Section: New Results
Dirichlet is Natural
Participants : Vincent Danos [correspondant] , Ilias Garnier.
In [32] the authors reconstruct a family of higher-order probabilities known as the Dirichlet process.
Giry and Lawvere's categorical treatment of probabilities, based on the
probabilistic monad
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 on