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 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](https://mathoverflow.net/questions/315897/the-category-of-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](https://mathoverflow.net/questions/310704/map-from-the-multiset-monad-to-the-giry-monad-from-data-to-probabilities), [and here](https://mathoverflow.net/questions/312148/what-is-the-category-of-algebras-for-the-finitely-supported-measures-monad).  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](https://arxiv.org/pdf/0903.5522.pdf) as $\mathcal{G}_{fin}$