@ARTICLE{TreeBASE2Ref16254, author = {Nicolas Lartillot and Herv? Philippe}, title = {Computing Bayes Factors Using Thermodynamic Integration.}, year = {2006}, keywords = {}, doi = {10.1080/10635150500433722}, url = {}, pmid = {}, journal = {Systematic Biology}, volume = {55}, number = {2}, pages = {195--207}, abstract = {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 paper, we propose to employ another method, based on an analogy with statistical physics, called thermodynamic integration. We describe the method, propose an implementation, and show on two analytical examples that this numerical method yields reliable estimates. In contrast, the harmonic mean estimator leads to a strong over-estimation 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 which 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.} }

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