Identifying the Best Approximating Model in Bayesian Phylogenetics: Bayes Factors, Cross-Validation or wAIC?

Abstract

There is still no consensus as to how to select models in Bayesian phylogenetics, and more generally in applied Bayesian statistics. Bayes factors are often presented as the method of choice, yet other approaches have been proposed, such as cross-validation or information criteria. Each of these paradigms raises specific computational challenges, but they also differ in their statistical meaning, being motivated by different objectives: either testing hypotheses or finding the best-approximating model. These alternative goals entail different compromises, and as a result, Bayes factors, cross-validation, and information criteria may be valid for addressing different questions. Here, the question of Bayesian model selection is revisited, with a focus on the problem of finding the best-approximating model. Several model selection approaches were re-implemented, numerically assessed and compared: Bayes factors, cross-validation (CV), in its different forms (k-fold or leave-one-out), and the widely applicable information criterion (wAIC), which is asymptotically equivalent to leave-one-out cross-validation (LOO-CV). Using a combination of analytical results and empirical and simulation analyses, it is shown that Bayes factors are unduly conservative. In contrast, CV represents a more adequate formalism for selecting the model returning the best approximation of the data-generating process and the most accurate estimates of the parameters of interest. Among alternative CV schemes, LOO-CV and its asymptotic equivalent represented by the wAIC, stand out as the best choices, conceptually and computationally, given that both can be simultaneously computed based on standard Markov chain Monte Carlo runs under the posterior distribution. [Bayes factor; cross-validation; marginal likelihood; model comparison; wAIC.].

Figure 1.

Theoretical fit of $M_2$ relative to $M_1$ (a) and mean squared estimation error (b) as a function of data size, under the normal model, and for two alternative priors (${\delta }^2=10$ and $1000$); vertical lines indicate the values of $n$ for which the two models have same quadratic LOO-CV score or same marginal likelihood.

Figure 2.

Fit of GTR, relative to LG, as a function of data size (number of aligned positions), on empirical data (10 random jackknife subsamples of the metazoan data set), using BF and LOO-CV. Error bars: standard deviation across jackknife replicates.

Figure 3.

Fit of GTR, relative to JTT, as a function of data size under BF and LOO-CV (a), and mean quadratic error on relative exchangeability estimates (b) on data simulated under the LG model (using the metazoan data set as a template). Error bars: standard deviation across four simulation replicates.

Figure 4.

LOO-CV and wAIC estimates (de-biased) for the GTR, CAT-Poisson, and CAT-GTR models (relative to LG), as a function of data size (number of aligned positions), on empirical data (metazoan data set). Error bars: standard deviation across four jackknife replicates.