© 2004 Kluwer Academic Publishers. Printed in the Netherlands. Chapter 9 BAYESIAN SUPERTREES

Abstract: In this chapter, we develop a Bayesian approach to supertree construction. Bayesian inference requires that prior knowledge be specified in terms of a probability distribution and incorporates this evidence in new analyses. This provides a natural framework for the accumulation of phylogenetic evidence, but it requires that phylogenetic results be expressed as probability distributions on trees. Because there are so many possible trees, it is usually not feasible to estimate the probability of each individual tree. Therefore, Bayesians summarize the distribution typically in terms of taxon-bipartition frequencies instead. However, bipartition frequencies are related only indirectly to tree probabilities. We discuss two ways in which Bayesian-bipartition frequencies can be translated into sets of multiplicative factors that function as keys to the probability distribution on trees. The Weighted Independent Binary (WIB) method associates factors to the presence or absence of taxon bipartitions, whereas the Weighted Additive Binary (WAB) method has factors with graded responses dependent on the degree of conflict between the tree and the partition. Although the methods are similar, we found that WAB is superior to WIB. We discuss several ways of estimating WAB factors from partition frequencies or directly from the data. One of these methods suggests a similarity between WAB factors and the decay index; indeed, the WAB factors represent a more natural measure of clade support than the bipartition frequencies themselves or the decay index and its probabilistic analog. WAB factors provide an efficient and convenient way of retrieving prior tree probabilities and WAB supermatrices accurately describe fully statistically specified supertree spaces, the latter of which can be sampled using MCMC algorithms with the computational efficiency of parsimony. This should allow construction of Bayesian supertrees with thousands of taxa