Is there any categorical version of central limit theorem?

Question

I'm not sure if the question even makes sense, but I wonder if there's any categorical reason that explains importance of Gaussian/normal distribution. In the ordinary probability theory, I guess central limit theorem explains why the normal distribution is important (or at least shows up a lot). I just found there's something called categorical probability theory (and also this) that I don't fully understand. In this viewpoint, do we still have any central limit "theorem"? Otherwise, can we still give any meaningful result about "categorical" normal distribution (whatever it is) - would be great if it satisfy certain universal property.

Answer 1

$\newcommand{\R}{\mathbb{R}}$ Although this doesn't answer the question, it may be worth noting that there is a Markov category of Gaussian distributions and Gaussian Markov kernels in in Section 6 of Fritz's paper A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics.

The objects in the category are the natural numbers. The morphisms $n\to m$ are the triples of matrices $(M,C,s)\in\R^{m\times n}\times\R^{m\times m}\times\R^{m\times1}$. Such a triple can be thought of as the linear statistical model $$Y:=MX+\xi, \tag{1}\label{1}$$ where $X$ is a random vector in $\R^{n\times1}$ and $\xi$ is a Gaussian random vector in $\R^{m\times1}$ (independent of $X$) with mean $s$ and covariance matrix $C$. This statistical model can alternatively be viewed as a transition kernel as they appear for example in Gauss–Markov models (where usually $s = 0$ is assumed).

The composition of two morphisms $$(M,C,s)\colon n\to m\quad\text{and}\quad(N,D,t)\colon m\to k$$ is then obtained by recalling \eqref{1} and similarly writing $Z:=NY+\eta$, with $\eta$ independent of $\xi$ — to get $Z=NMX+N\xi+\eta$, with $E(N\xi+\eta)=Ns+t$ and $\operatorname{Cov}(N\xi+\eta)=NCN^T+D$, thus resulting in the following definition of the composition of the two morphisms: $$(N,D,t)\circ(M,C,s):=(NM,NCN^T+D,Ns+t).$$

To see how this category is relevant in probability and statistics, consider for example the Kalman filter. If one formulates the theory of hidden Markov models and the Bayes filter for Markov categories in general, then this theory can be instatiated in $\mathsf{Gauss}$, where it specializes to the usual formulas that define the Kalman filter. This is part of a preprint that should appear very soon, see also this talk at ACT 2023.

Final remark: One can also construct a larger Markov category containing $\mathsf{Gauss}$ by dropping the assumption of $\xi$ being Gaussian, and taking general linear statistical models as morphisms that compose in a similar way.