marginal likelihood as exhaustive X validation

In the June issue of Biometrika (for which I am deputy editor) Edwin Fong and Chris Holmes have a short paper (that I did not process!) on the validation of the marginal likelihood as the unique coherent updating rule. Marginal in the general sense of Bissiri et al. (2016). Coherent in the sense of being invariant to the order of input of exchangeable data, if in a somewhat self-defining version (Definition 1). As a consequence, marginal likelihood arises as the unique prequential scoring rule under coherent belief updating in the Bayesian framework. (It is unique given the prior or its generalisation, obviously.)

“…we see that 10% of terms contributing to the marginal likelihood come from out-of-sample predictions, using on average less than 5% of the available training data.”

The paper also contains the interesting remark that the log marginal likelihood is the average leave-p-out X-validation score, across all values of p. Which shows that, provided the marginal can be approximated, the X validation assessment is feasible. Which leads to a highly relevant (imho) spotlight on how this expresses the (deadly) impact of the prior selection on the numerical value of the marginal likelihood. Leaving outsome of the least informative terms in the X-validation leads to exactly the log geometric intrinsic Bayes factor of Berger & Pericchi (1996). Most interesting connection with the Bayes factor community but one that depends on the choice of the dismissed fraction of p‘s.