrap measures of phylogenetic reliability

Owing to the exponential growth of genome databases, phylogenetic trees are now widely used to test a variety of evolutionary hypotheses. Nevertheless, computation time burden limits the application of methods such as maximum likelihood nonparametric bootstrap to assess reliability of evolutionary trees. As an alternative, the much faster Bayesian inference of phylogeny, which expresses branch support as posterior probabilities, has been introduced. However, marked discrepancies exist between nonparametric bootstrap proportions and Bayesian posterior probabilities, leading to difficulties in the interpretation of sometimes strongly conflicting results. As an attempt to reconcile these two indices of node reliability, we apply the nonparametric bootstrap resampling procedure to the Bayesian approach. The correlation between posterior probabilities, bootstrap maximum likelihood percentages, and bootstrapped posterior probabilities was studied for eight highly diverse empirical data sets and were also investigated using experimental simulation. Our results show that the relation between posterior probabilities and bootstrapped maximum likelihood percentages is highly variable but that very strong correlations always exist when Bayesian node support is estimated on bootstrapped character matrices. Moreover, simulations corroborate empirical observations in suggesting that, being more conservative, the bootstrap approach might be less prone to strongly supporting a false phylogenetic hypothesis. Thus, apparent conflicts in topology recovered by the Bayesian approach were reduced after bootstrapping. Both posterior probabilities and bootstrap supports are of great interest to phylogeny as potential upper and lower bounds of node reliability, but they are surely not interchangeable and cannot be directly compared.     

Measures of clade confidence do not correlate with accuracy of phylogenetic trees

Metrics of phylogenetic tree reliability, such as parametric bootstrap percentages or Bayesian posterior probabilities, represent internal measures of the topological reproducibility of a phylogenetic tree, while the recently introduced aLRT (approximate likelihood ratio test) assesses the likelihood that a branch exists on a maximum-likelihood tree. Although those values are often equated with phylogenetic tree accuracy, they do not necessarily estimate how well a reconstructed phylogeny represents cladistic relationships that actually exist in nature. The authors have therefore attempted to quantify how well bootstrap percentages, posterior probabilities, and aLRT measures reflect the probability that a deduced phylogenetic clade is present in a known phylogeny. The authors simulated the evolution of bacterial genes of varying lengths under biologically realistic conditions, and reconstructed those known phylogenies using both maximum likelihood and Bayesian methods. Then, they measured how frequently clades in the reconstructed trees exhibiting particular bootstrap percentages, aLRT values, or posterior probabilities were found in the true trees. The authors have observed that none of these values correlate with the probability that a given clade is present in the known phylogeny. The major conclusion is that none of the measures provide any information about the likelihood that an individual clade actually exists. It is also found that the mean of all clade support values on a tree closely reflects the average proportion of all clades that have been assigned correctly, and is thus a good representation of the overall accuracy of a phylogenetic tree.     

Branch-length prior influences Bayesian posterior probability of phylogeny

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 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%.      

Understanding mammalian evolution using Bayesian phylogenetic inference

1. Phylogenetic trees are critical in addressing evolutionary hypotheses; however, the reconstruction of a phylogeny is no easy task. This process has recently been made less arduous by using a Bayesian statistical approach. This method offers the advantage that one can determine the probability of some hypothesis (i.e. a phylogeny), conditional on the observed data (i.e. nucleotide sequences). 2. By reconstructing phylogenies using Bayes’ theorem in combination with Markov chain Monte Carlo, phylogeneticists are able to test hypotheses more quickly compared with using standard methods such as neighbour-joining, maximum likelihood or parsimony. Critics of the Bayesian approach suggest that it is not a panacea, and argue that the prior probability is too subjective and the resulting posterior probability is too liberal compared with maximum likelihood. 3. These issues are currently debated in the arena of mammalian evolution. Recently, proponents and opponents of the Bayesian approach have constructed the mammalian phylogeny using different methods under different conditions and with a variety of parameters. These analyses showed the robustness (or lack of) of the Bayesian approach. In the end, consensus suggests that Bayesian methods are robust and give essentially the same answer as maximum likelihood methods but in less time. 4. Approaches based on fossils and molecules typically agree on ordinal-level relationships among mammals but not on higher-level relationships, as Bayesian analyses recognize the African radiation, Afrotheria, and the two Laurasian radiations, Laurasiatheria and Euarchontoglires, whereas fossils did not predict Afrotheria.     

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.     

A New Method of Inference of Ancestral Nucleotide and Amino Acid Sequences

A statistical method was developed for reconstructing the nucleotide or amino acid sequences of extinct ancestors, given the phylogeny and sequences of the extant species. A model of nucleotide or amino acid substitution was employed to analyze data of the present-day sequences, and maximum likelihood estimates of parameters such as branch lengths were used to compare the posterior probabilities of assignments of character states (nucleotides or amino acids) to interior nodes of the tree; the assignment having the highest probability was the best reconstruction at the site. The lysozyme c sequences of six mammals were analyzed by using the likelihood and parsimony methods. The new likelihood-based method was found to be superior to the parsimony method. The probability that the amino acids for all interior nodes at a site reconstructed by the new method are correct was calculated to be 0.91, 0.86, and 0.73 for all, variable, and parsimony-informative sites, respectively, whereas the corresponding probabilities for the parsimony method were 0.84, 0.76, and 0.51, respectively. The probability that an amino acid in an ancestral sequence is correctly reconstructed by the likelihood analysis ranged from 91.3 to 98.7% for the four ancestral sequences.     

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.     

Tree-based maximal likelihood substitution matrices and hidden Markov models

There has been considerable interest in the problem of making maximum likelihood (ML) evolutionary trees which allow insertions and deletions. This problem is partly one of formulation: how does one define a probabilistic model for such trees which treats insertion and deletion in a biologically plausible manner? A possible answer to this question is proposed here by extending the concept of a hidden Markov model (HMM) to evolutionary trees. The model, called a tree-HMM, allows what may be loosely regarded as learnable affine-type gap penalties for alignments. These penalties are expressed in HMMs as probabilities of transitions between states. In the tree-HMM, this idea is given an evolutionary embodiment by defining trees of transitions. Just as the probability of a tree composed of ungapped sequences is computed, by Felsenstein's method, using matrices representing the probabilities of substitutions of residues along the edges of the tree, so the probabilities in a tree-HMM are computed by substitution matrices for both residues and transitions. How to define these matrices by a ML procedure using an algorithm that learns from a database of protein sequences is shown here. Given these matrices, one can define a tree-HMM likelihood for a set of sequences, assuming a particular tree topology and an alignment of the sequences to the model. If one could efficiently find the alignment which maximizes (or comes close to maximizing) this likelihood, then one could search for the optimal tree topology for the sequences. An alignment algorithm is defined here which, given a particular tree topology, is guaranteed to increase the likelihood of the model. Unfortunately, it fails to find global optima for realistic sequence sets. Thus further research is needed to turn the tree-HMM into a practical phylogenetic tool.     

Robustness of maximum likelihood tree estimation against different patterns of base substitutions

Summary In the maximum likelihood (ML) method for estimating a molecular phylogenetic tree, the pattern of nucleotide substitutions for computing likelihood values is assumed to be simpler than that of the actual evolutionary process, simply because the process, considered to be quite devious, is unknown. The problem, however, is that there has been no guarantee to endorse the simplification.To study this problem, we first evaluated the robustness of the ML method in the estimation of molecular trees against different nucleotide substitution patterns, including Jukes and Cantor's, the simplest ever proposed. Namely, we conducted computer simulations in which we could set up various evolutionary models of a hypothetical gene, and define a true tree to which an estimated tree by the ML method was to be compared. The results show that topology estimation by the ML method is considerably robust against different ratios of transitions to transversions and different GC contents, but branch length estimation is not so. The ML tree estimation based on Jukes and Cantor's model is also revealed to be resistant to GC content, but rather sensitive to the ratio of transitions to transversions.We then applied the ML method with different substitution patterns to nucleotide sequence data ontax gene from T-cell leukemia viruses whose evolutionary process must have been more complicated than that of the hypothetical gene. The results are in accordance with those from the simulation study, showing that Jukes and Cantor's model is as useful as a more complicated one for making inferences about molecular phylogeny of the viruses.     

BIOGEOGRAPHY OF NEW WORLD TAIGA-DWELLING MICROTUS (MAMMALIA: ARVICOLIDAE): A HYPOTHESIS TEST THAT ACCOUNTS FOR PHYLOGENETIC UNCERTAINTY

If we adopt a statistical approach to systematics and recognize that phylogenies are estimated with error, then we can begin to explore statistically justified methods for testing a variety of comparative hypotheses, including those concerning the evolution of life-history characters and biogeography. In this paper I examine two biogeographic hypotheses concerning the rodent genus Microtus. Like many comparative hypotheses, these can be phrased so that each predicts the existence of a particular monophyletic group. Neither of the predicted groups appear on the single best phylogeny as determined by both Dollo parsimony and maximum likelihood analysis of restriction site maps of mitochondrial DNA. Simulation studies, however, suggest that often the best phylogeny from a single data set has only a low probability of being exactly correct. We must also examine those trees that, while not the single best-supported tree, are not rejected by the data. If we find the best phylogeny for which a hypothesis is satisfied, then likelihood methods can be used to test whether that phylogeny is significantly worse then the best tree overall. If that tree can be rejected, then so can the hypothesis. Computational constraints limit the use of likelihood methods for searching among topologies, so parsimony is used as a data exploratory tool. One of the predicted groups cannot be rejected, even though the most parsimonious tree which includes that group requires 11 more steps than does the most parsimonious tree.     

Maximum likelihood alignment of DNA sequences

The optimal alignment problem for pairs of molecular sequences under a probabilistic model of evolutionary change is equivalent to the problem of estimating the maximum likelihood time required to transform one sequence to the other. When this time has been estimated, various alignments of high posterior probability may be written down. A simple model with two parameters is presented and a method is described by which the likelihood may be computed. Maximum likelihood estimates for some pairs of tRNA genes illustrate the method and allow us to obtain the best alignments under the model.     

Recognizing the forest for the trees: testing temporal patterns of cladogenesis using a null model of stochastic diversification

Computer simulations are developed and employed to examine the expected temporal distributions of nodes under a null model of stochastic lineage bifurcation and extinction. These Markovian models of phylogenetic process were constructed so as to permit direct comparisons against empirical phylogenetic trees generated from molecular or other information available solely from extant species. For replicate simulated phylads with n extant species, cumulative distribution functions (cdf's) of branching times were calculated, and compared (using the Kolmogorov-Smirnov test statistic D) to those from three published empirical trees. Molecular phylogenies for columbine plants and avian cranes showed statistically significant departures from the null expectations, in directions indicating recent and ancient species' radiations, respectively, whereas a molecular phylogeny for the Drosophila virilis species group showed no apparent historical clustering of branching events. Effects of outgroup choice and phylogenetic frame of reference were investigated for the columbines and found to have a predictable influence on the types of conclusions to be drawn from such analyses. To enable other investigators to statistically test for nonrandomness in temporal cladogenetic pattern in empirical trees generated from data on extant species, we present tables of mean cdf's and associated probabilities under the null model for expected branching times in phylads of varying size. The approaches developed in this report complement and extend those of other recent methods for employing null models to assess the statistical significance of pattern in evolutionary trees.      

Codon and rate variation models in molecular phylogeny

This article generalizes previous models for codon substitution and rate variation in molecular phylogeny. Particular attention is paid to (1) reversibility, (2) acceptance and rejection of proposed codon changes, (3) varying rates of evolution among codon sites, and (4) the interaction of these sites in determining evolutionary rates. To accommodate spatial variation in rates, Markov random fields rather than Markov chains are introduced. Because these innovations complicate maximum likelihood estimation in phylogeny reconstruction, it is necessary to formulate new algorithms for the evaluation of the likelihood and its derivatives with respect to the underlying kinetic, acceptance, and spatial parameters. To derive the most from maximum likelihood analysis of sequence data, it is useful to compute posterior probabilities assigning residues to internal nodes and evolutionary rate classes to codon sites. It is also helpful to search through tree space in a way that respects accepted phylogenetic relationships. Our phylogeny program LINNAEUS implements algorithms realizing these goals. Readers may consult our companion article in this issue for several examples.     

A Bayesian perspective on a non-parsimonious parsimony model

Several stochastic models of character change, when implemented in a maximum likelihood framework, are known to give a correspondence between the maximum parsimony method and the method of maximum likelihood. One such model has an independently estimated branch-length parameter for each site and each branch of the phylogenetic tree. This model--the no-common-mechanism model--has many parameters, and, in fact, the number of parameters increases as fast as the alignment is extended. We take a Bayesian approach to the no-common-mechanism model and place independent gamma prior probability distributions on the branch-length parameters. We are able to analytically integrate over the branch lengths, and this allowed us to implement an efficient Markov chain Monte Carlo method for exploring the space of phylogenetic trees. We were able to reliably estimate the posterior probabilities of clades for phylogenetic trees of up to 500 sequences. However, the Bayesian approach to the problem, at least as implemented here with an independent prior on the length of each branch, does not tame the behavior of the branch-length parameters. The integrated likelihood appears to be a simple rescaling of the parsimony score for a tree, and the marginal posterior probability distribution of the length of a branch is dependent upon how the maximum parsimony method reconstructs the characters at the interior nodes of the tree. The method we describe, however, is of potential importance in the analysis of morphological character data and also for improving the behavior of Markov chain Monte Carlo methods implemented for models in which sites share a common branch-length parameter.     

Maximum likelihood inference of protein phylogeny and the origin of chloroplasts

Summary A maximum likelihood method for inferring protein phylogeny was developed. It is based on a Markov model that takes into account the unequal transition probabilities among pairs of amino acids and does not assume constancy of rate among different lineages. Therefore, this method is expected to be powerful in inferring phylogeny among distantly related proteins, either orthologous or parallogous, where the evolutionary rate may deviate from constancy. Not only amino acid substitutions but also insertion/deletion events during evolution were incorporated into the Markov model. A simple method for estimating a bootstrap probability for the maximum likelihood tree among alternatives without performing a maximum likelihood estimation for each resampled data set was developed. These methods were applied to amino acid sequence data of a photosynthetic membrane protein,psbA, from photosystem II, and the phylogeny of this protein was discussed in relation to the origin of chloroplasts.     

Network models for sequence evolution

We introduce a general class of models for sequence evolution that includes network phylogenies. Networks, a generalization of strictly tree-like phylogenies, are proposed to model situations where multiple lineages contribute to the observed sequences. An algorithm to compute the probability distribution of binary character-state configurations is presented and statistical inference for this model is developed in a likelihood framework. A stepwise procedure based on likelihood ratios is used to explore the space of models. Starting with a star phylogeny, new splits (nontrivial bipartitions of the sequence set) are successively added to the model until no significant change in the likelihood is observed. A novel feature of our approach is that the new splits are not necessarily constrained to be consistent with a treelike mode of evolution. The fraction of invariable sites is estimated by maximum likelihood simultaneously with other model parameters and is essential to obtain a good fit to the data. The effect of finite sequence length on the inference methods is discussed. Finally, we provide an illustrative example using aligned VPl genes from the foot and mouth disease viruses (FMDV). The different serotypes of the FMDV exhibit a range of treelike and network evolutionary relationships.Correspondence to: A. von Haeseler     

Experimental phylogeny of neutrally evolving DNA sequences generated by a bifurcate series of nested polymerase chain reactions.

A known phylogeny was generated using a four-step serial bifurcate PCR method. The ancestor sequence (SSU rDNA) evolved in vitro for 280 nested PCR cycles, and the resulting 15 ancestor and 16 terminal sequences (2,238 bp each) were determined. Parsimony, distance, and maximum likelihood analysis of the terminal sequences reconstructed the topology of the real phylogeny and branch lengths accurately. Divergence dates and ancestor sequences were estimated with very small error, particularly at the base of the phylogeny, mostly due to insertion and deletion changes. The substitution patterns along the known phylogeny are not described by reversible models, and accordingly, the probability substitution matrix, based on the observed substitutions from ancestor to terminal nodes along the known phylogeny, was calculated. This approach is an extension of previous studies using bacteriophage serial propagation, because here mutations were allowed to occur neutrally rather than by addition of a mutagenic agent, which produced biased mutational changes. These results provide for the first time biochemical experimental support for phylogenies, divergence date estimates, and an irreversible substitution model based on neutrally evolving DNA sequences. The substitution preferences observed here (A to G and T to C) are consistent with the high G+C content of the Thermus aquaticus genome. This suggests, at least in part, that the method here described, which explores the high Taq DNA polymerase error rate, simulates the evolution of a DNA segment in a thermophilic organism. These organisms include the bacterial rod T. aquaticus and several Archaea, and thus, the method and data set described here may well contribute new insights about the genome evolution of these organisms.     

Phylogeographic analysis of the cornsnake (Elaphe guttata) complex as inferred from maximum likelihood and Bayesian analyses

Most phylogeographic studies have used maximum likelihood or maximum parsimony to infer phylogeny and bootstrap analysis to evaluate support for trees. Recently, Bayesian methods using Marlov chain Monte Carlo to search tree space and simultaneously estimate tree support have become popular due to its fast search speed and ability to create a posterior distribution of parameters of interest. Here, I present a study that utilizes Bayesian methods to infer phylogenetic relationships of the cornsnake (Elaphe guttata) complex using cytochrome b sequences. Examination of the posterior probability distributions confirms the existence of three geographic lineages. Additionally, there is no support for the monophyly of the subspecies of E. guttata. Results suggest the three geographic lineages partially conform to the ranges of previously defined subspecies, although Shimodaira-Hasegawa tests suggest that subspecies-constrained trees produce significantly poorer likelihood estimates than the most likely trees reflecting the evolution of three geographic assemblages. Based on molecular support, these three geographic assemblages are recognized as species using evolutionary species criteria: E. guttata, Elaphe slowinskii, and Elaphe emoryi [phylogeographic, maximum likelihood, maximum parsimony, bootstrap, Bayesian, Markov chain Monte Carlo, cornsnake, Cytochrome b, geographic lineages, E. guttta, E. slowinskii, and E. emoryi].