Presheaves on Markov Categories and Expectation Values

Categorical probability is the study of the structural aspects of probability, statistics and related fields by means of category-theoretic methods. Two categorical structures abound in the field:

1. 1.

   Via probability monads [1, 4], one models probabilistic maps as morphisms in Kleisli categories. The structure of algebra of a probability monad amounts to equipping the underlying space with a notion of forming expectation values. Therefore algebras are interesting because expectation values are among the most commonly used and fruitful ideas in probability theory.

2. 2.

   Markov categories [3] can be seen as an axiomatization of categories of probabilistic maps, including but not limited to the Kleisli categories of probability monads. Via their monoidal structure, they allow one to talk about stochastic independence, conditioning, sufficient statistics, etc.

In a plain Markov category, there is no general and simple way to talk about expectation values. In this work [2], we aim to address this shortcoming.

For example on $\mathrm{𝖥𝗂𝗇𝖲𝗍𝗈𝖼𝗁}$, the category of finite sets and stochastic matrices, there is a presheaf with $\mathrm{\Phi }(X)={ℝ}^X$, and where the action on morphisms is given by

$$\mathrm{\Phi }(f)(\varphi )(x)≔\sum _{y\in Y}f(x|y)\varphi (y).$$

When $f$ is deterministic, this reduces to the usual precomposition of functions. When the domain of $f$ is singleton, then this is precisely the usual notion of expectation value of $\varphi$ with respect to $f$.

We will argue that this results in a general framework for expectation values that is better behaved than algebras of a probability monad.¹¹1For example, with $\mathrm{𝖲𝗍𝗈𝖼𝗁}$ being the category of measurable spaces and Markov kernels, for every $n\in \mathbb{N}$ the hom-functors $\mathrm{𝖲𝗍𝗈𝖼𝗁}(\mathrm{\_},[-n,n])$ for $n\in \mathbb{N}$ are presheaves that model bounded measurable functions and their expectation values. The filtered colimit of these presheaves in the functor category $[{\mathrm{𝖲𝗍𝗈𝖼𝗁}}^{\mathrm{op}},\mathrm{𝖲𝖾𝗍}]$ is the presheaf of all bounded measurable functions, which plays a central role in traditional probability theory. But in the category of algebras of the Giry monad, the filtered colimit of the intervals $[-n,n]$ is trivial, i.e. the singleton space. Thus Idea 1 broadens the scope of categorical probability to certain quantitative aspects, such as conditional expectations, variance and covariance, and several inequalities, bringing the theory of categorical probability one step closer to the mathematical structures and methods used by practitioners. We expect that it opens up the possibility of developing many further aspects of traditional probability theory in the categorical setting.