© 2005 Society of Systematic Biologists
Branch-Length Prior Influences Bayesian Posterior Probability of Phylogeny
Edited by Paul Lewis: Associate Editor
1 Department of Biology, University College London Darwin Building, Gower Street London WC1E 6BT United Kingdom E-mail: z.yang{at}ucl.ac.uk
2 Genome Center and Section of Evolution and Ecology, University of California Davis One Shields Avenue, Davis, California 95616, U.S.A
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Abstract |
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 Top Abstract Simulation ExperimentSimulation ResultsAnalysis of the Land...DiscussionAcknowledgmentsReferences |
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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.
Keywords: Fair-balance paradox; Lindley's paradox; model selection; molecular phylogenetics; posterior probabilities; prior; star tree paradox
Received May 17, 2004; Revised October 2, 2004; Accepted January 21, 2005
For both reconstruction of phylogenetic relationships and use of phylogenies to understand molecular evolution, it is essential to quantify the statistical uncertainty in inferred phylogenies. Yet the phylogeny differs from a conventional statistical parameter and this difference poses obstacles to straightforward application of statistical estimation theory (Yang et al., 1995). Although maximum likelihood (ML) (Felsenstein, 1981) appears to be efficient for obtaining point estimates of phylogenies, determining statistical confidence has proven more difficult (Goldman et al., 2000).
A recent advance in molecular phylogenetics has been the development of the Bayesian approach (Rannala and Yang, 1996; Mau and Newton, 1997; Li et al., 2000), which circumvents some of the controversies surrounding the nonparametric bootstrap, the most commonly used procedure for assessing phylogenetic uncertainty (Felsenstein, 1985). Implementations of efficient Markov chain Monte Carlo (MCMC) algorithms (Larget and Simon, 1999; Huelsenbeck and Ronquist, 2001) have made the method very popular, and it is now widely used to infer species relationships such as the radiation of mammalian orders (Murphy et al., 2001) or the origin of land plants (Karol et al., 2001). However, posterior probabilities for trees or clades produced by the Bayesian method have often appeared surprisingly high (e.g., Suzuki et al., 2002), as was noted in the very first Bayesian phylogenetic analysis (Rannala and Yang, 1996). Several recent studies comparing posterior probabilities and bootstrap proportions using computer simulation suggest that bootstrap proportions tend to be too conservative, whereas posterior probabilities are too liberal (Suzuki et al., 2002; Cummings et al., 2003; Erixon et al., 2003; Simmons et al., 2004). However, most of those studies are hard to interpret as they did not simulate the trees and branch lengths under the same prior as was used in the Bayesian analysis, and thus theoretical expectations for the results are unavailable.
Here we examine the problem of spuriously high posterior probabilities by studying the simplest case of phylogeny reconstruction, namely estimation of the rooted tree for three species using binary characters evolving at a constant rate (Yang, 2000). The analysis of this simple case does not require the use of MCMC algorithms, and thus computational problems such as lack of convergence and poor mixing of the Markov chain are avoided. To establish the relevance of our analysis of the simple case to real data analysis, we corroborate our results by analyzing an empirical data set concerning the origin of land plants.
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Simulation Experiment |
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TopAbstractSimulation ExperimentSimulation ResultsAnalysis of the Land...DiscussionAcknowledgmentsReferences |
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Bayesian Estimation of Rooted Tree for Three Species
Here we describe our simulation study of the simple case of three species. Analysis of the real data set concerning land plant divergences is described later. Let the three binary rooted trees for species 1, 2, 3 be T1 = ((1, 2), 3), T2 = ((2, 3), 1), and T3 = ((3, 1), 2) (see Fig. 1). Each tree has an internal branch length t0 and an external branch length t1, measured by the expected number of changes per site. The data consist of three sequences of binary characters, evolving according to a continuous-time Markov process with equal substitution rates between the two characters. The molecular clock (rate constancy over time) is also assumed. This is the binary equivalent of the constant-rate Jukes and Cantor (1969) model for nucleotide substitution. For example, we could imagine the two states as representing purines and pyrimidines in a DNA sequence. The sequence data are summarized as the counts of four site patterns: n0 for the constant pattern xxx, and n1, n2, and n3 for the variable patterns xxy, yxx, and xyx, where x and y are two different states. Let n = n0, n1, n2, n3.
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The Bayesian approach to phylogeny estimation places prior distributions on trees and their branch lengths. The prior can represent either objective information or personal beliefs about the parameters before the data are collected and analyzed. We leave it open whether the prior should be interpreted in an objective or subjective Bayesian framework. We assume a uniform prior probability (1/3) for the three trees, and, given the tree topology, exponential priors for t0 and t1: f(t0| µ0) = exp(–t0/µ0)/µ0 and f(t1| µ1) = exp(–t1/µ1)/µ1, where µ0 and µ1 are the means. We also explore a few other priors for t0 and t1, such as uniform and gamma distributions, as described later.
From Yang (2000) (see also Newton, 1996), the likelihood is the multinomial probability of observing data given the tree and branch lengths:
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| (1) |
where C = n!/(n0! n1! n2! n3!) is a proportionality constant, and p0, p1, p2, p3 are probabilities of observing the four site patterns, respectively, under tree T1:
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The posterior probability for tree Ti, i = 1, 2, 3, is
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Computer Simulation
We simulated data sets to examine the properties of Bayesian posterior probabilities for trees. Except where stated otherwise, we conducted Bayesian simulation, sampling values of parameters from the prior. Each data set is generated by sampling branch lengths t0 and t1 from their prior distributions, calculating the probabilities of the four site patterns p0, p1, p2, p3 according to Equation 2, and then generating counts of site patterns (n0, n1, n2, n3) by sampling from the multinomial distribution M(n, p0, p1, p2, p3). The procedure is repeated to generate multiple data sets. We use T1 as the correct tree in the simulation, but interpret the results as if the data are simulated from a random tree chosen from T1, T2, and T3 with equal probability.
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Simulation Results |
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TopAbstractSimulation ExperimentSimulation ResultsAnalysis of the Land...DiscussionAcknowledgmentsReferences |
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Effect of Branch Length Prior in Simulated Data
We simulated data sets using the trees of Figure 1 to examine the effect of the prior for the internal branch length on Bayesian inference of tree topology. Each simulated data set is analyzed using the Bayesian method to calculate the posterior probabilities for the three trees (Equation 3): P1 for the correct tree, and P2 and P3 for the two wrong trees. We contrast the simulation model, the model used to generate the data, and the analysis model, the model used to analyze the data. The term model refers to the full model, including both the prior (for tree topology and branch lengths) and the likelihood (substitution) model. When the simulation and analysis models match, we say that the analysis model is correct. The only possible mismatch between the simulation and analysis models considered here is the prior for internal branch lengths; the correct prior for topology and the correct substitution model are assumed in the analysis.
For this simple case, the ML tree is T1, T2, or T3, depending on whether n1, n2, or n3 is the greatest. More precisely, T1 is the ML tree if and only if n1 > max(n2, n3) and n0 + n1 > n2+ n3 (Yang, 2000). When n1 > max(n2, n3) but n0+n1 
n2+n3, the sequences are more divergent than random sequences, and then none of the binary trees has a higher likelihood than the star tree. The maximum posterior probability tree is similarly determined as long as the prior mean for internal branch lengths µ0 > 0: that is, if n1 > max(n2, n3), we have P1 > max (P2, P3). Changing µ0 affects the magnitudes of the three posterior probabilities but not their order.
We use the case where the full model is correct—that is, where the analysis model matches the simulation model—to illustrate the interpretation of posterior probabilities for trees. When the data are simulated under the prior and when the full analysis model is correct, the posterior probability for a tree is the probability that the tree is true. Figure 2a ("correct" prior) shows results of such a simulation. Each data set is generated by choosing a tree from T1, T2, and T3 at random and by sampling t0 and t1 from exponential priors with means µ0 = 0.02 and µ1 = 0.2, respectively. The sequence length is n = 500. For those parameter values, the true tree is recovered by the likelihood or Bayesian methods with probability 0.86. When the data are analyzed (Equation 3), the correct exponential priors with the correct means µ0 = 0.02 and µ1 = 0.2 are assumed for t0 and t1, respectively. Each data set produces posterior probabilities P1, P2, P3 for the three trees, and these are collected into 50 bins, at 2% width for each bin. Then in the bin with posterior probability around P, the tree should be the true tree with probability P. For example, trees in the 94% to 96% bin all have posterior probabilities close to 95%. Among them, about 95% are the true tree while others (about 5%) are either of the two alternative (incorrect) trees (Fig. 2a).
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Such a match does not exist when the prior assumed in the analysis model does not match the prior assumed in the simulation. We considered the effect of the prior mean µ0 for the internal branch length only and used µ1 = 0.2 as in the simulation model. When the internal branch assumed in the prior is too short (µ0 = 0.002, "conservative" prior; Fig. 2b), low Ps (say, P < 1/3) overestimate the probability of the correct tree, whereas high Ps (say, P > 1/2 ) underestimate the probability of the correct tree. Thus, the method too often fails to reject or support any tree, and is too conservative. In contrast, when the mean internal branch length assumed in the prior is too large (µ0 = 0.2, "liberal" prior; Fig. 2c), low Ps underestimate and high Ps overestimate the probability of the correct tree, and the method is too liberal. The bootstrap method is also too liberal in these data sets if the bootstrap proportion is interpreted as the probability that the tree is correct.
Figure 2 also shows the distribution of posterior probabilities P over replicate data sets. The three posterior probabilities from each data set are grouped into the 50 bins, and the frequencies in the bins are used to plot the histogram. This procedure ignores the constraint that P1 + P2+ P3 = 1 and is not a proper way of representing the distribution of P1, P2 and P3 (which is shown in Fig. 3 below). With the correct prior (µ0 = 0.02; Fig. 2a), most posterior probabilities are near 0 or 1, although there is a third peak near 1/3. Use of the conservative prior (µ0 = 0.002; Fig. 2b) shifted the posterior probabilities towards 1/3. Note that in the extreme case where µ0
0, all three Ps will approach 1/3. In contrast, the liberal prior (µ0 = 0.2; Fig. 2c) shifts the density towards the two tails near 0 or 1, and polarizes the probabilities.
The joint density of posterior probabilities, f(P1, P2, P3), is shown in Figure 3, estimated from the same simulated data sets as used in Figure 2. The correct tree is recovered in the data set if and only if P1 > max(P2, P3). With the correct prior (µ0 = 0.02; Fig. 3a), there are many data sets in which P1 is near 1 and many data sets in which all three Ps are near 1/3. Note that data sets in which one of P1, P2, P3 is 0.80 are represented by three line segments in the plot, corresponding to each of the three trees T1, T2, T3 being the ML/Bayes tree. As the full analysis model is correct in this set of simulations (Fig. 3a), exactly 80% of the total density mass on those three line segments is located on the one corresponding to T1. The conservative prior (µ0 = 0.002; Fig. 3b) shifts the density towards the center of the plot, where all three Ps are close to 1/3. If µ0 0 in the analysis model, the density reduces to a point mass at P1 = P2 = P3 = 1/3. In contrast, the liberal prior (µ0 = 0.2; Fig. 3c) shifts the density to the three corners of the plot, so that one of P1, P2, P3 is near 1 whereas the other two are near 0, and high probabilities (say > 95%) are produced for wrong trees too often (say, > 5% of the time).
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Two additional sets of simulations were conducted, using n = 200 and 1000 sites, respectively, and using prior means µ0 = 0.1 and µ1 = 0.2. The data were analyzed in the same way as in Figure 2 and Figure 3, assuming the correct prior (µ0 = 0.1, µ1 = 0.2), a conservative prior (µ0 = 0.01, µ1 = 0.2), and a liberal prior (µ0 = 1, µ1 = 0.2). The results (not shown) were very similar to those of Figure 2 and Figure 3. In both sets of simulations, use of the correct prior produced a perfect match between the calculated posterior probability of a tree and the probability that the tree is true. Use of the conservative prior caused the posterior probabilities to become less extreme and the method to become too conservative. In contrast, the liberal prior made the posterior probabilities more extreme and the method too liberal. For n = 1000, the effect of the liberal prior was noted to be minor, because the posterior probabilities under the correct prior were already very extreme; for that sequence length, the correct tree is recovered in 96% of the simulated replicates. The effect of the conservative prior is always apparent. Furthermore, in both sets of simulations, the bootstrap proportions are noted to be too liberal, as in Figure 2d.
To examine which aspects of the prior for internal branch lengths affect posterior tree probabilities, we analyzed a fixed data set in Figure 4. The data are n = n0, n1, n2, n3 = 300, 80, 65, 55. For tree T1, the maximum likelihood estimates (MLEs) are 
0 = 0.04176, 1 = 0.16348, with log likelihood 
1 = –554.2858, whereas both trees T2 and T3 reduce to the star tree T0, with estimates 0 = 0, 1 = 0.19054, and 2 = 3 = 0 = – 556.2283. The bootstrap proportions for the three trees are (0.887, 0.104, 0.009). The results of Bayesian analysis are shown in Fig. 4, using exponential (Fig. 4a), uniform (Fig. 4b), and gamma (Fig. 4c) priors. In Figure 4a, exponential priors are used for t0 and t1, with the means µ0 varying and µ1 = 0.1 fixed. When µ0 increases from 0 to 
, the posterior probabilities (P1, P2, P3) change from (1/3, 1/3, 1/3) to (0.925, 0.052, 0.023). For this data set, the Ps are most sensitive in the region 0.001 < µ0 < 0.1. The prior mean µ1 for the external branch length is found to be much less important than is µ0 (results not shown). In Figure 4b, uniform priors are used for the branch lengths: t0
U(0, µ0) and t1 U(0, 1). The posterior probabilities for the three trees become more extreme when the upper bound µ0 increases. In Figure 3c, t1 has an exponential prior with mean µ1 = 0.1, but t0 has a gamma prior with mean µ0 and standard deviation 
0. The contours represent P1 as a function of µ0 and 0. The prior mean µ0 has much greater effect than the standard deviation 0 or variance of the gamma prior.
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Distribution of Posterior Probabilities in Data Sets Simulated under the Star Phylogeny
We examine how posterior probabilities P1, P2, P3 change with the increase of the sample size n when the data are simulated under a star phylogeny. The branch lengths are fixed at t0 = 0 and t1 = 0.2 in the simulation, which correspond to site-pattern probabilities p0 = 0.58700 for the constant pattern xxx, and p1 = p2 = p3 = 0.13767 for the three variable patterns (Equation 2). In the Bayesian analysis, we assumed µ0 = 0.1 and µ1 = 0.2 in the exponential priors for t0 and t1. The results are shown in Figure 5. In small samples (e.g., n = 20), the probabilities are most often close to 1/3, reflecting the paucity of the data. When the sample size increases (n = 200 or 1000), however, the probabilities shift to the corners of the plot, with one of the three probabilities close to 1 and the other two close to 0. We encountered problems with numerical integration using Mathematica for large n, and it is unclear what the limiting distribution f(P1, P2, P3) is when n
. Note that for those data, the bootstrap proportions are more extreme than the Bayesian probabilities (Fig. 5d).
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Similar simulations were conducted by Suzuki et al. (2002) using nucleotide-substitution models without the molecular clock to estimate unrooted trees for four species. The authors observed variable and occasionally very high posterior probabilities for the trees, similar to the pattern for n = 1000 in Figure 5. It is important to note that in the simulations of Figure 5 and of Suzuki et al. (2002), the data are generated using fixed branch lengths so that we are examining the frequentist sampling properties of the Bayesian method. Although it is reasonable to use frequentist criteria to evaluate a Bayesian method, there is no theory to guarantee its performance. Suzuki et al. (2002: 16139) incorrectly stated that "Bayesian ... trees were judged as false-positives when the posterior ... probability was > 95%. ... Note that the expected false-positive rate (type-I error) is 5% ... because the confidence level is 95%." These authors have confused posterior probabilities with frequentist P-values. Nevertheless, Suzuki et al. (2002) argued that a good method should give about equal probability (1/3) for the three bifurcating trees when the star tree is used to simulate data and when the amount of data is large. In this study, we take this viewpoint as well, as did Lewis et al. (2005). The concern is that if the interior branches are short in the real world, the real data may appear similar to data sets generated under the star tree, and then the posterior probabilities will be highly variable among data sets, sometimes strongly supporting the true tree and other times strongly supporting wrong trees. Lewis et al. (2005) called the phenomenon a star-tree paradox.
Fair-coin and fair-balance paradoxes
Lewis et al. (2005) drew an insightful parallel between Bayesian phylogeny reconstruction when the data are simulated under the star tree and a coin-tossing experiment. Suppose a coin is fair with the probability of heads to be exactly 
= 1/2, but we are required to compare two hypotheses that the coin is either negatively or positively biased: H1: < 1/2 and H2: > 1/2. The truth = 1/2 is considered impossible in the analysis. The data are the number of heads x out of n tosses of the coin, with the likelihood given by the binomial probability, x | bino(n, ). Lewis et al. (2005) argued that one would like the posterior model probability P1 = Pr(H1| x) to approach 1/2 when n . Assuming a uniform prior U(0, 1) and prior probability 1/2 for each model, the authors found that P1 instead converged to a uniform distribution. They referred to the phenomenon as the fair-coin paradox. Note that the posterior is given by | x beta(x + 1, n – x + 1), which converges to N(y, y(1 – y)/n), where y = x/n, when n . Let 
(·), 
(·), and -1(·) be the density function, the cumulative density function (CDF) and the inverse CDF (quantile) of the standard normal distribution. We have P1 = Pr( < 1/2| x) 
Also dP1/dy = 
,where a = –1(P1). Since y N(1/2, 1/(4n)), P1 has the density
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A simpler argument can be constructed using the normal distribution. Suppose n measurement errors x1, x2, ..., xn are observed on a balance, which is fair (calibrated) so that the xi are independent draws from N(0, 2) with mean 0 = 0 and known variance 2. We are required to test two hypotheses that the balance has either negative or positive bias: H1: < 0 and H2: > 0. The truth = 0 is not allowed in the analysis. We assume a normal prior N(0, 
2), with larger 2 representing more diffuse priors. Equivalently, H1 and H2 each has prior probability 1/2, and under each model the prior on is N(0, 2), truncated to the range (–, 0) under H1 or (0, ) under H2. The likelihood is given by 
, since the sample mean 
is a sufficient statistic. The posterior of given data x = {x1, x2, ..., xn} is given by | x N (
/(2 + n2), 22/(2 + n2)). Thus the posterior model probability is
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,
which is a standard normal variable since the sample is from N(0, 2) and 
(0, 2/n). Thus, (z) or P1 U(0, 1) when n (see Ripley, 1987: 59). Note that using an increasingly diffuse prior (that is, letting 2 ) has a similar effect as increasing the sample size.
To follow Suzuki et al. (2002) and Lewis et al. (2005), we would like a good method to give equal support for H1 and H2 when n . The Bayesian method does not achieve this; instead, the posterior model probability P1 converges to U(0, 1). This may be termed the fair-balance paradox. When n , the sample mean 
will be closer and closer to 0 = 0; in 99% of data sets, will be in the narrow interval 
. Also, the confidence interval for from each data set will be narrower and narrower around the true value 0. However, when forced to decide whether < 0 or > 0, the posterior model probability varies widely among data sets, just like a random variable from U(0, 1), sometimes strongly supporting one of the two hypotheses.
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Analysis of the Land Plant Data of Karol Et Al. |
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TopAbstractSimulation ExperimentSimulation ResultsAnalysis of the Land...DiscussionAcknowledgmentsReferences |
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Sequence alignment
To demonstrate the relevance of our analysis of the simple three-species case (Fig. 1) to real data sets used in molecular phylogenetics, we examined the impact of the prior for internal branch lengths on the posterior clade probabilities using the data set of Karol et al. (2001) concerning land plant divergences. The 40 species are identified in Figure 6; see appendix 2 in Karol et al. (2001) for GenBank accession numbers for the sequences. The alignment was retrieved from the Science Web site (http://www.sciencemag.org/cgi/content/full/294/ 5550/2351/DC1/1) and includes four genes concatenated as a supergene. The four genes are atpB and rbcL from the chloroplast, nad5 from the mitochondria, and the small subunit rRNA gene (SSU rRNA) from the nuclear genome. We made a few minor corrections to the alignment of Karol et al., leaving 5141 sites in the sequence, compared with 5147 used by Karol et al. The alignment is available at the Systematic Biology web site, http://systematicbiology.org.
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MrBayes (Huelsenbeck and Ronquist, 2001; Ronquist and Huelsenbeck, 2003) was used to conduct the Bayesian analysis (see below for our modifications), with a Markov process model of nucleotide substitution used for likelihood calculation. Three of the four genes (atpB, rbcL, and nad5) are protein-coding, with huge differences among the three codon positions in the evolutionary dynamics, such as the evolutionary rate, the base compositions, the transition/transversion rate ratio, and the extent of rate variation among sites. Ideally, such heterogeneity should be taken into account in the analysis (Yang, 1996), and indeed MrBayes provides some models for combined analysis of such heterogeneous data. However, for the posterior probabilities calculated from our analysis to be directly comparable with those of Karol et al. (2001), we followed those authors and ignored the differences among codon positions. Thus, we used the HKY+G model, with five categories in the discrete gamma model of rates for sites (Yang, 1994a; Hasegawa et al., 1985). To reduce computation, parameters in the substitution model are fixed at their maximum likelihood estimates (MLEs) obtained from a few parsimony trees:
for the transition/transversion rate ratio; 
for nucleotide frequencies; and 
for the gamma shape parameter. Thus, the model accounts for the major features of the more complex model GTR+I+G, which Karol et al. used. Note that the gamma distribution with 
1 allows for virtually invariable sites with rates close to 0. We expect that fixing the substitution parameters to their MLEs will have little effect on tree reconstruction (Yang et al., 1995), as those parameters are reliably estimated from the large data set, with standard errors to be about 1% to 2% of the MLEs. The molecular clock is not assumed, and unrooted trees are considered.
Modifications to MrBayes and MCMC analysis
The current version of MrBayes (version 3.0) (Ronquist and Huelsenbeck, 2003) assumes the same prior, either uniform or exponential, for all branch lengths on the unrooted tree. To use exponential priors with different means µ0 and µ1 for the internal versus external branch lengths, the source code of the program was modified by Z.Y. The change of the prior only affects the calculation of the prior ratio in MCMC moves that alter branch lengths, and does not affect other parts of the program, such as likelihood calculation or proposals that change other parameters. Thus, the source code was analyzed and appropriate changes were made. Our extensive tests indicated that the modifications were correct. We assume uniform priors for topologies and exponential priors with different means for the internal and external branch lengths. The prior mean of external branch lengths is fixed at µ1 = 0.1, whereas the prior mean of internal branch lengths µ0 is varied to see how it affects posterior clade probabilities. We used preliminary runs to determine reasonable settings to ensure convergence of the MCMC algorithm. Results reported in Figure 6 and below were obtained by running two simultaneous chains, using a burn-in of 20,000 iterations, followed by sampling every 10 iterations for a total of 2,000,000 iterations. Each analysis is conducted at least twice, using different random numbers, to confirm consistency between runs.
The original analysis of Karol et al. (2001) using MrBayes (Huelsenbeck and Ronquist, 2001) produced very high posterior probabilities for the inferred clades. The results obtained from applying the modified version of the program, assuming exponential priors with different means µ0 and µ1 for the internal and external branch lengths, are summarized in Figure 6. When µ0 is in the range (0.00004, 0.001), we recover the same phylogeny as reported by Karol et al. (2001), with larger µ0 producing higher probabilities for the clades. When µ0 
0.01, posterior probabilities for all but one node on the tree of Karol et al. were calculated to be 100%, but surprisingly one node was not found in the sampled trees (Fig. 6a). Posterior clade probabilities calculated using µ0 = 0.0001 and 0.1 are listed on the tree of Figure 6a, whereas Figure 6b shows the posterior probabilities for four important clades on the phylogeny as functions of µ0. Posterior probabilities for other nodes show similar changes with the change of µ0 (results not shown).
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Discussion |
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TopAbstractSimulation ExperimentSimulation ResultsAnalysis of the Land...DiscussionAcknowledgmentsReferences |
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Bayesian Posterior Probabilities Versus Bootstrap Proportions
Bayesian posterior probabilities are conceptually straightforward to interpret. The posterior probability for a tree or clade is the probability that the tree or clade is true, given the data and the model (including the prior and the likelihood model). Using the simple case of phylogeny reconstruction for three species, we illustrated this interpretation (Fig. 2a). In contrast, the bootstrap proportion has been much harder to interpret. At least three interpretations have been offered in the literature (see, e.g., Berry and Gascuel, 1996). The first is that it means repeatability. A clade with bootstrap proportion P in the original data set is expected to be in the estimated tree with probability P if many new data sets are generated from the same data-generating process and if the same tree reconstruction method is used to analyze the data sets (Felsenstein, 1985). The rationale for resampling the original data by bootstrap is that the distribution of the bootstrap samples around the observed data set is a good approximation of the unknown distribution of observed data from the data-generating process (Efron, 1979; Efron et al., 1996). The simulations of Hillis and Bull (1993) suggest that the bootstrap proportion varies so much among replicate data sets that it is useless as a measure of repeatability. A second interpretation is the frequentist type-I error rate, using the star tree as the null hypothesis (Felsenstein and Kishino, 1993) or a confidence interval (Felsenstein and Kishino, 1993; Zharkikh and Li, 1995). If we generate many data samples under the star tree in which the concerned clade (with bootstrap proportion P from the original data set) is absent, then the clade will be in the estimated tree with probability < 1 – P. Efron et al. (1996) argued that this interpretation is only approximate, and suggested a more complex, two-step bootstrap procedure for transforming bootstrap proportions into standard frequentist confidence intervals. A third interpretation is phylogenetic accuracy: a clade with bootstrap proportion P is in the true tree with probability P. This interpretation equates bootstrap proportion with Bayesian posterior probability and appears to be the one that most empirical phylogeneticists use or would like to use (e.g., Hillis and Bull, 1993; Murphy et al., 2001). All studies comparing the two approaches appear to be using this interpretation, as otherwise the two measures are incomparable.
The fact that the posterior probabilities change drastically with the prior for internal branch lengths (e.g., Fig. 4) suggests that the posterior probability and bootstrap proportion are two fundamentally different measures of phylogenetic uncertainty. This result appears to contradict previous claims that the two should be theoretically close (Efron et al., 1996; Newton, 1996), and to agree with recent simulations demonstrating their differences (Suzuki et al., 2002; Alfaro et al., 2003; Cummings et al., 2003; Douady et al., 2003; Erixon et al., 2003). We observe that bootstrap proportions are more similar to posterior probabilities under some priors than under others. For example, for the data set of Figure 4a, the bootstrap proportions are close to the posterior probabilities under the exponential prior with mean µ0 = 0.06 but are more or less extreme than the posterior probabilities when µ0 < or > 0.06, respectively. In the data sets of Figure 2 simulated under the prior, the bootstrap proportions are on average comparable to posterior probabilities under µ0 = 0.2 but are more different from the posterior probabilities under µ0 = 0.02 or 0.002. Thus, the bootstrap will be too conservative or too liberal, that is, the bootstrap proportions will be too moderate or too extreme relative to the posterior probabilities under the correct prior, depending on whether the prior mean µ0 used to generate the replicate data sets is very small or very large. It is clear that the bootstrap proportion, if interpreted as the probability that the clade is correct, is not always conservative, as suggested previously (Hillis and Bull, 1993).
We asked the question whether certain priors can produce posterior probabilities that are close to the bootstrap proportions in all replicate data sets. For the data sets of Figure 2 and Figure 3 we used an iterative algorithm to adjust the means µ0 and µ1 in the exponential priors to minimize the difference between the posterior probabilities and the bootstrap proportions in each data set. We found that µ0 and µ1 "estimated" in this way vary considerably among data sets, suggesting that it is in general impossible for bootstrap proportions to match posterior probabilities under fixed priors. Similarly, we observed in the simple case of three species that the effect of overall sequence divergence is quite different on the two measures. For example, adding constant sites (n0) to the data tends to polarize the posterior probabilities while having little impact on bootstrap proportions. For example, Table 1 lists posterior probabilities calculated for the data set of Figure 4, but with different numbers of constant sites added. The bootstrap proportions are (0.887, 0.104, 0.009), estimated using 100,000 bootstrap pseudosamples. At this level of accuracy, we cannot detect any difference in bootstrap proportions among the data sets. The lack of effect of n0 on the bootstrap is understandable in this case, because in each data set, the maximum likelihood tree is determined by the counts of sites for the three variable patterns and largely independent of n0. In contrast, the posterior probabilities become more extreme when constant sites are added to the data. Intuitively, adding constant sites reduces the overall sequence divergence, and as a result, every observed change becomes less likely and is counted as stronger evidence in calculation of posterior probabilities. Those results also suggest that the posterior probability might be more sensitive to the substitution model, especially concerning rate variation among sites, than bootstrap proportions.
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Factors Inflating Posterior Probabilities for Trees
Bayesian posterior probability for a tree or clade is the probability that the tree or clade is true given the data and model (prior and substitution model). Thus, there can be only three possible reasons for spuriously high clade probabilities: (i) program errors and computational problems, (ii) misspecification of the likelihood (substitution) model, and (iii) misspecification and sensitivity of the prior. Lack of convergence and poor mixing in the MCMC algorithm can cause the chain to stay in an artificially small subset of the parameter space, leading to spuriously high support for the trees visited in the chain. This problem may be a serious concern in Bayesian analysis of large data sets, but in principle may be resolved by running longer chains and designing more efficient algorithms. Model misspecification, that is, use of an overly simple substitution model, is also known to cause spuriously high posterior probabilities (Buckley, 2002; Huelsenbeck and Rannala, 2004; Lemmon and Moriarty, 2004; Suzuki et al., 2002). The problem can in theory be resolved by implementing more-realistic substitution models or taking a model-averaging approach (Huelsenbeck et al., 2004). In this study, we examined the effect of the prior on internal branch lengths and demonstrated that the posterior probabilities are sensitive to the prior specification. We note that high posterior probabilities were observed in simulated data sets where the substitution model is correct (this study) and in analyses that did not use MCMC algorithms (Rannala and Yang, 1996). In those cases, the first two factors do not apply. The sensitivity of Bayesian inference to prior specification is more fundamental and difficult to deal with (see below). The uniform prior with a large upper bound such as 10 or 100 is often advocated as a "non-informative" or "diffuse" prior for branch lengths. However, such a prior causes inflated clade probabilities and is one of the worst in this regard. Exponential priors with small means appear preferable.
The Effect of Prior on Bayesian Model Comparison and Phylogeny Estimation
Phylogeny reconstruction can be viewed as a problem of model selection rather than parameter estimation (Yang et al., 1995). Different trees have different likelihood functions with different branch length parameters, and are equivalent to non-nested models. In contrast to Bayesian parameter estimation under a well-specified model, where the posterior will be dominated by the likelihood when more and more data are available, Bayesian hypothesis testing or model selection is a difficult area, and weak prior information for model parameters is known to cause problems (e.g., Bernardo, 1980; DeGroot, 1982; Berger, 1985: 144–157). Below, we briefly review the literature on Bayesian model selection in presence of weak prior information, partly because phylogeny estimation appears to be affected by similar difficulties but mainly because some of the suggested remedies appear useful to phylogeny estimation.
An extreme well-known case is Lindley's paradox, in which Bayesian analysis and traditional hypothesis testing approaches reach drastically different conclusions (Lindley, 1957; see also Jeffreys, 1939). Consider test of a simple null hypothesis H0: = 0 against the composite alternative hypothesis H1: 
0 using a random sample x1, x2, ..., xn from N(, 2) with 2 known. The usual test is based on the sufficient statistic having a normal distribution N(0, 2/n) under H0 and calculates the P-value as 
. In the Bayesian analysis, suppose the prior is Pr(H0) = Pr(H1) = 1/2, and N(0, 2) under H1. The likelihood is given by 
(0, 2/n) under H0 and by 
(, 2/n) under H1. Then the ratio of posterior model probabilities, which is also the Bayes factor since the prior model probability is uniform, is equal to the ratio of the marginal likelihoods
|
| (7) |
z/2, so that we reject H0 at the significance level , but as n , we see that B and Pr(H0| x) 1. Hence the paradox: while the significance test rejects H0 decisively at = 10–10, say, the Bayesian method strongly supports H0 with posterior model probability Pr(H0| x) approaching 1. We can also fix the sample size n but increase 2, making the prior more and more diffuse; again Pr(H0| x) 1 as 2 . In both cases, the prior distribution under H1 becomes more and more spread out relative to the likelihood, which is concentrated in a small region close to but different from 0.
We note that Lindley's paradox is controversial, even among Bayesian statisticians. Some view it as revealing logical flaws in traditional hypothesis testing (e.g., Good, 1982: 342; Berger, 1985: 144–157; Press, 2003: 220–225), whereas others consider the Bayesian approach to be misleading and suggested fixes (e.g., Bernardo, 1980; Shafer, 1982). However, all appear to agree that the extreme sensitivity of the posterior model probabilities to the prior means that an objective Bayesian analysis is impossible. As remarked by O'Hagan and Forster (2004: 78), "Lindley's paradox arises in a fundamental way whenever we wish to compare different models for the data, and where we wish to express weak prior information about parameters in one or more of the models. " For such difficulties to arise, the compared models can have one or more parameters, or one model can be sharp (with no parameters), and the prior can be proper and informative as increasing the size of data while keeping the prior fixed has the same effect. To appreciate the generality of the problem, we contrast Bayesian parameter estimation with model selection using uniform priors for parameters. First, consider estimation of parameter in a well-specified model, with the prior f() = 1/(2c), – c < < c. The posterior is
|
| (8) |
) is concentrated in a small region (inside the prior interval as long as the prior is diffuse enough to contain the true ), outside which f(x | ) is vanishingly small. Then the integral in the denominator is insensitive to c, and so is the posterior. In contrast, consider comparison between two models H1 involving parameter 1 with prior f1(1) = 1/(2c1), –c1 < < c1, and H2 involving parameter 2 with prior f2(2) = 1/(2c2), –c2 < < c2. The Bayes factor is
|
| (9) |
When the data are informative and the likelihood fi(x | i) under model i, i = 1, 2, is highly concentrated, the two integrals are more or less independent of ci. However, the Bayes factor or posterior model probability depends on c2/c1, and that sensitivity will not disappear with the increase of data. The difficulty is that when the prior information is weak, one may not be able to decide whether U(–10, 10) or U(–100, 100) is a more appropriate prior, even though the Bayes factor differs by 10-fold between the two.
We note some differences between Lindley's paradox and the phylogeny estimation problem. First, in tree estimation, the mean of the prior for internal branch lengths is important, whereas in Lindley's paradox, it is the variance. In both cases, increasing the sample size has the same effect of exacerbating the problem. Second, if we view tree estimation as a problem of hypothesis testing, with the binary trees being the alternative hypothesis (hypotheses) and the star tree being the null hypothesis, the pattern in phylogeny estimation is opposite to that in Lindley's paradox. In the former, posterior probabilities are high for the binary tree, which is considered the "alternative" hypothesis, whereas in the latter, the effect of increasing amounts of data or a progressively diffuse prior is a strong support for the null hypothesis. Lewis et al. (2005) argued that the phylogeny problem, especially the star-tree paradox, is more similar to the fair-coin paradox they constructed (or the equivalent fair-balance paradox discussed in this study). However, this formulation has difficulties as well. First, a simple hypothesis test considers one alternative hypothesis, but we have many binary trees and such composite hypothesis tests can have complex properties. Second, a conventional hypothesis test makes assumptions about parameters in a general model, whereas different trees are equivalent to different models with different parameter spaces. Third, rejection of the star tree is not an appropriate measure of the statistical support for the ML/Bayes tree; one can construct data sets in which all three binary trees are significantly better than the star tree but it is ridiculous to claim that all three binary trees are significantly supported by the same data (Tateno et al., 1994; Yang, 1994b). Here we consider Lindley's paradox, the fair-coin or fair-balance paradoxes, and phylogeny estimation as three distinct manifestations of the deeper problem of sensitivity of Bayesian model selection to the prior for model parameters.
If we use an extreme prior µ0 = 0, all binary trees will have the same small probability. Thus, high clade probabilities in any data set can be made small by assuming a very small µ0 in the analysis model, and therefore there must always be a region of µ0 over which the posterior probabilities are sensitive to changes in the prior. However, in large data sets, this sensitive region may include only very small values of µ0. In our analysis of the land plant data set, the sensitive region is (10–5, 10–3) (Fig. 6b). Such values may seem unrealistically small if we consider estimated internal branch lengths in published trees. The question arises as to whether the prior for the internal branch lengths is relevant for the high posterior probabilities reported in many real data sets. We suggest that the answer is "Yes." In Bayesian model comparison, parameters in different models with different definitions are usually assigned different priors. In phylogeny reconstruction, branch lengths in different phylogenies have different biological meanings, and one can envisage assigning different priors for them. For example, a biologist's information or belief about the internal branch length in the tree ((human, chimpanzee), gorilla) may well be different than about the internal branch length in the tree (human, (chimpanzee, gorilla)), in each case the respective tree topology being assumed to be true. The branch in the latter tree may be expected to be shorter if the tree is considered less likely to be true than the former tree. Estimates of internal branch lengths in wrong or poor trees are typically small and often 0. If we specify the prior to represent our prior knowledge of branch lengths in all binary trees, the majority of which are wrong or poor trees, a very small µ0 is necessary. This argument also suggests that µ0 should be smaller in larger trees with more species.
Possible Remedies to Deal with the Sensitivity to the Prior
In a traditional parameter estimation problem, two approaches can be taken when the posterior is sensitive to parameters in the prior such as µ0. The first is the hierarchical or full Bayesian approach, which assigns a hyperprior for µ0 and integrates µ0 out in the MCMC algorithm. Suchard et al. (2001) implemented such an approach. Adding a hyperprior to µ0 is equivalent to specifying a different prior for t0, in the same way that the gamma prior considered in Figure 4 is an extension of the exponential prior. From our results (Fig. 4), the mean of the prior for t0 appears more important than the variance.
The second approach to dealing with the prior parameter µ0 is the empirical Bayes approach, which estimates µ0 from the data and uses the estimate to calculate posterior probabilities. We implemented this approach for the three-species case (Fig. 1). We estimate µ1 as the single branch length in the star phylogeny (i.e., with t0 = 0): 
log(4n0/n – 1)/3/4, as an overall measure of sequence divergence. We estimate the prior mean µ0 by maximizing the marginal likelihood: 
of Equation 4. The estimates µ0 and µ1 are then used to calculate Pi's in Equation 3. Application of this approach to the data analyzed in Figure 4 gives parameter estimates µ1 = 0.19054 and µ0 = 0.02746, with the marginal log likelihood = – 558.3644. The posterior probabilities are (0.8278, 0.1120, 0.0602) (c.f. Fig. 4a). We note that use of the star tree to estimate µ1 leads to overestimates of µ1 and underestimates of µ0, and may be problematic in large trees. However, use of the marginal likelihood function to estimate µ0 means that the estimate will be dominated by the ML tree, which typically has longer internal branches than poor trees. An alternative strategy is to estimate parameters such as µ0 and µ1 from the data for each possible tree topology, and then choose values representative of the collection of estimates among the trees, for use in Bayesian calculation. This strategy is computationally demanding because of the great number of trees, but will produce very small estimates of µ0 in real data analysis since most trees are wrong trees with small or zero internal branch lengths. As far as we are aware, the empirical Bayes approach has been used only in estimation of parameters in a well-specified model and not in dealing with sensitivity of Bayesian model comparison to the prior.
We note that Bayesian model comparison is an extremely active research area, with much controversy. A number of modifications have been introduced to deal with the sensitivity of Bayes factors to the prior on model parameters, resulting in a plethora of Bayes factors: such as intrinsic, partial, fractional, and pseudo-Bayes factors (see, e.g., O'Hagan and Forster, 2004 pp. 183–191). In discussions of Lindley's paradox, several possible remedies were suggested in the literature. We now discuss their potential use in the phylogeny problem. All such remedies are to some degree subjective and none are generally accepted. (a) Blame the question (e.g., Hill, 1982). It has been suggested that one cannot reasonably expect a parameter to take a fixed value = 0; instead, one should consider the null hypothesis that lies within a narrow interval (–
,). (b) Blame the data and avoid using large data sets (Bartlett, 1957). Those two options are unlikely to be relevant or appealing to molecular phylogeneticists. (c) Assume that both hypotheses are wrong and add a pinch of probability 
for errors, i.e., for possibilities unaccounted for in the model (DeGroot, 1982; Hill, 1982; see also Jeffreys, 1961: 129). How to determine is subjective. In the phylogeny problem, one of the binary trees should be correct, so that this idea does not appear logically sound. Factors such as lineage sorting or horizontal gene transfers may cause different genes to have different tree topologies, but this should best be dealt with by allowing data partitions to have different phylogenies, rather than by considering the composite phylogeny to be a star phylogeny. Nevertheless, one might assume a prior probability for the star tree, or, equivalently, assume a point mass at t0 = 0 in the prior for internal branch lengths t0, with the rest of the density coming from a distribution. This should have an effect similar to that achieved by assuming a small µ0 in the exponential prior and will reduce high posterior probabilities for trees. Lewis et al. (2005) has implemented this strategy, using a reversible-jump MCMC algorithm to move between trees with different numbers of branch lengths. (d) Use data or at least the size of data to specify the prior (Bernardo, 1980; Davison, 2003). As the prior is supposed to reflect information or beliefs before the data are gathered, this idea is outrageous to many Bayesian statisticians but considered useful by others. In the case of Lindley's paradox, one suggestion is to let the variance 2 be proportional to 1/n, that is, N(0, c2/n), so that increasingly informative priors are used for under H1 in larger data sets. Values of c in the range 5 to 20 appear to produce results comparable to the traditional significance test (Davison, 2003: 586–587).
A similar strategy can be applied to the fair-balance problem. We can let the prior become increasingly informative as n increases by specifying the prior variance 2 = c2/nk. The posterior model probability is then (cf: Equation 6)
|
| (10) |
Note that if z N(0, 1), then y = (az+ b), where a and bare constants, has the density
|
| (11) |
N(0, 1) the density of the posterior model probability P1 becomes
|
| (12) |
If k = 1, the prior variance becomes 2 = c2/n, which decreases at the rate 1/n. The density of P1 then peaks at 1/2, so that P1 is more likely to be around 1/2 than close to 0 or 1. However, the density is independent of n, and will not become more concentrated around 1/2 with the increase of n. When k > 1, the distribution converges to the point mass P1 = 1/2 when n , as we wanted, and at a faster rate for larger k.
To choose an appropriate k, we would also want f(P1) to converge to the point mass at 1 (or 0) at a reasonable rate when n , if the true < 0 and H1 is the true model (or if > 0 and H2 is the true model). From Equation 11, the density of P1 if the true parameter value is 0 is given as
|
| (13) |
, f(P1| 0) degenerates to a point mass at 1 (or 0) if 0 < 0 (or if 0 > 0) irrespective of k. However, the convergence is faster if k is smaller. To avoid the fair-balance paradox when 0 = 0 and to achieve a fast convergence when 0 0, k should be greater than 1 but not too much greater. Figure 7 plots the densities when 
for 0 = 0 and 0.01 and for two sample sizes n = 1,000 and 1,000,000, with c = 2 fixed.
|
For the phylogeny problem, we note that the objectives of both strategies (c) and (d), discussed above, can be achieved by applying a small mean in the prior for the internal branch length. The prior mean should be increasingly smaller for longer sequences (greater n) and larger trees, and should also reflect the overall information content (e.g., as indicated by overall sequence divergences), in the same way that
2 is used in the priors discussed above. Incorporating those factors in the prior appears hard, and merits further research. |
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TopAbstractSimulation ExperimentSimulation ResultsAnalysis of the Land...DiscussionAcknowledgmentsReferences |
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We thank Paul Lewis, Fredrik Ronquist, and Marc Suchard for many constructive criticisms, and Paul Lewis for making the Lewis, Holder, and Holsinger paper available to us before its publication. This study is supported by a grant from the Biotechnological and Biological Sciences Research Council (UK) to Z.Y. and National Institutes of Health grant HG01988 to B.R.
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TopAbstractSimulation ExperimentSimulation ResultsAnalysis of the Land...DiscussionAcknowledgmentsReferences |
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C. P. Randle and K. M. Pickett Are Nonuniform Clade Priors Important in Bayesian Phylogenetic Analysis? A Response to Brandley et al. Syst Biol, February 1, 2006; 55(1): 147 - 151. [Full Text] [PDF]
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