A domain theory for statistical probabilistic programming

We give an adequate denotational semantics for languages with recursive higher-order types, continuous probability distributions, and soft constraints. These are expressive languages for building Bayesian models of the kinds used in computational statistics and machine learning. Among them are untyp...

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Bibliographic Details
Published in:Proceedings of ACM on programming languages Vol. 3; no. POPL; pp. 1 - 29
Main Authors: Vákár, Matthijs, Kammar, Ohad, Staton, Sam
Format: Journal Article
Language:English
Published: New York, NY, USA ACM 02.01.2019
Subjects:
Bayesian analysis
Compilers
Computing methodologies
Context specific languages
Denotational semantics
Domain specific languages
Functional languages
General programming languages
Interpreters
Language types
Machine learning
Machine learning theory
Models of computation
Probabilistic computation
Program semantics
Semantics and reasoning
Software and its engineering
Software notations and tools
Theory and algorithms for application domains
Theory of computation
adequacy
probability
denotational semantics
recursion
domain theory
ISSN:2475-1421, 2475-1421
Online Access:Get full text
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Summary:We give an adequate denotational semantics for languages with recursive higher-order types, continuous probability distributions, and soft constraints. These are expressive languages for building Bayesian models of the kinds used in computational statistics and machine learning. 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. 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. As we show, quasi-Borel predomains form both a model of Fiore's axiomatic domain theory and a model of Kock's synthetic measure theory.
ISSN:2475-1421
2475-1421
DOI:10.1145/3290349