A convenient category for higher-order probability theory

Conference item

Higher-order probabilistic programming languages allow programmers to write sophisticated models in machine learning and statistics in a succinct and structured way, but step outside the standard measure-theoretic formalization of probability theory. Programs may use both higher-order functions and continuous distributions, or even define a probability distribution on functions. But standard probability theory does not handle higher-order functions well: the category of measurable spaces is not cartesian closed. Here we introduce quasi-Borel spaces. We show that these spaces: form a new formalization of probability theory replacing measurable spaces; form a cartesian closed category and so support higher-order functions; form a well-pointed category and so support good proof principles for equational reasoning; and support continuous probability distributions. We demonstrate the use of quasi-Borel spaces for higher-order functions and probability by: showing that a well-known construction of probability theory involving random functions gains a cleaner expression; and generalizing de Finetti’s theorem, that is a crucial theorem in probability theory, to quasi-Borel spaces.

Authors: Heunen, C; Kammar, O; Staton, S; Yang, H

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: IEEE
• Host title: 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
• Journal: Logic in Computer Science (LICS)
• Publication date: 2017-08-18
• Acceptance date: 2017-03-22
• DOI: 10.1109/LICS.2017.8005137
• ISSN: 1043-6871
• ISBN: 9781509030194