The Bayes factor is commonly used for comparing different evolutionary rate models and different topologies in phylogeny. It is crucial to develop efficient Monte Carlo methods for estimating the marginal likelihoods in the Bayes factor. The Monte Carlo methods currently advocated in the phylogenetic literature include the harmonic mean (HM) method and the thermodynamic integration or path sampling (PS) method. However, these two methods may not be able to provide accurate estimates of the marginal likelihoods due to the complexity of the phylogenetic models. In this research work, we develop several new Monte Carlo methods including stepping stone (SS) method and bridge stepping stone (BSS) method, as well as better choices of the path parameter to overcome the limitations of current available methods. In addition to Bayes factor, we also investigate other attractive model comparison criteria, such as deviance information criterion (DIC) and conditional predictive ordinate (CPO) to compare different models and examine sensitivity of priors in phylogenetics. We further extend the SS method to other statistical applications. One of such applications is to compute marginal likelihoods of regression models for binary response data with different links, including logit, complementary log-log, and generalized t links. The marginal likelihoods are used for guiding the choice of links.
