Markov categories provide a synthetic, axiom-based framework in which conditional independence, sufficiency, completeness, and classical theorems such as Basu and Bahadur hold uniformly across many probability theories.
From Kleisli Categories to Commutative C*-algebras: Probabilistic Gelfand Duality
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abstract
C*-algebras form rather general and rich mathematical structures that can be studied with different morphisms (preserving multiplication, or not), and with different properties (commutative, or not). These various options can be used to incorporate various styles of computation (set-theoretic, probabilistic, quantum) inside categories of C*-algebras. At first, this paper concentrates on the commutative case and shows that there are functors from several Kleisli categories, of monads that are relevant to model probabilistic computations, to categories of C*-algebras. This yields a new probabilistic version of Gelfand duality, involving the "Radon" monad on the category of compact Hausdorff spaces. We then show that the state space functor from C*-algebras to Eilenberg-Moore algebras of the Radon monad is full and faithful. This allows us to obtain an appropriately commuting state-and-effect triangle for C*-algebras.
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2019 1verdicts
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A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics
Markov categories provide a synthetic, axiom-based framework in which conditional independence, sufficiency, completeness, and classical theorems such as Basu and Bahadur hold uniformly across many probability theories.