0
$\begingroup$

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, 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 $\mathcal{G}_{fin}$

Improve this question
$\endgroup$
5
  • $\begingroup$ Please let me know how I can improve the question before voting to close. $\endgroup$
    – Ben Sprott
    Commented Nov 27, 2018 at 19:39
  • 2
    $\begingroup$ I'm trying to be as nice as I can since I believe you may not be a mathematician. I base this on the rather imprecise and disorganized presentation. You seek a probability measure on a set of functions with no particular purpose in mind other than to compute its entropy. This in itself requires more care because the entropy is associated to a pair of probability measures, one of which is absolutely continuous with respect to the other. It would help if you discussed a concrete situation that inspired you. $\endgroup$
    – Liviu Nicolaescu
    Commented Nov 27, 2018 at 20:13
  • 1
    $\begingroup$ @LiviuNicolaescu Thanks. I will try to improve with your suggestions. $\endgroup$
    – Ben Sprott
    Commented Nov 27, 2018 at 20:31
  • 3
    $\begingroup$ I got stuck at "How can we have data about a function?"What does that mean? What does it mean for data to be about a function? $\endgroup$
    – Dan Piponi
    Commented Nov 27, 2018 at 22:55
  • $\begingroup$ @DanPiponi Hi Dan, I am revisting this question. Do you have any thoughts on my question? The example I am working on are databases. In particular, one may have two columns of a database connected by a foreign key. This is a span. Do the two columns form a function? Probably not, however, they may contain data about a function. Take the example I give above where we see the columns as multisets. If every set element is pointed to the same set element via the span, then I am saying that the span exactly specifies a function. The span gives "data about a function" $\endgroup$
    – Ben Sprott
    Commented Dec 21, 2018 at 17:45

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .