Dirichlet is Natural

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 DX on X . We prove that the family DX 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 DX [17,26].