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## A Convenient Category for Higher-Order Probability Theory

Resource type

Authors/contributors

- Heunen, Chris (Author)
- Kammar, Ohad (Author)
- Staton, Sam (Author)
- Yang, Hongseok (Author)

Title

A Convenient Category for Higher-Order Probability Theory

Abstract

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.

Publication

arXiv:1701.02547 [cs, math]

Date

2017-01-10

Accessed

2019-10-10T11:48:09Z

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Extra

arXiv: 1701.02547

Citation

Heunen, C., Kammar, O., Staton, S., & Yang, H. (2017). A Convenient Category for Higher-Order Probability Theory.

*ArXiv:1701.02547 [Cs, Math]*. Retrieved from http://arxiv.org/abs/1701.02547
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