Bayesian Model Checking by Betting: A Game-Theoretic Alternative to Bayesian p-values and Classical Bayes Factors

A strictly Bayesian model consists of a set of possible data distributions and a prior distribution over that set. If there are other models available, how well they predicted the data may be compared using Bayes factors. If not, a model may be checked using a Bayesian p-value such as a prior predictive p-value or a posterior predictive p-value. However, recent criticisms of ordinary p-values apply with equal force against Bayesian p-values. Many of those criticisms are overcome by e-values, martingales interpreted as the amount of evidence discrediting a null hypothesis, measured as a payoff for betting against it. This article proposes the use of e-values to check Bayesian models by testing their prior predictive distributions as null hypotheses. Two generally applicable methods for checking strictly Bayesian models are provided. The first method calibrates Bayesian p-values by transforming them into Bayesian e-values. The second method uses Bayes factors or their approximations as Bayesian e-values. A robust Bayesian model, a set of strictly Bayesian models, may be checked using various functions that use the e-values of those strictly Bayesian models. Other functions measure how much the data support a Bayesian model. Relations to possibility theory are discussed.