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

Abstract

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.

Full text

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Syst. Biol. 53(6):904–913, 2004
Copyright c ? Society of Systematic Biologists
ISSN: 1063-5157 print / 1076-836X online
DOI: 10.1080/10635150490522629
Frequentist Properties of Bayesian Posterior Probabilities of Phylogenetic Trees Under
Simple and Complex Substitution Models
JOHN P. HUELSENBECK1AND BRUCE RANNALA2
1Section of Ecology, Behavior and Evolution, Division of Biological Sciences, University of California, San Diego, La Jolla, California 92093-0116, USA
E-mail: johnh@biomail.ucsd.edu
2Genome Center & Section of Evolution and Ecology, University of California Davis, One Shields Avenue, Davis, California 95616, USA
Abstract.— What does the posterior probability of a phylogenetic tree mean? This simulation study shows that Bayesian
posteriorprobabilitieshavethemeaningthatistypicallyascribedtothem;theposteriorprobabilityofatreeistheprobability
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
inferredtreesviaeitherbootstrapping(formaximumlikelihoodestimates)orposteriorprobabilities(forBayesianestimates)
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.
[Bayesian estimation; Markov chain Monte Carlo; posterior probability; prior probability.]
Quantifyingtheuncertaintyofaphylogeneticestimate
is at least as important a goal as obtaining the phyloge-
netic estimate itself. Measures of phylogenetic reliability
not only point out what parts of a tree can be trusted
wheninterpretingtheevolutionofagroup,butcanguide
future efforts in data collection that can help resolve re-
maining uncertainties. The reliability of a tree can be as-
sessedusingfrequentistandBayesianapproaches.Inthe
frequentistapproach,thereliabilityofaphylogenetictree
could be described by a confidence set of trees, in which
the confidence level or coverage probability would be
100(1 − α)%(hereαisanumberbetween0and1thatcon-
trols the size of the confidence set of trees). The interpre-
tation of a 95% (α = 0.05) confidence set of trees (simply
a list of trees) requires a thought experiment: if we could
takerandomsamplesofthesamesizeastheoriginaldata
matrixandcreateconfidencesetsoftreesinthesameway
foreachsample,then95%ofthoseconfidencesetswould
containthetruetree.Analternativeinterpretationisthat
if an investigator calculated a 100(1 − α)% confidence
set of trees every time he or she performed a phyloge-
neticanalysis,thenatmostaproportionα ofthemwould
not contain the true tree over the investigator’s lifetime.
In practice, phylogeneticists do not calculate confidence
sets of trees, but assign measures of phylogenetic relia-
bility to individual branches of a phylogenetic tree. The
general idea, however, is the same; if the investigator’s
confidence in a particular group is 95%, then ideally the
groupingoftaxawouldbeincorrect5%ofthetime(onre-
peated sampling). Assessing the uncertainty in individ-
ual clades on a tree has the benefit that it allows the sys-
tematist to evaluate specific hypotheses of monophyly.
The task of developing good frequentist measures
of phylogenetic reliability, however, has proven par-
ticularly difficult, and has resulted in many different
approaches being taken. One group of methods calcu-
lates a test statistic for a branch on the phylogeny, such
as the Bremer decay index (Bremer, 1988, 1994), and
then uses permutation procedures to generate a null
distribution for the test statistic. The T-PTP test (Faith,
1991), for example, follows this strategy and aims at
testing the hypothesis of monophyly for a group. The
problem with this approach, however, is that the null
distribution for the test statistic has nothing to do with
the null hypothesis of monophyly (Swofford et al., 1996;
DeBry, 2001); if a group truly were monophyletic, then
one would not expect the observations supporting that
monophyly to be random. An alternative method for
assessingthereliabilityofaphylogeneticgroupinvolves
testing the null hypothesis that the branch supporting
the monophyly of the group is zero in length. Although
this type of test is clearly related to the support of a
group (a long branch subtending a group on a tree
means that many characters evolved on that branch), it
does not directly assess the reliability of a group.
The most widely used, and generally accepted,
method for assessing phylogenetic reliability is the non-
parametric bootstrap method (Efron, 1979; Efron and
Tibshirani, 1993) introduced to the phylogenetics litera-
ture by Felsenstein (1985). The nonparametric bootstrap
method constructs new bootstrapped data matrices of
the same size as the original data matrix by randomly
sampling characters (e.g., columns in an alignment of
DNA sequences) with replacement. The same method
of estimating phylogeny that was applied in the analysis
of the original data matrix is then applied to each of the
bootstrapped data matrices. The fraction of the time a
particular clade appears in analysis of the bootstrapped
data sets represents the confidence value for that group-
ing of taxa; if a group appeared in 88% of the analyses
of the bootstrapped data matrices, then its confidence
level is approximately 88%. The jackknife method is
similar in spirit to the bootstrap method (Mueller and
Ayala, 1982; Farris et al., 1996). Instead of resampling
the original data matrix with replacement, however,
a fixed number of characters (columns) are randomly
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HUELSENBECK AND RANNALA—POSTERIOR PROBABILITIES
905
deleted from the original analysis in construction of the
jackknifed data matrices. Each of these jackknifed data
matrices is then analyzed using the same phylogenetic
method as was applied to the original data. Although
the end goal of the jackknife method is the same as the
bootstrapmethod,itisnotclearwhatfractionofthesites
should be dropped in any particular analysis so that
the jackknife fraction for a group matches the desired
confidence level (Farris et al., 1996; Felsenstein, 2003).
There are three problems with using the nonparamet-
ric bootstrap method to assess phylogenetic reliability.
None of these problems is necessarily fatal, but taken
together they greatly complicate the application of the
bootstrap method in phylogenetics and the interpreta-
tion of bootstrap proportions on trees. The first problem
concerns the computational complexity of the method,
especially when applied with the method of maximum
likelihood to estimate phylogeny. Simply put, bootstrap
analyses can be computationally prohibitive when the
models of sequence analysis are complicated or the data
sets are large. This problem can be alleviated by clever
programming and fast computers, but bootstrap anal-
yses will always take many times longer than analysis
of the original data. Another problem is that it is diffi-
cult to apply the bootstrap method under some sorts of
models. For example, the standard bootstrap method in
which individual sites are sampled is inappropriate for
models that incorporate correlation in rates (Yang, 1995;
Felsenstein and Churchill, 1996) because the sampling
procedure destroys the autocorrelation structure of rates
in the bootstrap matrices. Potentially, this problem can
be dealt with by sampling blocks of characters (K¨ unsch,
1989), but this solution remains unexplored. Finally, the
interpretation of the bootstrap proportion as applied in
phylogenetics is problematic.
In most statistical applications, the bootstrap proce-
dure is used to approximate the sampling distribution
of a statistic. We imagine that we have some observa-
tions from which we estimate the value of some param-
eter that has a true value of θT. The estimated value of
the parameter is denotedˆθ. The observations are treated
as random variables (variables that take their values by
chance), so that the parameter, which is a function of
the observations, is also a random variable.The proba-
bility distribution of the parameter estimate is called the
sampling distribution, and several very useful quanti-
ties can be calculated from it. Importantly, the sampling
distribution can be used to assess the variability in an
estimate. For example, one can construct a confidence
interval, which is a range of parameter values that con-
tains the true value of the parameter with some prespec-
ified coverage probability (usually 95%). The bootstrap
provides a computer intensive, but straightforward way
to approximate the sampling distribution of a parame-
ter even when the modeling assumptions are quite com-
plex. Unfortunately, the application of the bootstrap in
phylogenetics is not as simple as it is for most statistical
problems. One of the main problems is that there is not a
natural way to measure the variability of trees (Holmes,
2003). Another problem is that the parameter estimates
change discontinously; small changes in the data can re-
sult in different tree topologies being chosen as best. The
bootstrap proportions have been variously interpreted
as the probability that a clade is correct (a notion ex-
plored by Hillis and Bull, 1993), the robustness of the re-
sultsofaphylogeneticanalysistoperturbation(Holmes,
2003), and the probability of incorrectly rejecting a hy-
pothesis of monophyly (Felsenstein and Kishino, 1993).
Hillis and Bull (1993) performed a simulation study that
investigated the coverage probability of the bootstrap
proportion. In their simulation analyses, the bootstrap
proportion for a clade usually underestimated the true
probability that the clade is correct. The general results
of the Hillis and Bull study have been replicated in other
studies (e.g., Alfaro et al., 2002; Zharkikh and Li, 1992a,
1992b). The bootstrap proportions can be interpreted in
ahypothesistestingframework.Inthiscase,thehypoth-
esis to be tested is one of monophyly for a group. One
minus the bootstrap proportion for a clade can be inter-
preted as the probability of incorrectly rejecting the hy-
pothesis of monophyly. The iterated bootstrap (Rodrigo,
1993),full-and-partialbootstrap(ZharkikhandLi,1995),
and corrected bootstrap of Efron et al. (1996) all attempt
to improve the accuracy of the bootstrap as an estimate
ofoneminustheprobabilityofincorrectlyrejectingahy-
pothesis of monophyly (Sanderson and Wojciechowski,
2000). These correction procedures, however, add con-
siderablecomplexitytothebootstrapprocedure.Todate,
only Sanderson and Wojciechowski (2000) have applied
the corrected bootstrap, and only to a single clade. Al-
though the best interpretation of the bootstrap propor-
tion on a phylogenetic tree appears to be in a hypothe-
sis testing framework, where the bootstrap proportion is
related to the type I error, most practicing systematists
appear to interpret bootstrap support as the probability
thatthecladeiscorrect.Althoughthisinterpretationand
thehypothesistestingoneareclearlyrelated(ahighboot-
strapproportionshouldbeassociatedwithcorrectclades
more often than low bootstrap proportions), the ques-
tions are different. To our knowledge, only the Bayesian
approach directly addresses the probability that a clade
is correct, conditional on the observations.
The methods discussed above all use the frequen-
tist interpretation of probability. The complication in
frequentist statistics is that the parameter (e.g., phylo-
genetic tree) is considered to take a fixed but unknown
value; the parameter is not treated as a random variable,
and hence cannot be directly assigned a probability. To
obtain an idea of the variability of an estimated phy-
logeny, then, one must resort to the thought experiment
of sampling data sets and reconstructing phylogeny on
each. The distribution of phylogenetic trees obtained in
this manner is an approximation of the sampling distri-
bution of phylogeny. In a Bayesian analysis, on the other
hand, the parameters are treated as random variables
and can be directly assigned probabilities. Bayesian in-
ferenceofphylogenysuggestsanaturalwaytoassessthe
uncertainty in a phylogeny: the probability that a tree is
correctissimplytheposteriorprobabilityofthetree.The
posterior probability of a tree is conditioned on the data

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SYSTEMATIC BIOLOGYVOL. 53
and the model used in the analysis both being correct.
The use of posterior probabilities has some intuitive ap-
peal. For one, the interpretation of posterior probabili-
ties is direct and simple; one does not need to invoke a
thoughtexperimentofrepeatedsamplingtointerpretthe
results of a Bayesian analysis. More practically, posterior
probabilities can be approximated using Markov chain
MonteCarloevenundercomplexmodelsofsubstitution
in a fraction of the time that would be required for max-
imum likelihood bootstrapping (the closest analog to a
Bayesiananalysisofphylogeny;LargetandSimon,1999).
The proper interpretation of posterior probabilities on
trees has recently attracted a lot of attention from biolo-
gists. Many systematists have noted that posterior prob-
abilities on clades tend to be higher than the bootstrap
proportions on the same clades (Douady et al., 2003;
Erixon et al., 2003; but see Cummings et al., 2003), and
Murphy et al. (2001) speculated that posterior probabil-
ities do not suffer from being too conservative like the
nonparametric bootstrap (Hillis and Bull, 1993). The re-
sults of simulation studies comparing posterior proba-
bilities and bootstrap proportions have been mixed. One
important problem that can be addressed in simulation
is the robustness of posterior probabilities to violation
of model assumptions. This can be done by simulating
data under a complicated model of evolution, and then
usinganincorrect(oversimplified)modelofevolutionin
the Bayesian analysis. Suzuki et al. (2002) took this ap-
proach and argued that posterior probabilities are more
sensitive to model misspecification than the bootstrap
method. Their method for violating the assumptions of
the method, however, was extreme; they simulated data
sets on the three possible trees for four species, and then
concatenated the data matrices. This means that each
third of the concatenated data set had a different under-
lyingphylogenetictree.TheBayesiananalysis,however,
assumed a common tree for the data matrix. Although
there are many cases in which evolutionary biologists
think that different phylogenies may underlie different
parts of the genome (e.g., for coalescence trees), this type
of model violation is extreme and does not mimic the
more universal concern that the substitution model as-
sumed in the analysis is incorrect. Indeed, analysis of
model adequacy suggests that models currently used in
phylogenetic analysis fail to capture important evolu-
tionary processes (Goldman, 1993). It is also important
to analyze the behavior of Bayesian posterior probabili-
ties (and other methods for assessing phylogenetic reli-
ability) when all of the assumptions of the analysis are
satisfied. This gives a “best case” picture of the statis-
tical behavior of a method. It is already known that it
is difficult to interpret the uncorrected bootstrap sup-
port for a clade as the probability that the clade is cor-
rect (Hillis and Bull, 1993; Holmes, 2003), even when all
of the assumptions of the method are satisfied. A few
studieshaveexaminedthestatisticalpropertiesofposte-
rior probabilities when the assumptions of the Bayesian
analysis are satisfied. Lemmon and Moriarty (2004) ex-
aminedthebehaviorofposteriorprobabilitiesinsimula-
tion. They examined cases in which the Bayesian model
was over- and underspecified. In general, underspecifi-
cation of the phylogenetic modelled to biased estimates
ofparameters.Alfaroetal.(2003)andWilcoxetal.(2002)
compared posterior probabilities and bootstrap propor-
tionsinsimulationandfoundthatposteriorprobabilities
gaveamoreaccuraterepresentationofphylogeneticcon-
fidence than the bootstrap method (when the assump-
tionsofthemethodweresatisfied).Ingeneral,aposterior
probability of, say, 0.91 was more likely to correspond to
a probability that the tree was correct of 0.91 than the
bootstrap method. Importantly, the posterior probabili-
ties did not perfectly match the probability that a clade
is correct in these simulations. For example, Wilcox et al.
(2002: 369) point out that “...Bayesian support values
provide much closer estimates of phylogenetic accuracy
(even though they are still somewhat conservative) than
the estimates provided by corresponding bootstrap pro-
portions.” A similar result is seen in Figure 4 of Alfaro
etal.(2003).Thisresultisworrisome,becauseitsuggests
that posterior probabilities may not have the meaning
that is ascribed to them (i.e., that the posterior probabil-
ity of a clade is the probability that the clade is correct).
In this study, we examine the statistical properties
of Bayesian posterior probabilities on small (six taxon)
trees. We examine the behavior of the method when all
of the assumptions are satisfied, and also when the as-
sumptionsofthemethodareviolated.Weshowthatpos-
terior probabilities do have a well-defined and easily in-
terpreted meaning when the assumptions of the method
are satisfied. We also show that posterior probabilities
can be sensitive to model misspecification, suggesting
that care should be taken to use models of sequence evo-
lutionthatareasrealisticaspossibleinBayesiananalysis.
METHODS
We evaluate the statistical properties of posterior
probabilities using computer simulation. In a tradi-
tional computer simulation, one fixes parameters of
the phylogenetic model or varies them systematically
over a parameter space (e.g., Huelsenbeck and Hillis,
1993; Huelsenbeck, 1995). For example, one might
decide to evaluate the behavior of some phylogenetic
method on many data matrices that were simulated on a
common tree and set of branch lengths. This traditional
simulation procedure is how earlier studies examined
thepropertiesofBayesianposteriorprobabilities(Alfaro
et al., 2003; Suzuki et al., 2002; Wilcox et al., 2002).
However, if the goal is to study posterior probabilities
when all of the assumptions of the Bayesian analysis
are satisfied, then the traditional approach cannot be
used. The model in a Bayesian analysis of phylogeny
has two parts: One part involves assumptions about
how the substitutions occur on the tree; the other part of
the model describes the prior probability distribution
of the parameters. The traditional simulation approach
doesnottreattheprioroftheBayesiananalysisseriously.
In effect, a traditional simulation treats the parameters
as fixed, whereas a Bayesian analysis treats the pa-
rameters as random variables. We use a modification

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HUELSENBECK AND RANNALA—POSTERIOR PROBABILITIES
907
FIGURE 1.
of the evolutionary model (such as the tree, branch lengths, and sub-
stitution model parameters), simulating many data matrices on the
fixed tree. The procedure used in this study first draws the tree, branch
lengths, and substitution model parameters from the prior probability
distribution before simulating data. This is done for each replicate in
the simulation.
of the traditional approach, drawing the phylogenetic
parameters such as the tree and branch lengths from
a probability distribution and then simulating a single
data matrix on every such draw from the prior (Fig. 1).
In other words, instead of fixing the parameters of the
simulation, we fix the prior probability distribution
from which the phylogenetic parameters are drawn.
We simulated data under the JC69 (Jukes and Cantor,
1969)modelandunderthegeneraltimereversiblemodel
with gamma distributed rate variation (GTR+?; Tavar´ e,
1986; Yang, 1993, 1994). The JC69 model is very simple,
assumingthattherateofsubstitutionisequalacrosssites,
that the four nucleotide frequencies are the same, and
that the rates of change among nucleotides are equal.
The GTR+? model is considerably more complex. The
GTR+? model of DNA substitution allows the rates of
substitution to differ among sites in the sequence (they
are random variables drawn from a gamma distribution
withshapeparametera),thenucleotidefrequenciestobe
different, and the rate of substitution among nucleotides
to be different. We simulated data under the JC69 and
GTR+? models,butanalyzedthedataunderthesemod-
els as well as many of the models in between the JC69
and GTR+? models. Table 1 shows the models used in
this study and the prior probability distribution of the
parameters for each.
We simulated data matrices of c = 100, c = 500, and
c = 1000sitesontreesofsixspecies.Wechosetoexamine
The traditional simulation procedure fixes parameters
the six species case for a number of reasons. For one, six
species is the smallest number of species for which there
are several tree shapes for unrooted trees. More impor-
tantly, there are only 105 possible unrooted trees for six
species, and the posterior probability of each can be rea-
sonablyapproximatedusingMarkovchainMonteCarlo.
Ifwehaddecidedtoexaminemorespecies,thenthepos-
terior probabilities of individual trees would have been
more difficult to accurately estimate. We used MrBayes
v3.0 (Huelsenbeck and Ronquist, 2001) to approximate
posteriorprobabilitiesusingMarkovchainMonteCarlo.
We ran a single Markov chain for 200,000 cycles for each
simulated data matrix, discarding samples taken during
the first 50,000 cycles. For each set of model conditions,
we simulated a total of 10,000 matrices.
Consider just one of the simulations that was per-
formed in this study. In one set of simulations, we sim-
ulated data matrices of c = 100 sites under the GTR+?
model of DNA substitution and analyzed the simulated
data matrices under the JC69 model. Each of the 10,000
simulated data matrices was generated as follows: First,
we picked an unrooted tree from the prior probability
distributionoftrees.Weconsideredalltreestobeequally
probable. Hence, each of the 105 trees had a probability
of 1/105 of being chosen. Second, once the tree was cho-
sen, we assigned branch lengths to the tree. The branch
lengths were randomly assigned by drawing each from
an exponential distribution with parameter λ = 5. The
exponential distribution has parameter λ. The mean of
the exponential is 1/λ and the variance is 1/λ2. Hence,
the average branch length on the trees in this study was
1/5 = 0.2expectedsubstitutionspersite.Third,wechose
the nucleotide frequencies from a Dirichlet (5, 5, 5, 5)
distribution. Fourth, we chose the gamma-shape param-
eter for among-site rate variation from an exponential
distribution with parameter 2 (resulting in an average
shape parameter of a = 0.5 over simulations). Fifth, we
chose the rates of substitution by generating six inde-
pendent exponential random variables with parameter
1. These six exponential random variables correspond
to the rates rAC, rAG, rAT, rCG, rCT, and rGT. Sixth, after
the parameters of the model had been chosen, 100 sites
were simulated on the tree. Finally, a Bayesian analysis
was performed on the simulated data matrix by running
a Markov chain for 200,000 cycles. The output from the
Markov chain Monte Carlo allowed the posterior proba-
bilitiesofindividualtreestobeapproximated.Ofcourse,
because the tree is known, we can ask whether a high
posterior probability also corresponded to a high proba-
bility that the tree was correct. The entire procedure de-
scribedherewasrepeated10,000times.Notethatnotwo
of the 10,000 simulated data matrices had precisely the
same parameter values (i.e., they may have differed in
topologybutcertainlydifferedinthebranchlengthsand
the substitution model parameters). Each of the figures
thatsummarizetheresultsofthesimulationsisbasedon
105 × 10000 = 1050000 data points, and these million
plus data points were assigned (according to posterior
probability) to 20 bins.

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SYSTEMATIC BIOLOGYVOL. 53
TABLE 1.
probability distribution from which parameters for simulated data sets were drawn, or the settings for the Bayesian analysis of the data. (JC69:
Jukes and Cantor, 1969; F81: Felsenstein, 1981; SYM: symmetric model; GTR: general time reversible model, Tavar´ e, 1986.)
Parameter settings for the models examined in this study. The prior probability distributions for the parameters specify either the
Model Nucleotide frequenciesSubstitution ratesGamma rate variation TopologyBranch lengths
JC69
F81
SYM
JC69+?
SYM+?
F81+?
GTR
GTR+?
Fixed (1/4, 1/4, 1/4, 1/4)
Dirichlet (5, 5, 5, 5)
Fixed (1/4, 1/4, 1/4, 1/4)
Fixed (1/4, 1/4, 1/4, 1/4)
Fixed (1/4, 1/4, 1/4, 1/4)
Dirichlet (5, 5, 5, 5)
Dirichlet (5, 5, 5, 5)
Dirichlet (5, 5, 5, 5)
Fixed (1, 1, 1, 1, 1, 1)
Fixed (1, 1, 1, 1, 1, 1)
Dirichlet (1, 1, 1, 1, 1, 1)
Fixed (1, 1, 1, 1, 1, 1)
Dirichlet (1, 1, 1, 1, 1, 1)
Fixed (1, 1, 1, 1, 1, 1)
Dirichlet (1, 1, 1, 1, 1, 1)
Dirichlet (1, 1, 1, 1, 1, 1)
Fixed (∞)
Fixed (∞)
Fixed (∞)
Exponential (2)
Exponential (2)
Exponential (2)
Fixed (∞)
Exponential (2)
Uniform
Uniform
Uniform
Uniform
Uniform
Uniform
Uniform
Uniform
Exponential (5)
Exponential (5)
Exponential (5)
Exponential (5)
Exponential (5)
Exponential (5)
Exponential (5)
Exponential (5)
RESULTS AND DISCUSSION
The Meaning of Posterior Probabilities
In the best-case scenario in which all of the assump-
tions of the Bayesian analysis are satisfied, the posterior
probabilityofatreeisequaltotheprobabilitythatthetree
is correct. Figure 2 shows the relationship between the
posterior probability for a phylogenetic tree and the fre-
quency at which trees with that posterior probability are
correct (identical to the true tree used in the simulation)
forthefoursimulationsthatwereperformedinwhichall
of the assumptions of the Bayesian analysis are satisfied.
Forlargenumbersofreplicatesimulations,thefrequency
of correct trees will approximate the “frequentist” prob-
FIGURE 2.
Bayesian analysis are satisfied.
The relationship between posterior probabilities on trees and the probability that the tree is correct when the assumptions of the
ability that a tree is correct. Thus, we subsequently refer
to the frequency of correct trees as the probability the
tree is correct in all figures. The relationship is linear,
indicating that posterior probabilities have the meaning
ascribed to them: the posterior probability of a tree is the
probability that the tree is correct (assuming that the model
is correct). Importantly, the Bayesian method is currently
the only phylogenetic method that has this property.
Robustness of Posterior Probabilities
Two sets of simulations were performed in which the
assumptions of the Bayesian analysis were violated. In
the first set, the assumptions of the Bayesian analysis