Posterior predictive Bayesian phylogenetic model selection

We present two distinctly different posterior predictive approaches to Bayesian phylogenetic model selection, and illustrate these methods using examples from green algal protein-coding cpDNA sequences and flowering plant rDNA sequences. The Gelfand-Ghosh (GG) approach allows dissection of an overall measure of model fit into components due to posterior predictive variance (Pm) and goodness-of-fit (Gm), which distinguishes this method from the posterior predictive P-value approach. The conditional predictive ordinate (CPO) method provides a site-specific measure of model fit useful for exploratory analyses and can be combined over sites yielding the log pseudomarginal likelihood (LPML), which is useful as an overall measure of model fit. CPO provides a useful cross-validation approach that is computationally efficient, requiring only a sample from the posterior distribution (no additional simulation is required). Both GG and CPO add new perspectives to Bayesian phylogenetic model selection based on the predictive abilities of models, and complement the perspective provided by the marginal likelihood (including Bayes Factor comparisons) based solely on the fit of competing models to observed data.