Magdalena Wache comments on From Finite Factors to Bayes Nets

[Magdalena Wache]

Are you sure the “arrow forgetting” operation is correct?

You say there should be an arrow when

for all probability distributions $P$ over the elements of $N_i$, the approximation of $N_j$ over $P$ is no worse than the approximation of $N_i$ over $P$

wherein $\left[N_i\right]=[A\to B\leftarrow C]$ here, and $\left[N_j\right]=[A-B-C]$. If we take a distribution in which $A\perp /C|B$ , then $N_j$ is actually a worse approximation, because $A\perp C|B$ must hold in any distribution which can be expressed by a graph in $\left[N_j\right]$ right?

Also, I really appreciate the diagrams and visualizations throughout the post!

[J Bostock]

Thanks for the feedback. There’s a condition which I assumed when writing this which I have realized is much stronger than I originally thought, and I think I should’ve devoted more time to thinking about its implications.

When I mentioned “no information being lost”, what I meant is that in the interaction $A\to B$, each value $b\in B$ (where $B$ is the domain of $P_B$) corresponds to only one value of $a\in A$. In terms of FFS, this means that each variable must be the maximally fine partition of the base set which is possible with that variable’s set of factors.

Under these conditions, I am pretty sure that $A\perp C⟹A\perp C|B$