De Finetti’s Theorem in Categorical Probability

Towards a Categorical Account of Conditional Probability

This paper presents a categorical account of conditional probability, covering both the classical and the quantum case. Classical conditional probabilities are expressed as a certain “triangle-fill-in” condition, connecting marginal and joint probabilities, in the Kleisli category of the distribution monad. The conditional probabilities are induced by a map together with a predicate (the condit...

A Categorical Basis for Conditional Probability

This paper identifies several key properties of a monad that allow us to formulate the basics of conditional probability theory, using states for distributions/measures and predicates for events/probability density functions (pdf’s). The distribution monad for discrete probability and the Giry monad for continuous probability are leading examples. Our categorical description handles discrete an...

A Categorical Approach to Probability Theory

Computably Categorical Fields via Fermat's Last Theorem

We construct a computable, computably categorical field of infinite transcendence degree over Q, using the Fermat polynomials and assorted results from algebraic geometry. We also show that this field has an intrinsically computable (infinite) transcendence basis.

Cumulants in Noncommutative Probability Theory Iv. Noncrossing Cumulants: De Finetti’s Theorem, L-inequalities and Brillinger’s Formula

In this paper we collect a few results about exchangeability systems in which crossing cumulants vanish, which we call noncrossing exchangeability systems. De Finetti’s theorem states that any exchangeable sequence of random variables is conditionally i.i.d. with respect to some σ-algebra. In this paper we prove a “free” noncommutative analog of this theorem, namely we show that any noncrossing...