Abstract

I will describe our recent work on statistical probabilistic programming languages. These are expressive languages for describing generative Bayesian models of the kinds used in computational statistics and machine learning. We give an adequate denotational semantics for a calculus with recursive higher-order types, continuous probability distributions, and soft constraints. Among them are untyped languages, similar to Church and WebPPL, because our semantics allows recursive mixed-variance datatypes. Our semantics justifies important program equivalences including commutativity.

Our new semantic model is based on `quasi-Borel predomains’. These are a mixture of chain-complete partial orders (cpos) and quasi-Borel spaces. Quasi-Borel spaces are a recent model of probability theory that focuses on sets of admissible random elements. I will give a brief introduction to quasi-Borel spaces and predomains, and their properties.

Probability is traditionally treated in cpo models using probabilistic powerdomains, but these are not known to be commutative on any class of cpos with higher-order functions. By contrast, quasi-Borel predomains do support both a commutative probabilistic powerdomain and higher-order functions, which I will describe.

For more details on this joint work with Matthijs Vákár and Sam Staton, see:

Matthijs Vákár, Ohad Kammar, and Sam Staton. 2019. A Domain Theory for Statistical Probabilistic Programming. Proc. ACM Program. Lang. 3, POPL , Article 36 (January 2019), 35 pages., DOI : 10.1145/3290349.