This post assumes knowledge of category theory, finite factored sets, and Bayes nets.

The Setup

I've already talked about DAGs and factor overlap Venn diagrams in a previous post, where I studied them within a category-theoretic framework. Here I'll also perform an explicit construction of them using set theory.

DAGs

I have already discussed the set of directed acyclic graphs over $n$ elements. We will denote the set of all DAGs of $n$ elements as $D A G (n)$ . Each Bayes net can be thought of as a set of pairs of elements ${(i \in {1, . . ., n}, j \neq i \in {1, . . ., n}), . . .}$ .

This set can be converted into the category of Bayes nets over $n$ elements by the addition of morphisms corresponding to "bookkeeping"-type relationships, which we will denote ${D A G}_{n}$ . From this category, we can form a category whose elements are sets of the elements of ${D A G}_{n}$ , subject to the following condition on a set $S$ :

(I'll slightly go against standard notation by using calligraphic acronyms for my category names i.e. ${D A G}_{n}$ . I don't feel like any of my categories are definitely natural or useful enough to earn a "proper" bold name like ${D A G}_{n}$ )

$\forall s \in S, b \in D A G (n), s \to b ⟹ b \in S$

This means that, if we can reach a given Bayes net $b$ from any element $s$ of our set $S$ , that Bayes net $b$ must also be in $S$ . I refer to these as compatible sets and denote the category ${C S B}_{n}$ (standing for compatible sets of Bayes nets) as the category whose elements are compatible sets of $n$ elements and for any two elements. To write it out fully:

$O b ({C S B}_{n}) = {S \in P (D A G (n)) ∣ ∣ \forall s \in S, d \in D A G (n), {D A G}_{n} (s, d) = {\to} ⟹ d \in S}$

${C S B}_{n} (A, B) = {\begin{matrix} {\to} & A \supseteq B {} & A \subset B \end{matrix}$

This should be read as "There is a unique morphism from $A$ to $B$ if and only if $A$ is a weak superset of $B$ , otherwise there is no morphism". Orderings of this form always follow the rules required to create a category.

(Aside: sometimes we think about equivalence classes of Bayes nets. If we choose, we can first convert our Bayes nets to equivalence classes, then convert them to compatible sets, but this is not needed here)

This category has an initial object ${D i s}_{n}$ which is a set consisting of the discrete Bayes net (with no arrows at all) and therefore all Bayes nets. Less trivially it has a terminal object ${I n d}_{n}$ which contains exactly the Bayes nets which ...

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