# On the marginal likelihood and cross-validation

## TL;DR

This paper reveals the formal equivalence between the Bayesian marginal likelihood and exhaustive leave-$p$-out cross-validation with the log posterior predictive, offering new insights into model evaluation methods.

## Contribution

It demonstrates the equivalence between marginal likelihood and cross-validation, and proposes an alternative cumulative cross-validation approach for model assessment.

## Key findings

- Marginal likelihood equals exhaustive leave-$p$-out cross-validation averaged over all $p$ and test sets.
- The log posterior predictive is the only coherent scoring rule under data exchangeability.
- The approach highlights the sensitivity of marginal likelihood to prior choices.

## Abstract

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 partitioning 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 posterior 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 choice of the prior. We suggest an alternative approach using cumulative 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.