I did a quick literature search and found nothing on "statistical models of functions". Let me explain what I am looking for. Given the category of Sets and Function, we have arbitrary functions defined: $f:A \rightarrow B$. How can we have data about a function? How can we have a statistical model of this function? This data and its corresponding probability theory would allow us to, say, compute an entropy of the model. Let us say that $\mathcal{M}_f$ is a model of $f$. Then we can compute the entropy $H(\mathcal{M}_f)$. The entropy would be zero for models of that can, arguably, be said to simply be functions. Something like this exists when $g: \mathbb{R}^n \rightarrow \mathbb{R}^m$ for some $m, n$, and this is known as Gaussian Processes. You could model the data about a function in several ways. For instance, the category of sets and relations has a special subcategory that is just sets and functions. That is, under certain circumstances, a relation is just a function: zero entropy. Anything else could have higher entropy. A span could be seen as a function, when, for every term $a_i$ and its set element $s_i$, the span always "connects" the same set elements to the same set elements. These would have zero entropy. Anything else would have nonzero entropy. I believe we need a probability theory for this.

If you have seen my other post about multisets and spans, you can see that I am trying to build this into a category. My ultimate goal would be to do the following: define a monad, $\mathcal{M}$ on Set that sends a set to its set of multisets with morphisms (and in time, the composition) defined in the post above. With this monad, we treat it like the multiset monad and find a probability measures monad for it as we see here, and here. We also find the Eilenberg-Moore category for this, $EM(\mathcal{M})$, and then analyze it to understand its probability theory.

How far off am I? Does any of this sound reasonable? Is it possible that I am just looking for the monad of measures of finite support, along with its EM category, defined here as $\mathcal{G}_{fin}$