Computing Bayes Factors Using Thermodynamic Integration

Nicolas Lartillot, Hervé Philippe, Paul Lewis
2006 Systematic Biology  
In the Bayesian paradigm, a common method for comparing two models is to compute the Bayes factor, defined as the ratio of their respective marginal likelihoods. In recent phylogenetic works, the numerical evaluation of marginal likelihoods has often been performed using the harmonic mean estimation procedure. In the present article, we propose to employ another method, based on an analogy with statistical physics, called thermodynamic integration. We describe the method, propose an
more » ... on, and show on two analytical examples that this numerical method yields reliable estimates. In contrast, the harmonic mean estimator leads to a strong overestimation of the marginal likelihood, which is all the more pronounced as the model is higher dimensional. As a result, the harmonic mean estimator systematically favors more parameter-rich models, an artefact that might explain some recent puzzling observations, based on harmonic mean estimates, suggesting that Bayes factors tend to overscore complex models. Finally, we apply our method to the comparison of several alternative models of amino-acid replacement. We confirm our previous observations, indicating that modeling pattern heterogeneity across sites tends to yield better models than standard empirical matrices. [Bayes factor; harmonic mean; mixture model; path sampling; phylogeny; thermodynamic integration.] As for model fit, the normalization constant, p(D \ M), is of primary importance. As a function of M, it can literaly read as the likelihood of model M, given the data D. Accordingly, among several models, one is led to choose the one of greatest marginal likelihood. When two models Mo and Mi are being compared, one usually defines the Bayes factor in favor of M\ over Mo as the ratio of their respective marginal likelihoods (Jeffreys, 1935; Kass and 195 by guest on
doi:10.1080/10635150500433722 pmid:16522570 fatcat:iw6dunyexrcxxdso7pvqeakyoe