In Bayesian statistics, the marginal likelihood, also known as the evidence, is used to evaluate model ﬁt as it quantiﬁes the joint probability of the data under the prior. In contrast, non-Bayesian models are typically compared using cross-validation on held-out data, either through k-fold par-titioning or leave-p-out subsampling. We show that the marginal likelihood is formally equivalent to exhaustive leave-p-out cross-validation averaged over all values of p and all held-out test sets when using the log posterior predictive probability as the scoring rule. Moreover, the log pos-terior predictive is the only coherent scoring rule under data exchangeability. This offers new insight into the marginal likelihood and cross-validation and highlights the potential sensitivity of the marginal likelihood to the setting of the prior. We suggest an alternative approach using aggregate cross-validation following a preparatory training phase. Our work has connections to prequential analysis and intrinsic Bayes factors but is motivated through a different course.
Fong, Edwin & Holmes, Chris. (2019). On the marginal likelihood and cross-validation. arXiv:1905.08737.