Dirichlet is Natural
Vincent Danos, Ilias Garnier · Electronic Notes in Theoretical Computer Science · 2015
Giry and Lawvere's categorical treatment of probabilities, based on the probabilistic monad G , offer an elegant and hitherto unexploited treatment of higher-order probabilities. The goal of this paper is to follow this formulation to reconstruct a family of higher-order probabilities known as the Dirichlet process . This family is widely used in non-parametric Bayesian learning. Given a Polish space X , we build a family of higher-order probabilities in G ( G ( X ) ) indexed by M ⁎ ( X ) the set of non-zero finite measures over X . The construction relies on two ingredients. First, we develop a method to map a zero-dimensional Polish space X to a projective system of finite approximations, the limit of which is a zero-dimensional compactification of X . Second, we use a functorial version of Bochner's probability extension theorem adapted to Polish spaces, where consistent systems of probabilities over a projective system give rise to an actual probability on the limit. These ingredients are combined with known combinatorial properties of Dirichlet processes on finite spaces to obtain the Dirichlet family D X on X . We prove that the family D X is a natural transformation from the monad M ⁎ to G ∘ G over Polish spaces, which in particular is continuous in its parameters. This is an improvement on extant constructions of D X [17,26].