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.     

Moving beyond noninformative priors: why and how to choose weakly informative priors in Bayesian analyses

Throughout the last two decades, Bayesian statistical methods have proliferated throughout ecology and evolution. Numerous previous references established both philosophical and computational guidelines for implementing Bayesian methods. However, protocols for incorporating prior information, the defining characteristic of Bayesian philosophy, are nearly nonexistent in the ecological literature. Here, I hope to encourage the use of weakly informative priors in ecology and evolution by providing a ‘consumer's guide’ to weakly informative priors. The first section outlines three reasons why ecologists should abandon noninformative priors: 1) common flat priors are not always noninformative, 2) noninformative priors provide the same result as simpler frequentist methods, and 3) noninformative priors suffer from the same high type I and type M error rates as frequentist methods. The second section provides a guide for implementing informative priors, wherein I detail convenient ‘reference’ prior distributions for common statistical models (i.e. regression, ANOVA, hierarchical models). I then use simulations to visually demonstrate how informative priors influence posterior parameter estimates. With the guidelines provided here, I hope to encourage the use of weakly informative priors for Bayesian analyses in ecology. Ecologists can and should debate the appropriate form of prior information, but should consider weakly informative priors as the new ‘default’ prior for any Bayesian model.     

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.     

Properties of phylogenetic trees generated by Yule-type speciation models

We investigate some discrete structural properties of evolutionary trees generated under simple null models of speciation, such as the Yule model. These models have been used as priors in Bayesian approaches to phylogenetic analysis, and also to test hypotheses concerning the speciation process. In this paper we describe new results for three properties of trees generated under such models. Firstly, for a rooted tree generated by the Yule model we describe the probability distribution on the depth (number of edges from the root) of the most recent common ancestor of a random subset of k species. Next we show that, for trees generated under the Yule model, the approximate position of the root can be estimated from the associated unrooted tree, even for trees with a large number of leaves. Finally, we analyse a biologically motivated extension of the Yule model and describe its distribution on tree shapes when speciation occurs in rapid bursts.     

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.     

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.     

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.     

An evaluation of prior influence on the predictive ability of Bayesian model averaging

Model averaging is gaining popularity among ecologists for making inference and predictions. Methods for combining models include Bayesian model averaging (BMA) and Akaike’s Information Criterion (AIC) model averaging. BMA can be implemented with different prior model weights, including the Kullback–Leibler prior associated with AIC model averaging, but it is unclear how the prior model weight affects model results in a predictive context. Here, we implemented BMA using the Bayesian Information Criterion (BIC) approximation to Bayes factors for building predictive models of bird abundance and occurrence in the Chihuahuan Desert of New Mexico. We examined how model predictive ability differed across four prior model weights, and how averaged coefficient estimates, standard errors and coefficients’ posterior probabilities varied for 16 bird species. We also compared the predictive ability of BMA models to a best single-model approach. Overall, Occam’s prior of parsimony provided the best predictive models. In general, the Kullback–Leibler prior, however, favored complex models of lower predictive ability. BMA performed better than a best single-model approach independently of the prior model weight for 6 out of 16 species. For 6 other species, the choice of the prior model weight affected whether BMA was better than the best single-model approach. Our results demonstrate that parsimonious priors may be favorable over priors that favor complexity for making predictions. The approach we present has direct applications in ecology for better predicting patterns of species’ abundance and occurrence.     

Calibrated tree priors for relaxed phylogenetics and divergence time estimation

The use of fossil evidence to calibrate divergence time estimation has a long history. More recently, Bayesian Markov chain Monte Carlo has become the dominant method of divergence time estimation, and fossil evidence has been reinterpreted as the specification of prior distributions on the divergence times of calibration nodes. These so-called "soft calibrations" have become widely used but the statistical properties of calibrated tree priors in a Bayesian setting hashave not been carefully investigated. Here, we clarify that calibration densities, such as those defined in BEAST 1.5, do not represent the marginal prior distribution of the calibration node. We illustrate this with a number of analytical results on small trees. We also describe an alternative construction for a calibrated Yule prior on trees that allows direct specification of the marginal prior distribution of the calibrated divergence time, with or without the restriction of monophyly. This method requires the computation of the Yule prior conditional on the height of the divergence being calibrated. Unfortunately, a practical solution for multiple calibrations remains elusive. Our results suggest that direct estimation of the prior induced by specifying multiple calibration densities should be a prerequisite of any divergence time dating analysis.     

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.     

Identifying the rooted species tree from the distribution of unrooted gene trees under the coalescent

Gene trees are evolutionary trees representing the ancestry of genes sampled from multiple populations. Species trees represent populations of individuals—each with many genes—splitting into new populations or species. The coalescent process, which models ancestry of gene copies within populations, is often used to model the probability distribution of gene trees given a fixed species tree. This multispecies coalescent model provides a framework for phylogeneticists to infer species trees from gene trees using maximum likelihood or Bayesian approaches. Because the coalescent models a branching process over time, all trees are typically assumed to be rooted in this setting. Often, however, gene trees inferred by traditional phylogenetic methods are unrooted. We investigate probabilities of unrooted gene trees under the multispecies coalescent model. We show that when there are four species with one gene sampled per species, the distribution of unrooted gene tree topologies identifies the unrooted species tree topology and some, but not all, information in the species tree edges (branch lengths). The location of the root on the species tree is not identifiable in this situation. However, for 5 or more species with one gene sampled per species, we show that the distribution of unrooted gene tree topologies identifies the rooted species tree topology and all its internal branch lengths. The length of any pendant branch leading to a leaf of the species tree is also identifiable for any species from which more than one gene is sampled.     

Quantification of prior impact in terms of effective current sample size

Bayesian methods allow borrowing of historical information through prior distributions. The concept of prior effective sample size (prior ESS) facilitates quantification and communication of such prior information by equating it to a sample size. Prior information can arise from historical observations; thus, the traditional approach identifies the ESS with such a historical sample size. However, this measure is independent of newly observed data, and thus would not capture an actual “loss of information” induced by the prior in case of prior-data conflict. We build on a recent work to relate prior impact to the number of (virtual) samples from the current data model and introduce the effective current sample size (ECSS) of a prior, tailored to the application in Bayesian clinical trial designs. Special emphasis is put on robust mixture, power, and commensurate priors. We apply the approach to an adaptive design in which the number of recruited patients is adjusted depending on the effective sample size at an interim analysis. We argue that the ECSS is the appropriate measure in this case, as the aim is to save current (as opposed to historical) patients from recruitment. Furthermore, the ECSS can help overcome lack of consensus in the ESS assessment of mixture priors and can, more broadly, provide further insights into the impact of priors. An R package accompanies the paper.     

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.     

New Analytic Results for Speciation Times in Neutral Models

In this paper, we investigate the standard Yule model, and a recently studied model of speciation and extinction, the “critical branching process.” We develop an analytic way—as opposed to the common simulation approach—for calculating the speciation times in a reconstructed phylogenetic tree. Simple expressions for the density and the moments of the speciation times are obtained. Methods for dating a speciation event become valuable, if for the reconstructed phylogenetic trees, no time scale is available. A missing time scale could be due to supertree methods, morphological data, or molecular data which violates the molecular clock. Our analytic approach is, in particular, useful for the model with extinction, since simulations of birth-death processes which are conditioned on obtaining n extant species today are quite delicate. Further, simulations are very time consuming for big n under both models.     

Molecular dating for phylogenies containing a mix of populations and species by using Bayesian and RelTime approaches

Simultaneous molecular dating of population and species divergences is essential in many biological investigations, including phylogeography, phylodynamics and species delimitation studies. In these investigations, multiple sequence alignments consist of both intra‐ and interspecies samples (mixed samples). As a result, the phylogenetic trees contain interspecies, interpopulation and within‐population divergences. Bayesian relaxed clock methods are often employed in these analyses, but they assume the same tree prior for both inter‐ and intraspecies branching processes and require specification of a clock model for branch rates (independent vs. autocorrelated rates models). We evaluated the impact of a single tree prior on Bayesian divergence time estimates by analysing computer‐simulated data sets. We also examined the effect of the assumption of independence of evolutionary rate variation among branches when the branch rates are autocorrelated. Bayesian approach with coalescent tree priors generally produced excellent molecular dates and highest posterior densities with high coverage probabilities. We also evaluated the performance of a non‐Bayesian method, RelTime, which does not require the specification of a tree prior or a clock model. RelTime's performance was similar to that of the Bayesian approach, suggesting that it is also suitable to analyse data sets containing both populations and species variation when its computational efficiency is needed.     

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%.      

Practical considerations for using functional uniform prior distributions for dose‐response estimation in clinical trials

Estimating nonlinear dose‐response relationships in the context of pharmaceutical clinical trials is often a challenging problem. The data in these trials are typically variable and sparse, making this a hard inference problem, despite sometimes seemingly large sample sizes. Maximum likelihood estimates often fail to exist in these situations, while for Bayesian methods, prior selection becomes a delicate issue when no carefully elicited prior is available, as the posterior distribution will often be sensitive to the priors chosen. This article provides guidance on the usage of functional uniform prior distributions in these situations. The essential idea of functional uniform priors is to employ a distribution that weights the functional shapes of the nonlinear regression function equally. By doing so one obtains a distribution that exhaustively and uniformly covers the underlying potential shapes of the nonlinear function. On the parameter scale these priors will often result in quite nonuniform prior distributions. This paper gives hints on how to implement these priors in practice and illustrates them in realistic trial examples in the context of Phase II dose‐response trials as well as Phase I first‐in‐human studies.     

Elicitation of a beta prior for Bayesian inference in clinical trials

When making Bayesian inferences we need to elicit an expert's opinion to set up the prior distribution. For applications in clinical trials, we study this problem with binary variables. A critical and often ignored issue in the process of eliciting priors in clinical trials is that medical investigators can seldom specify the prior quantities with precision. In this paper, we discuss several methods of eliciting beta priors from clinical information, and we use simulations to conduct sensitivity analyses of the effect of imprecise assessment of the prior information. These results provide useful guidance for choosing methods of eliciting the prior information in practice.     

The devil in the details: interactions between the branch-length prior and likelihood model affect node support and branch lengths in the phylogeny of the Psoraceae

In popular use of Bayesian phylogenetics, a default branch-length prior is almost universally applied without knowing how a different prior would have affected the outcome. We performed Bayesian and maximum likelihood (ML) inference of phylogeny based on empirical nucleotide sequence data from a family of lichenized ascomycetes, the Psoraceae, the morphological delimitation of which has been controversial. We specifically assessed the influence of the combination of Bayesian branch-length prior and likelihood model on the properties of the Markov chain Monte Carlo tree sample, including node support, branch lengths, and taxon stability. Data included two regions of the mitochondrial ribosomal RNA gene, the internal transcribed spacer region of the nuclear ribosomal RNA gene, and the protein-coding largest subunit of RNA polymerase II. Data partitioning was performed using Bayes' factors, whereas the best-fitting model of each partition was selected using the Bayesian information criterion (BIC). Given the data and model, short Bayesian branch-length priors generate higher numbers of strongly supported nodes as well as short and topologically similar trees sampled from parts of tree space that are largely unexplored by the ML bootstrap. Long branch-length priors generate fewer strongly supported nodes and longer and more dissimilar trees that are sampled mostly from inside the range of tree space sampled by the ML bootstrap. Priors near the ML distribution of branch lengths generate the best marginal likelihood and the highest frequency of "rogue" (unstable) taxa. The branch-length prior was shown to interact with the likelihood model. Trees inferred under complex partitioned models are more affected by the stretching effect of the branch-length prior. Fewer nodes are strongly supported under a complex model given the same branch-length prior. Irrespective of model, internal branches make up a larger proportion of total tree length under the shortest branch-length priors compared with longer priors. Relative effects on branch lengths caused by the branch-length prior can be problematic to downstream phylogenetic comparative methods making use of the branch lengths. Furthermore, given the same branch-length prior, trees are on average more dissimilar under a simple unpartitioned model compared with a more complex partitioned models. The distribution of ML branch lengths was shown to better fit a gamma or Pareto distribution than an exponential one. Model adequacy tests indicate that the best-fitting model selected by the BIC is insufficient for describing data patterns in 5 of 8 partitions. More general substitution models are required to explain the data in three of these partitions, one of which also requires nonstationarity. The two mitochondrial ribosomal RNA gene partitions need heterotachous models. We found no significant correlations between, on the one hand, the amount of ambiguous data or the smallest branch-length distance to another taxon and, on the other hand, the topological stability of individual taxa. Integrating over several exponentially distributed means under the best-fitting model, node support for the family Psoraceae, including Psora, Protoblastenia, and the Micarea sylvicola group, is approximately 0.96. Support for the genus Psora is distinctly lower, but we found no evidence to contradict the current classification.     

Potential applications and pitfalls of Bayesian inference of phylogeny

Only recently has Bayesian inference of phylogeny been proposed. The method is now a practical alternative to the other methods; indeed, the method appears to possess advantages over the other methods in terms of ability to use complex models of evolution, ease of interpretation of the results, and computational efficiency. However, the method should be used cautiously. The results of a Bayesian analysis should be examined with respect to the sensitivity of the results to the priors used and the reliability of the Markov chain Monte Carlo approximation of the probabilities of trees.