Predictive inference under exchangeability and the Imprecise Dirichlet Multinomial Model
This talk (and the extremely long paper to go with it) marks the happy end of a spontaneous and informal research project that started in 2004, branched off in a surprising number of directions, and taught me so much about mathematics in general, and the foundations of probability theory in particular. Along the way, I needed all the help I could get, and got it from Enrique Miranda, Erik Quaeghebeur, Jasper De Bock and Márcio Diniz.
Here’s the 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 is 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 an 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.