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review: John P. Huelsenbeck, Fredrik Ronquist, Rasmus Nielsen, and Jonathan P. Bollback Bayesian Inference of Phylogeny and Its Impact on Evolutionary Biology Science 2001; 294: 2310-2314 [Abstract] [Full text] [PDF]

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Reverend Bayes Reports Back to Popper

Daniel P. Faith, J. W. Trueman, Australian National University   (15 February 2002)

Reverend Bayes Reports Back to Popper 15 February 2002
Daniel P. Faith, Research Scientist The Australian Museum, J. W. Trueman, Australian National University Respond to this E-Letter: Re: Reverend Bayes Reports Back to Popper	Popperian corroboration assessment is a missing ingredient in Huelsenbeck et al.’s (1, 2) framework for Bayesian inference of phylogeny. We doubt that Bayesian posterior probabilities of phylogenetic hypotheses would be reliable on their own as measures of support and argue that these probabilities can be treated as Popperian evidence statements subject to corroboration assessment. Even a high posterior probability may not provide corroboration for the phylogenetic hypothesis (3, 4). Consider hypothesized phylogenetic relationships among three taxa, A, B, and C, with ancestor taxon O, and data consisting of a single character with states 0, 1, 1, 0, respectively. Suppose that the evolutionary model states that a character-state change from 0 to 1 can occur in any branch segment with probability, q. Take the prior probability of each of the three possible rooted trees to be 1/3. Then, for any small value of q, the posterior probability of the “BC” tree (B and C monophyletic) based on this single character will be close to 1.0 (5). But should we believe that this is the true tree? Huelsenbeck et al. emphasize that “the posterior probability of a tree can be interpreted as the probability that the tree is correct” (2) only if the model is accurate. The posterior probability that the BC tree hypothesis is correct is indeed close to 1.0, IF our model truly describes how such data might arise. But this issue involves more than just consideration of evolutionary models - these might not include all possible explanations for the observed data. Popperian “background knowledge” considers factors, including chance, that might account for the evidence (3). The high posterior probability of the BC tree, taken as evidence, has a 1/3 probability of occurring by chance (6). Corroboration is low (7). In a corroboration framework, a Bayesian posterior probability is just another form of goodness-of-fit of data and hypothesis. Bayesian inference then is analogous to other phylogenetic approaches that provide fit values as evidence (8), diminishing some claimed distinctions between Bayesian inference and other approaches (9). For example, combining different analysis results using Bayes theorem (2) resembles combination based on any fit measure – complete with the usual danger of weighting biases (8). Such limitations, paradoxically, point to an expanded role for Bayes theorem in phylogenetics. By replacing the “data” term with “evidence” equal to fit, Bayes theorem is linked to corroboration/severity calculations, and separate analyses may be combined at this level (8, 10). A greater role for Bayes theorem, linked to corroboration, may mean greater impact for Bayesian phylogenetic inference. References and Notes 1. After “Reverend Bayes meets Darwin”, the title of the supplemental material to Huelsenbeck et al. (2); see http://www.sciencemag.org/cgi/content/full/294/5550/2310/DC1 2. J. P. Huelsenbeck et al., Science 294, 2310 (2001). 3. Popper [Realism and the aim of science (Routledge, London, 1983)] argues that we have a high degree of corroboration and a “severe” test to the extent that the evidence is improbable given only background knowledge (including “coincidence,” “chance,” or by “mere accident”). 4. Our concerns about Bayesian phylogenetic inference parallel ongoing general debates between Bayesian and frequentist advocates. For example, D. G. Mayo [Error and the growth of experimental knowledge (Univ. of Chicago Press, Chicago, IL, 1996)] argues that Bayesian support does not capture the errors involved if the data-generation had been repeated many times. Mayo argues for severe tests, a criterion linked to Popper’s severity/corroboration [D. P. Faith, Syst. Biol. 48, 675 (1999)]. 5. Let the BC tree hypothesis be denoted by “h” and the data by “d.” Bayes theorem is calculated as (1): P(h, d) = P(d, h) X P(h) / P(d, any tree), where “,” indicates “given”. For a small value of q, this approximates q(1/3) / [ q(1/3) + q2(2/3) ] and is close to 1.0. 6. An a posteriori PTP test for corroboration assessment of the BC tree [D. P. Faith, Syst. Zool. 3, 366 (1991)], permuting the original data with ancestor character state fixed, implies that the observed evidence for the BC tree has probability 1/3. 7. The BC example also relates to corroboration of monophyletic groups, and any such corroboration contrasts with posterior probabilities as nodal support. The posterior probability is regarded as “a natural measure of nodal support” [P. O. Lewis, Trends Ecol. Evol. 16, 30 (2001)], but we propose an alternative measure of support, providing a limited form of corroboration (8), that would use the MCMC to estimate the Bayes factor relevant to the hypothesis of monophyly. This would provide for a rapid test akin to a likelihood-ratio-test for monophyly [J. P. Huelsenbeck et al., Syst. Biol. 45, 546 (1996)]. Such support assessment might be useful for exploring cases of low and sometimes conflicting nodal posterior probabilities - for example, those reported for hyrax-elephant versus hyrax-sirenian in supplementary material to W. J. Murphy et al., Science 294, 2348 (2001). 8. D. P. Faith, J. W. H. Trueman, Syst. Biol. 50, 331 (2001). 9. Huelsenbeck et al. argue that Bayesian inference, but not cladistic parsimony, examines near-optimal trees, but the conventional examination of a set of near-most-parsimonious trees using the decay-index is analogous to examining a range of trees having good posterior probabilities. 10. A general form of Bayes theorem is: P(h,e.k) = P(e,h.k) X P(h,k) / P(e,k) where h is the hypothesis, e is evidence, and k is background knowledge. P(e,h.k) / P(e,k) indicates degree of corroboration provided by the test corresponding to evidence e; for discussion see (8). 11. We thank the members of the Coopers and Cladistics Discussion Groups (Canberra and Sydney) for discussion.