johnswentworth comments on Some Rules for an Algebra of Bayes Nets

[johnswentworth]

Proof that the quoted bookkeeping rule works, for the exact case:

• The original DAG $G$ asserts $P\left[X\right]={\prod }_{i}P\left[X_i|X_{p{a}^G\left(i\right)}\right]$

• If $G^{\prime }$ just adds an edge from $j$ to $k$, then $G^{\prime }$ says $P\left[X\right]=P[X_k|X_{p{a}^G\left(k\right)},X_j]{\prod }_{i\ne k}P\left[X_i|X_{p{a}^G\left(i\right)}\right]$

• The original DAG’s assertion $P\left[X\right]={\prod }_{i}P\left[X_i|X_{p{a}^G\left(i\right)}\right]$ also implies $P[X_k|X_{p{a}^G\left(k\right)},X_j]=P\left[X_k|X_{p{a}^G\left(k\right)}\right]$, and therefore implies $G^{\prime }$’s assertion $P\left[X\right]=P[X_k|X_{p{a}^G\left(k\right)},X_j]{\prod }_{i\ne k}P\left[X_i|X_{p{a}^G\left(i\right)}\right]$.

The approximate case then follows by the new-and-improved Bookkeeping Theorem.

Not sure where the disconnect/​confusion is.

[Lorxus]

Let me see if I’ve understood point 3 correctly here. (I am not convinced I have actually found a flaw, I’m just trying to reconcile two things in my head here that look to conflict, so I can write down a clean definition elsewhere of something that matters to me.)

$P$ factors over $G$. In $G$, $X_j,X_k$ were conditionally independent of each other, given $X_{p{a}^G\left(k\right)}$. Because $P$ factors over $G$ and because in $G$, $X_j,X_k$ were conditionally independent of each other, given $X_{p{a}^G\left(k\right)}$, we can very straightforwardly show that $P$ factors over $G^{\prime }$, too. This is the stuff you said above, right?

But if we go the other direction, assuming that some arbitrary $P^{\prime }$ factors over $G^{\prime }$, I don’t think that we can then still derive that $P^{\prime }$ factors over $G$ in full generality, which was what worried me. But that break of symmetry (and thus lack of equivalence) is… genuinely probably fine, actually—there’s no rule for arbitrarily deleting arrows, after all.

That’s cleared up my confusion/​worries, thanks!