Abstract
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].
Highlights
It has been argued that exact bisimulations between Markovian systems are better conceptualized using the more general notion of bisimulation metrics [29]
We keep the idea of using a robust means of comparison, but we add a second idea: namely to introduce a way of quantifying the uncertainty in the chains being compared
To quantify uncertainty in the Markov chains, we propose to explore in the longer term concepts of “uncertain Markov chains” as elements of type X → G2(X), where X is an object of Pol, the category of Polish spaces and G is the Giry probability functor
Summary
It has been argued that exact bisimulations between Markovian systems are better conceptualized using the more general notion of bisimulation metrics [29]. To quantify uncertainty in the Markov chains, we propose to explore in the longer term concepts of “uncertain Markov chains” as elements of type X → G2(X), where X is an object of Pol, the category of Polish spaces (separable and completely metrisable spaces) and G is the Giry probability functor. This is to say that the chain takes values in “random probabilities” (ie probabilities of probabilities). For finite Xs this setup poses no difficulty, but for more general spaces, one needs to construct a computational handle on G2(X) - the space of uncertain or higher-order probabilities This is what we do in this paper. Weak convergence of probability measures is treated in [7,27]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
