Borel Kernels and their Approximation, Categorically

Fredrik Dahlqvist, Alexandra Silva, Vincent Danos, Ilias Garnier

Research output: Contribution to journalArticlepeer-review

Abstract

This paper introduces a categorical framework to study the exact and approximate semantics of probabilistic programs. We construct a dagger symmetric monoidal category of Borel kernels where the dagger-structure is given by Bayesian inversion. We show functorial bridges between this category and categories of Banach lattices which formalize the move from kernel-based semantics to predicate transformer (backward) or state transformer (forward) semantics. These bridges are related by natural transformations, and we show in particular that the Radon-Nikodym and Riesz representation theorems - two pillars of probability theory - define natural transformations.

With the mathematical infrastructure in place, we present a generic and endogenous approach to approximating kernels on standard Borel spaces which exploits the involutive structure of our category of kernels. The approximation can be formulated in several equivalent ways by using the functorial bridges and natural transformations described above. Finally, we show that for sensible discretization schemes, every Borel kernel can be approximated by kernels on finite spaces, and that these approximations converge for a natural choice of topology.

We illustrate the theory by showing that our approximation scheme can be used in practice as an approximate Bayesian inference algorithm and as an approximation scheme for programs in the probabilistic network specification language ProbNetKAT.
Original languageEnglish
Pages (from-to)91-119
Number of pages29
JournalElectronic Notes in Theoretical Computer Science
Volume341
DOIs
Publication statusPublished - 11 Dec 2018
Event34th Conference on the Mathematical Foundations of Programming Semantics (MFPS 2018) - Dalhousie University, Halifax, Canada
Duration: 6 Jun 20189 Jun 2018
https://www.mathstat.dal.ca/mfps2018/

Keywords

  • Probabilistic programming
  • probabilistic semantics
  • Markov process
  • Bayesian inference
  • approximation

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