nLab category of couplings

Idea

The category of couplings, sometimes denoted Krn or ProbStoch, is a category of probability spaces and transport plans between them.

This category is naturally a dagger category, as transport plans can be interpreted as going in either direction. (One can see that as an instance of Bayesian inversion.)

This category is used in categorical probability to model, abstractly, properties which only hold almost surely, i.e. only up to an event of zero probability (zero measure).

Construction

There are two equivalent definitions of the category of couplings, either in terms of couplings, or in terms of equivalence classes of Markov kernels.

In terms of couplings

The category Krn has

• As objects, standard Borel probability spaces, i.e. triples $(X,\mathcal{A},p)$ where $X$ is a set $\mathcal{A}$ is a sigma-algebra on $X$ making $(X,\mathcal{A})$ a standard Borel space, and $p$ is a probability measure on $(X,\mathcal{A})$;

• as morphisms, couplings between probability spaces.

The identity and composition couplings are described here.

In terms of Markov kernels

The category Krn has

• As objects, standard Borel probability spaces (see above);

• as morphisms, equivalence classes of measure-preserving Markov kernels under almost sure equality.

The identity and composition are constructed as in Stoch.

Equivalence of the definitions

Given a measure-preserving Markov kernel $k:(X,\mathcal{A},p)\to(Y,\mathcal{B},q)$, one can define a coupling canonically as follows,

$$r_k(A\times B) = \int_A k(B|x)\,p(dx)$$

for all $A\in\mathcal{A}$ and $B\in\mathcal{B}$. (See also here.)

Conversely, whenever $(X,\mathcal{A},p)$ and $(Y,\mathcal{A},q)$ are standard Borel, given a coupling $r$ on $(X\times Y,\mathcal{A}\otimes\mathcal{B})$ one can form the regular conditional distribution $r':(X,\mathcal{A},p)\to(Y,\mathcal{A},q)$, which is a measure-preserving kernel, defined up to almost sure equality.

As one can check, these two assignment are mutually inverse, so that the two definitions of Krn give isomorphic categories.

Basic structures and properties

Dagger structure

(…)

Generalizations and extensions

For non-standard-Borel measurable spaces

(…)

For abstract Markov categories

(…)

References

The category Krn was originally defined in

• Fredrik Dahlqvist, Vincent Danos, Ilias Garnier, and Alexandra Silva, Borel kernels and their approximation, categorically. In MFPS 34: Proceedings of the Thirty-Fourth Conference on the Mathematical Foundations of Programming Semantics, volume 341, 91–119, 2018. arXiv.

It also appears in the following works:

• Tobias Fritz, A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics. Adv. Math., 370:107239, 2020. arXiv:1908.07021.

• Noé Ensarguet, Paolo Perrone, Categorical probability spaces, ergodic decompositions, and transitions to equilibrium. arXiv.

• Dexter Kozen, Alexandra Silva, Erik Voogd, Joint Distributions in Probabilistic Semantics, MFPS 2023. (arXiv)

• Paolo Perrone and Ruben Van Belle, Convergence of martingales via enriched dagger categories, 2024. (arXiv)

Related concepts

• Markov kernel
• coupling (probability)
• dagger category
• Stoch
• categorical probability

category: category, probability