On the Distributions of Bootstrap Support and Posterior Distributions for a Star Tree

Several authors have recently noted that when data are generated from a star topology, posterior probabilities can often be very large, even with arbitrarily large sequence lengths. This is counter to intuition, which suggests convergence to the limit of equal probability for each topology. Here the limiting distributions of bootstrap support and posterior probabilities are obtained for a four-taxon star tree. Theoretical results are given, providing confirmation that this counterintuitive phenomenon holds for both posterior probabilities and bootstrap support. For large samples the limiting results for posterior probabilities are the same regardless of the prior. With equal-length terminal edges, the limiting distribution is similar but not the same across different choices for the lengths of the edges. In contrast to previous results, the case of unequal lengths of terminal edges is considered. With two long edges, the posterior probability of the tree with long edges together tends to be much larger. Using the neighbor-joining algorithm, with equal edge lengths, the distribution of bootstrap support tends to be qualitatively comparable to posterior probabilities. As with posterior probabilities, when two of the edges are long, bootstrap support for the tree with long branches together tends to be large. The bias is less pronounced, however, as the distribution of bootstrap support gets close to uniform for this tree, whereas posterior probabilities are much more likely to be large. Our findings for maximum likelihood estimation are based entirely on simulation and in contrast suggest that bootstrap support tends to be fairly constant across edge-length choices.