Bayes and Gibbs Estimations, Empirical Processes, and Resolution of Singularities

Abstract. This paper clarifies the generalization errors of Bayes and Gibbs estimations in non-identifiable statistical models based on algebraic geometry and empirical process theory. First, the concrete results are proven for statistical models with normal crossing singularities. Second, they are generalized for all models by using Hironaka’s desingularization theorem. The Bayes generalization error is given by the largest pole of the zeta function of the statistical model, whereas the Gibbs one is represented by the empirical process.