Variable Tree Topology Stepping-Stone Marginal Likelihood Estimation

Department of Ecology and Evolutionary Biology, University of Kansas, Lawrence, Kansas, USAPaul O. LewisDepartment of Ecology and Evolutionary Biology, University of Connecticut, Storrs, Connecticut, USADavid L. SwoffordDepartment of Biology and Institute for Genome Sciences and Policy, Duke University, Durham, North Carolina, USADavid BryantDepartment of Mathematics and Statistics, University of Otago, Dunedin, New ZealandCONTENTS5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.2 The generalized stepping-stone (GSS) method . . . . . . . . . . . . . . . . . . 96 5.3 Reference distribution for tree topology . . . . . . . . . . . . . . . . . . . . . . . . . 985.3.1 Tree topology reference distribution . . . . . . . . . . . . . . . . . . . . . 98 5.3.1.1 Tree simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.3.1.2 Tree topology reference distribution . . . . . . . . . 1005.3.2 Edge length reference distribution . . . . . . . . . . . . . . . . . . . . . . 103 5.3.3 Comparison with CCD methods . . . . . . . . . . . . . . . . . . . . . . . . 1045.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.4.1 Model details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.4.2 Brute-force approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.4.3 GSS performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.6 Funding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Algorithms, andThe marginal likelihood is central to Bayesian model selection. It is the normalizing constant in Bayes' formula, and the Bayes factor used to compare two models is a ratio of marginal likelihoods. The marginal likelihood is defined as the expected value of the likelihood with respect to the prior. Methods for accurately estimating the marginal likelihood in phylogenetics were developed only recently (Lartillot and Philippe, 2006a; Xie et al., 2011; Fan et al., 2011; Arima and Tardella, 2012). Two of these methods-thermodynamic integration (TI; Lartillot and Philippe, 2006a) and the stepping-stone method (SS; Xie et al., 2011)—allow marginal likelihood estimation when the tree topology varies, but the most efficient methods to date-generalized stepping-stone (GSS; Fan et al., 2011) and the inflated density ratio method (IDR; Arima and Tardella, 2012)—have thus far remained restricted to estimating marginal likelihoods for a fixed tree topology. This chapter is concerned with updating GSS to allow variable tree topology, and the chapter by Wu et al. (Chapter 6) is concerned with updating the IDR method to allow variable tree topology.