A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics — Tobias Fritz (2019). arXiv:1908.07021 (v8, PDF); Adv. Math. 370, 107239 (2020).
The standard reference for Markov categories: symmetric monoidal categories with copy and (natural) delete, as a synthetic setting for probability and statistics. Develops many examples (finite and measurable stochastic maps, Gaussians, Kleisli categories of affine monads, diagram categories, hypergraph categories), deterministic morphisms, conditionals and disintegration, conditional independence with the semigraphoid laws, almost-sure equality, and synthetic proofs of the Fisher–Neyman factorisation theorem and the theorems of Basu and Bahadur.
Sources: the paper, arXiv:1908.07021v8, checked against the arXiv listing. Index: Papers.
Key definitions and results
- Definition 2.1: Markov category
- §3–6: Kleisli categories of monoidal monads, Stoch, the Radon monad, Gauss
- Definition 10.1: deterministic morphisms; Remark 10.13: they form a cartesian subcategory
- Definitions 11.1, 11.5, Proposition 11.17: conditionals and disintegration
- §12: conditional independence and the semigraphoid properties
- §13: almost surely; Remark 13.10: Bayesian inversion as a dagger
- §14–16: sufficient statistics, Basu, Bahadur
Concept notes
Markov Category, Copy-Discard Category, Conditionals and Disintegration, Conditional Independence, Almost-Sure Equality, Giry Monad, Bayesian Inversion
Used in Lenticulum.jl
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