Predictive inference under exchangeability, and the imprecise dirichlet multinomial model

Abstract

Coherent reasoning under uncertainty can be represented in a very general manner by coherent sets of desirable gambles. In this framework, and for a given finite category set, coherent predictive inference under exchangeability can be represented using Bernstein coherent cones of multivariate polynomials on the simplex generated by this category set. This is a powerful generalisation of de Finetti’s representation theorem allowing for both imprecision and indecision. We define an inference system as a map that associates a Bernstein coherent cone of polynomials with every finite category set. Many inference principles encountered in the literature can then be interpreted, and represented mathematically, as restrictions on such maps.We discuss two important inference principles: representation insensitivity— a strengthened version of Walley’s representation invariance—and specificity. We show that there is a infinity of inference systems that satisfy these two principles, amongst which we discuss in particular the inference systems corresponding to (a modified version of)Walley and Bernard’s imprecise Dirichlet multinomial models (IDMMs) and the Haldane inference system.