Exploring Synthetic Topology in Homotopy Type Theory for Probabilistic Programming 🚀

Talk at the Categorical Probability and Statistics workshop 2020:
http://perimeterinstitute.ca/personal/tfritz/2019/cps_workshop/

Title: Synthetic topology in Homotopy Type Theory for probabilistic programming
Speaker: Bas Spitters
Chair: Prakash Panangaden
Date: June 7th, 2020

Abstract:
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.

Martin E. Bidlingmaier, Florian Faissole, Bas Spitters
arXiv:1912.07339