The conflation of ignorance and knowledge in the inference of clade posteriors

The objective Bayesian approach relies on the construction of prior distributions that reflect ignorance. When topologies are considered equally probable a priori, clades cannot be. Shifting justifications have been offered for the use of uniform topological priors in Bayesian inference. These include: (i) topological priors do not inappropriately influence Bayesian inference when they are uniform; (ii) although clade priors are not uniform, their undesirable influence is negated by the likelihood function, even when data sets are small; and (iii) the influence of nonuniform clade priors is an appropriate reflection of knowledge. The first two justifications have been addressed previously: the first is false, and the second was found to be questionable. The third and most recent justification is inconsistent with the first two, and with the objective Bayesian philosophy itself. Thus, there has been no coherent justification for the use of nonflat clade priors in Bayesian phylogenetics. We discuss several solutions: (i) Bayesian inference can be abandoned in favour of other methods of phylogenetic inference; (ii) the objective Bayesian philosophy can be abandoned in favour of a subjective interpretation; (iii) the topology with the greatest posterior probability, which is also the tree of greatest marginal likelihood, can be accepted as optimal, with clade support estimated using other means; or (iv) a Bayes factor, which accounts for differences in priors among competing hypotheses, can be used to assess the weight of evidence in support of clades.
©The Willi Hennig Society 2009     

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.     

How meaningful are Bayesian support values?

In this study, we used an empirical example based on 100 mitochondrial genomes from higher teleost fishes to compare the accuracy of parsimony-based jackknife values with Bayesian support values. Phylogenetic analyses of 366 partitions, using differential taxon and character sampling from the entire data matrix of 100 taxa and 7,990 characters, were performed for both phylogenetic methods. The tree topology and branch-support values from each partition were compared with the tree inferred from all taxa and characters. Using this approach, we quantified the accuracy of the branch-support values assigned by the jackknife and Bayesian methods, with respect to each of 15 basal clades. In comparing the jackknife and Bayesian methods, we found that (1) both measures of support differ significantly from an ideal support index; (2) the jackknife underestimated support values; (3) the Bayesian method consistently overestimated support; (4) the magnitude by which Bayesian values overestimate support exceeds the magnitude by which the jackknife underestimates support; and (5) both methods performed poorly when taxon sampling was increased and character sampling was not increases. These results indicate that (1) the higher Bayesian support values are inappropriate (in magnitude), and (2) Bayesian support values should not be interpreted as probabilities that clades are correctly resolved. We advocate the continued use of the relatively conservative bootstrap and jackknife approaches to estimating branch support rather than the more extreme overestimates provided by the Markov Chain Monte Carlo-based Bayesian methods.     

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.     

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.     

Comparing bootstrap and posterior probability values in the four-taxon case

Assessment of the reliability of a given phylogenetic hypothesis is an important step in phylogenetic analysis. Historically, the nonparametric bootstrap procedure has been the most frequently used method for assessing the support for specific phylogenetic relationships. The recent employment of Bayesian methods for phylogenetic inference problems has resulted in clade support being expressed in terms of posterior probabilities. We used simulated data and the four-taxon case to explore the relationship between nonparametric bootstrap values (as inferred by maximum likelihood) and posterior probabilities (as inferred by Bayesian analysis). The results suggest a complex association between the two measures. Three general regions of tree space can be identified: (1) the neutral zone, where differences between mean bootstrap and mean posterior probability values are not significant, (2) near the two-branch corner, and (3) deep in the two-branch corner. In the last two regions, significant differences occur between mean bootstrap and mean posterior probability values. Whether bootstrap or posterior probability values are higher depends on the data in support of alternative topologies. Examination of star topologies revealed that both bootstrap and posterior probability values differ significantly from theoretical expectations; in particular, there are more posterior probability values in the range 0.85-1 than expected by theory. Therefore, our results corroborate the findings of others that posterior probability values are excessively high. Our results also suggest that extrapolations from single topology branch-length studies are unlikely to provide any general conclusions regarding the relationship between bootstrap and posterior probability values.     

Evaluating the clade size effect in alternative measures of branch support

The clade size effect refers to a bias that causes middle‐sized clades to be less supported than small or large‐sized clades. This bias is present in resampling measures of support calculated under maximum likelihood and maximum parsimony and in Bayesian posterior probabilities. Previous analyses indicated that the clade size effect is worst in maximum parsimony, followed by maximum likelihood, while Bayesian inference is the least affected. Homoplasy was interpreted as the main cause of the effect. In this study, we explored the presence of the clade size effect in alternative measures of branch support under maximum parsimony: Bremer support and symmetric resampling, expressed as absolute frequencies and frequency differences. Analyses were performed using 50 molecular and morphological matrices. Symmetric resampling showed the same tendency that bootstrap and jackknife did for maximum parsimony and maximum likelihood. Few matrices showed a significant bias using Bremer support, presenting a better performance than resampling measures of support and comparable to Bayesian posterior probabilities. Our results indicate that the problem is not maximum parsimony, but resampling measures of support. We corroborated the role of homoplasy as a possible cause of the clade size effect, increasing the number of random trees during the resampling, which together with the higher chances that medium‐sized clades have of being contradicted generates the bias during the perturbation of the original matrix, making it stronger in resampling measures of support.     

Polytomies and Bayesian phylogenetic inference

Bayesian phylogenetic analyses are now very popular in systematics and molecular evolution because they allow the use of much more realistic models than currently possible with maximum likelihood methods. There are, however, a growing number of examples in which large Bayesian posterior clade probabilities are associated with very short branch lengths and low values for non-Bayesian measures of support such as nonparametric bootstrapping. For the four-taxon case when the true tree is the star phylogeny, Bayesian analyses become increasingly unpredictable in their preference for one of the three possible resolved tree topologies as data set size increases. This leads to the prediction that hard (or near-hard) polytomies in nature will cause unpredictable behavior in Bayesian analyses, with arbitrary resolutions of the polytomy receiving very high posterior probabilities in some cases. We present a simple solution to this problem involving a reversible-jump Markov chain Monte Carlo (MCMC) algorithm that allows exploration of all of tree space, including unresolved tree topologies with one or more polytomies. The reversible-jump MCMC approach allows prior distributions to place some weight on less-resolved tree topologies, which eliminates misleadingly high posteriors associated with arbitrary resolutions of hard polytomies. Fortunately, assigning some prior probability to polytomous tree topologies does not appear to come with a significant cost in terms of the ability to assess the level of support for edges that do exist in the true tree. Methods are discussed for applying arbitrary prior distributions to tree topologies of varying resolution, and an empirical example showing evidence of polytomies is analyzed and discussed.     

Resolving an ancient, rapid radiation in Saxifragales

Despite the prior use of approximately 9000 bp, deep-level relationships within the angiosperm clade, Saxifragales remain enigmatic, due to an ancient, rapid radiation (89.5 to 110 Ma based on the fossil record). To resolve these deep relationships, we constructed several new data sets: (1) 16 genes representing the three genomic compartments within plant cells (2 nuclear, 10 plastid, 4 mitochondrial; aligned, analyzed length = 21,460 bp) for 28 taxa; (2) the entire plastid inverted repeat (IR; 26,625 bp) for 17 taxa; (3) "total evidence" (50,845 bp) for both 17 and 28 taxa (the latter missing the IR). Bayesian and ML methods yielded identical topologies across partitions with most clades receiving high posterior probability (pp = 1.0) and bootstrap (95% to 100%) values, suggesting that with sufficient data, rapid radiations can be resolved. In contrast, parsimony analyses of different partitions yielded conflicting topologies, particularly with respect to the placement of Paeoniaceae, a clade characterized by a long branch. In agreement with published simulations, the addition of characters increased bootstrap support for the putatively erroneous placement of Paeoniaceae. Although having far fewer parsimony-informative sites, slowly evolving plastid genes provided higher resolution and support for deep-level relationships than rapidly evolving plastid genes, yielding a topology close to the Bayesian and ML total evidence tree. The plastid IR region may be an ideal source of slowly evolving genes for resolution of deep-level angiosperm divergences that date to 90 My or more. Rapidly evolving genes provided support for tip relationships not recovered with slowly evolving genes, indicating some complementarity. Age estimates using penalized likelihood with and without age constraints for the 28-taxon, total evidence data set are comparable to fossil dates, whereas estimates based on the 17-taxon data are much older than implied by the fossil record. Hence, sufficient taxon density, and not simply numerous base pairs, is important in reliably estimating ages. Age estimates indicate that the early diversification of Saxifragales occurred rapidly, over a time span as short as 6 million years. Between 25,000 and 50,000 bp were needed to resolve this radiation with high support values. Extrapolating from Saxifragales, a similar number of base pairs may be needed to resolve the many other deep-level radiations of comparable age in angiosperms.     

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.     

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.].     

Testing a molecular clock without an outgroup: derivations of induced priors on branch-length restrictions in a Bayesian framework

We propose a Bayesian method for testing molecular clock hypotheses for use with aligned sequence data from multiple taxa. Our method utilizes a nonreversible nucleotide substitution model to avoid the necessity of specifying either a known tree relating the taxa or an outgroup for rooting the tree. We employ reversible jump Markov chain Monte Carlo to sample from the posterior distribution of the phylogenetic model parameters and conduct hypothesis testing using Bayes factors, the ratio of the posterior to prior odds of competing models. Here, the Bayes factors reflect the relative support of the sequence data for equal rates of evolutionary change between taxa versus unequal rates, averaged over all possible phylogenetic parameters, including the tree and root position. As the molecular clock model is a restriction of the more general unequal rates model, we use the Savage-Dickey ratio to estimate the Bayes factors. The Savage-Dickey ratio provides a convenient approach to calculating Bayes factors in favor of sharp hypotheses. Critical to calculating the Savage-Dickey ratio is a determination of the prior induced on the modeling restrictions. We demonstrate our method on a well-studied mtDNA sequence data set consisting of nine primates. We find strong support against a global molecular clock, but do find support for a local clock among the anthropoids. We provide mathematical derivations of the induced priors on branch length restrictions assuming equally likely trees. These derivations also have more general applicability to the examination of prior assumptions in Bayesian phylogenetics.     

A Phylogenetic Analysis of the Plastid matK Gene with Emphasis on Melanthiaceae sensu lato

Abstract: The presented mat K tree primarily agrees well with the previously presented rbc L tree and combined rbc L + atp B + 18SrDNA tree. According to the mat K tree, the monocotyledons are monophyletic with 100 % bootstrap support. Acorus diverges first from all other monocotyledons (90 % bootstrap support) in which two major clades are recognized: one (89 %) consisting of Alismatanae and Tofieldia (Nartheciaceae), and the other (< 50 %) comprising Lilianae, Commelinanae and Nartheciaceae other than Tofieldia. Within the latter major clade, Petrosavia and Japonolirion (Nartheciaceae) (82 %) diverge first from the remaining taxa (< 50 %) in which two clades are formed: one (81 %) consisting of Pandanales, Dioscoreales and Nartheciaceae-Narthecioideae, and the other (< 50 %) comprising Liliales, Asparagales and Commelinanae. In the former clade, Dioscoreales and Narthecioideae are grouped together (88 %). In the latter clade, Asparagales and Commelinanae are grouped together (< 50 %). Differences between the mat K and rbc L tree topologies appear in the positions of Tricyrtis (Calochortaceae) and Dracaenaceae. Differences between the mat K and combined rbc L + atp B + 18SrDNA tree topologies exist in the positions of the Petrosavia-Japonolirion pair (Nartheciaceae) and Pandanales. The stop codon position of the mat K gene appears to be highly variable among the monocotyledons, especially in the Liliales.     

Implications of uniformly distributed,empirically informed priors for phylogeographical model selection: A reply to Hickerson et al.

Establishing that a set of population‐splitting events occurred at the same time can be a potentially persuasive argument that a common process affected the populations. Recently, Oaks et al. ( 2013 ) assessed the ability of an approximate‐Bayesian model‐choice method (msBayes ) to estimate such a pattern of simultaneous divergence across taxa, to which Hickerson et al. ( 2014 ) responded. Both papers agree that the primary inference enabled by the method is very sensitive to prior assumptions and often erroneously supports shared divergences across taxa when prior uncertainty about divergence times is represented by a uniform distribution. However, the papers differ about the best explanation and solution for this problem. Oaks et al. ( 2013 ) suggested the method's behavior was caused by the strong weight of uniformly distributed priors on divergence times leading to smaller marginal likelihoods (and thus smaller posterior probabilities) of models with more divergence‐time parameters (Hypothesis 1); they proposed alternative prior probability distributions to avoid such strongly weighted posteriors. Hickerson et al. ( 2014 ) suggested numerical‐approximation error causes msBayes analyses to be biased toward models of clustered divergences because the method's rejection algorithm is unable to adequately sample the parameter space of richer models within reasonable computational limits when using broad uniform priors on divergence times (Hypothesis 2). As a potential solution, they proposed a model‐averaging approach that uses narrow, empirically informed uniform priors. Here, we use analyses of simulated and empirical data to demonstrate that the approach of Hickerson et al. ( 2014 ) does not mitigate the method's tendency to erroneously support models of highly clustered divergences, and is dangerous in the sense that the empirically derived uniform priors often exclude from consideration the true values of the divergence‐time parameters. Our results also show that the tendency of msBayes analyses to support models of shared divergences is primarily due to Hypothesis 1, whereas Hypothesis 2 is an untenable explanation for the bias. Overall, this series of papers demonstrates that if our prior assumptions place too much weight in unlikely regions of parameter space such that the exact posterior supports the wrong model of evolutionary history, no amount of computation can rescue our inference. Fortunately, as predicted by fundamental principles of Bayesian model choice, more flexible distributions that accommodate prior uncertainty about parameters without placing excessive weight in vast regions of parameter space with low likelihood increase the method's robustness and power to detect temporal variation in divergences.     

Molecular systematics of Middle American harvest mice Reithrodontomys (Muridae), estimated from mitochondrial cytochrome b gene sequences

We estimated phylogenetic relationships among 16 species of harvest mice using sequences from the mitochondrial cytochrome b (cyt b) gene. Gene phylogenies constructed using maximum parsimony (MP), maximum likelihood (ML) and Bayesian inference (BI) optimality criteria were largely congruent and arranged taxa into two groups corresponding to the two recognized subgenera (Aporodon and Reithrodontomys). All analyses also recovered R. mexicanus and R. microdon as polyphyletic, although greater resolution was obtained using ML and BI approaches. Within R. mexicanus, three clades were identified with high nodal support (MP and ML bootstrap, Bremer decay and Bayesian posterior probabilities). One represented a subspecies of R. mexicanus from Costa Rica (R. m. cherrii) and a second was distributed in the Sierra Madre Oriental of Mexico. The third R. mexicanus clade consisted of mice from southern Mexico southward to South America. Polyphyly between the two moieties of R. microdon corresponded to the Isthmus of Tehuantepec in southern Mexico. Populations of R. microdon microdon to the east of the isthmus (Chiapas, Mexico) grouped with R. tenuirostris, whereas samples of R. m. albilabris to the west in Oaxaca, Mexico, formed a clade with R. bakeri. Within the subgenus Reithrodontomys, all analyses recovered R. montanus and R. raviventris as sister taxa, a finding consistent with earlier studies based on allozymes and cyt b data. There was also strong support (ML and BI criteria) for a clade consisting of ((R. megalotis, R. zacatecae) (R. sumichrasti)). In addition, cytb gene phylogenies (MP, ML, and BI) recovered R. fulvescens and R. hirsutus (ML and BI) as basal taxa within the subgenus Reithrodontomys. Constraint analyses demonstrated that tree topologies treating the two subgenera (Aporodon and Reithrodontomys) as monophyletic (ML criterion) was significantly better (p>0.036) and supported polyphyly of R. mexicanus (both ML and MP criteria - p>0.013) and R. microdon (MP criterion only for certain topologies; p>0.02). Although several species-level taxa were identified based on multiple, independent data sets, we recommended a conservative approach which will involve thorough analyses of museum specimens including material from type localities together with additional sampling and data from multiple, nuclear gene markers.     

From terrestrial to aquatic habitats and back again — molecular insights into the evolution and phylogeny of Hydrophiloidea (Coleoptera) using multigene analyses

The phylogenetic relationships within Hydrophiloidea have been a matter of controversial discussion for many years and the supposedly repeated changes between aquatic and terrestrial lifestyles are not well understood. In order to address these issues we used an extensive molecular data set comprising sequences from six nuclear and mitochondrial genes. The analyses accomplished with the entire data set resulted in largely congruent tree topologies concerning the main branches, independent from the analytical procedures. However, only Bayesian analyses yielded sufficient high posterior probabilities, whereas bootstrap support values for most nodes were generally low. Our results are only partially congruent with hypotheses based on morphological analyses. Spercheidae were placed as the sister group of the remaining hydrophiloid subgroups. Hydrophiloidea excluding Spercheidae split into two clades: the 'helophorid lineage' comprising the small groups Epimetopidae, Hydrochidae, Georissidae and Helophoridae, and the largest family, Hydrophilidae. Within Hydrophilidae, Hydrophilinae do not form a monophylum. The predominantly terrestrial Sphaeridiinae were placed as a subordinate clade within this subfamily. Furthermore, our data suggest a single origin of the aquatic lifestyle in Hydrophiloidea, with numerous secondary changes to terrestrial habits and tertiary changes to aquatic habitats within Sphaeridiinae.     

Improving phylogenetic inference of mushrooms with RPB1 and RPB2 nucleotide sequences (Inocybe; Agaricales)

Approximately 3000 bp across 84 taxa have been analyzed for variable regions of RPB1, RPB2, and nLSU-rDNA to infer phylogenetic relationships in the large ectomycorrhizal mushroom genus Inocybe (Agaricales; Basidiomycota). This study represents the first effort to combine variable regions of RPB1 and RPB2 with nLSU-rDNA for low-level phylogenetic studies in mushroom-forming fungi. Combination of the three loci increases non-parametric bootstrap support, Bayesian posterior probabilities, and resolution for numerous clades compared to separate gene analyses. These data suggest the evolution of at least five major lineages in Inocybe-the Inocybe clade, the Mallocybe clade, the Auritella clade, the Inosperma clade, and the Pseudosperma clade. Additionally, many clades nested within each major lineage are strongly supported. These results also suggest the family Crepiodataceae sensu stricto is sister to Inocybe. Recognition of Inocybe at the family level, the Inocybaceae, is recommended.     

The prior probabilities of phylogenetic trees

Bayesian methods have become among the most popular methods in phylogenetics, but theoretical opposition to this methodology remains. After providing an introduction to Bayesian theory in this context, I attempt to tackle the problem mentioned most often in the literature: the “problem of the priors”—how to assign prior probabilities to tree hypotheses. I first argue that a recent objection—that an appropriate assignment of priors is impossible—is based on a misunderstanding of what ignorance and bias are. I then consider different methods of assigning prior probabilities to trees. I argue that priors need to be derived from an understanding of how distinct taxa have evolved and that the appropriate evolutionary model is captured by the Yule birth–death process. This process leads to a well-known statistical distribution over trees. Though further modifications may be necessary to model more complex aspects of the branching process, they must be modifications to parameters in an underlying Yule model. Ignoring these Yule priors commits a fallacy leading to mistaken inferences both about the trees themselves and about macroevolutionary processes more generally.     

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.     

Bayesian transmogrification of clade divergence dates: a critique

The crucial step in Bayesian dating of phylogenies is the selection of prior probability curves for clade ages. In studies on regions derived from Gondwana, many authors have used steep priors, stipulating that clades can only be a little older than their oldest known fossil. These studies have ruled out vicariance associated with Gondwana breakup, but only because of the particular priors that were adopted. The use of non‐flat priors for fossil‐based ages is not justified and is unnecessary. Tectonic calibrations can be integrated with fossil calibrations that are used to give minimum clade ages only.