I'm learning category theory and trying to boost my intuition by applying what I learn to the category of finite stochastic maps (FinStochMap, defined below).
FinStochMap seems a natural category for thinking about finite probability distributions, but I'm wondering how I can think about related concepts such as expectations. It seems we have to "go outside" the category in order to think about expected values, because in order to do so we have to associate a numerical value with each member of a set, and because the expected values themselves are neither objects nor morphisms in FinStochMap but numbers. Perhaps there is a way to do this by defining a functor from FinStochMap to some other category, but I don't see an obvious way to do it.
I'm particularly interested in picking out the set of all probability distributions with a particular expected value for some quantity. (Which is to say, I want to define mixture families in FinStochMap.) This would be a subset of $\mathrm{Hom}(\{*\},A)$.
Definition of FinStochMap. I'm sure this is a standard concept, but I include the definition just in case. By FinStochMap I mean the category where the objects are finite sets, and the morphisms are conditional probability distributions (or Markov kernels if you prefer), i.e. a morphism $f:A\to B$ is a function $p_f:A\times B \to [0,1]$, with elements written $p_f(b|a)$, such that $\sum_{b\in B} p(b|a)=1$ for all $a\in A$. The composition rule is that for $f:A\to B$ and $g:B\to C$, the composition $f;g:A\to C$ is given by $$ p_{f;g}(c|a) = \sum_{b\in B} p_f(b|a)p_g(c|b). $$ An identity morphism $I_A:A\to A$ is given by $p_{I_A}(a|a') = 1$ if $a=a'$, and 0 otherwise.
In FinStochMap, a morphism from the terminal object to an object $A$ can be seen as a probability distribution over $A$. To be concrete, consider a morphism in FinStochMap, $f:\{*\}\to A$. This gives us a set of numbers $p(a|*)$ for every $a\in A$, such that $\sum_{a\in A} p(a|*)=1$, so we can interpret $p(a|*)$ as the probability of $a$.
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