Synthetic topology in Homotopy Type Theory for probabilistic programming

The ALEA Coq library formalizes measure theory based on a variant of the Giry monad on the category of sets. This enables the interpretation of a probabilistic programming language with primitives for sampling from discrete distributions. However, continuous distributions have to be discretized because the corresponding measures cannot be defined on all subsets of their carriers. This paper proposes the use of synthetic topology to model continuous distributions for probabilistic computations in type theory. We study the initial σ-frame and the corresponding induced topology on arbitrary sets. Based on these intrinsic topologies, we define valuations and lower integrals on sets and prove versions of the Riesz and Fubini theorems. We then show how the Lebesgue valuation, and hence continuous distributions, can be constructed.

Weak bisimulations for the Giry monad

We study the existence of bisimulations for Kleisli morphisms associated with the Giry monad of subprobabilities over Polish spaces. We first investigate these morphisms and show that the problem can be reduced to the existence of bisimulations for objects in the base category of stochastic relations using simulation equivalent congruences. This leads us to a criterion for two objects to be bisimilar.