Abstract
Markov categories are a recent category-theoretic approach to the foundations of probability and statistics. Here we develop this approach further by treating infinite products and the Kolmogorov extension theorem. This is relevant for all aspects of probability theory in which infinitely many random variables appear at a time. These infinite tensor products ⨂i∈JXi come in two versions: a weaker but more general one for families of objects (Xi)i∈J in semicartesian symmetric monoidal categories, and a stronger but more specific one for families of objects in Markov categories.As a first application, we state and prove versions of the zero--one laws of Kolmogorov and Hewitt--Savage for Markov categories. This gives general versions of these results which can be instantiated not only in measure-theoretic probability, where they specialize to the standard ones in the setting of standard Borel spaces, but also in other contexts.
Highlights
Background on Markov categoriesWe assume familiarity with symmetric monoidal categories, up to and including string diagram notation and string diagram calculus for composite morphisms in symmetric monoidal categories.A symmetric monoidal category C is semicartesian if the monoidal unit I ∈ C is terminal
Applying the defining universal property of an infinite tensor product with respect to maps out of I implements the Kolmogorov extension theorem in C: probability measures on an infinite product XJ are in bijection with consistent families of probability measures on the finite products XF for F ⊆ J
If ((Xi, Σi))i∈N is a sequence of standard Borel spaces, the cartesian product XN = i∈N Xi, when equipped with the product σ-algebra ΣN, satisfies the universal property of an infinite tensor product with respect to maps out of I, since the marginalization maps implement a bijection between probability measures on XN = i∈N Xi and compatible families of probability measures on the finite subproducts XF = i∈F Xi, where the compatibility is with respect to marginalization to smaller subproducts specified by F ⊆ F. We prove that this implies the universal property in general: if A and all the Xi are standard Borel spaces, Markov kernels A → XJ are in bijection with compatible families of Markov kernels A → XF
Summary
We assume familiarity with symmetric monoidal categories, up to and including string diagram notation and string diagram calculus for composite morphisms in symmetric monoidal categories. Our goal is to use semicartesian monoidal categories in order to develop aspects of probability theory in categorical terms, in such a way that instantiating this theory in Stoch or BorelStoch recovers the standard theory As it turns out, doing so requires a bit more structure, in a form which has been axiomatized first by Cho and Jacobs [3] as affine CD-categories, very similar definitions occur in earlier work of Golubtsov [10]. In FinStoch and Stoch, the comultiplication morphisms are given by copying, i.e. by those stochastic matrices or Markov kernels which map an element x ∈ X to the Dirac delta measure at (x, x) ∈ X × X. If j∈F Xj is a finite product without any particular order on the factors, for p : A → j∈F Xj there is no ambiguity about whether p displays conditional independence or not, and we write ⊥j∈F Xj || A in case that it does
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