Probability monads with submonads of deterministic states - Extended version
arXiv:2204.07003 · doi:10.1145/3531130.3533355
Abstract
Probability theory can be studied synthetically as the computational effect embodied by a commutative monad. In the recently proposed Markov categories, one works with an abstraction of the Kleisli category and then defines deterministic morphisms equationally in terms of copying and discarding. The resulting difference between 'pure' and 'deterministic' leads us to investigate the 'sober' objects for a probability monad, for which the two concepts coincide. We propose natural conditions on a probability monad which allow us to identify the sober objects and define an idempotent sobrification functor. Our framework applies to many examples of interest, including the Giry monad on measurable spaces, and allows us to sharpen a previously given version of de Finetti's theorem for Markov categories. This is an extended version of the paper accepted for the Logic In Computer Science (LICS) conference 2022. In this document we include more mathematical details, including all the proofs, of the statements and constructions given in the published version.
arXiv comment16 pages. Extended version of paper accepted for LICS 2022 conference
References in corpus (10)
- A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics
- Disintegration and Bayesian Inversion via String Diagrams
- A Convenient Category for Higher-Order Probability Theory
- Denotational validation of higher-order Bayesian inference
- Commutative monads as a theory of distributions
- De Finetti's Theorem in Categorical Probability
- Infinite products and zero-one laws in categorical probability
- Bimonoidal Structure of Probability Monads
- Paradoxes of Probabilistic Programming
- Probability monads with submonads of deterministic states - Extended version