Mateusz Bagiński comments on Some Rules for an Algebra of Bayes Nets

[Mateusz Bagiński]

$D_{K}L\left(P\right[X\left]||{\prod }_{i\le k}P\right[X_{i-1}|X_i\left]\right)P\left[X_k\right]\left({\prod }_{k\le i}P\right[X_{i+1}|X_i\left]\right)$

$=D_{K}L\left(P\right[X\left]||{\prod }_{i\le k+1}P\right[X_{i-1}|X_i\left]\right)P\left[X_{k+1}\right]\left({\prod }_{k+1\le i}P\right[X_{i+1}|X_i\left]\right)$

I think the parentheses are off here. IIUC you want to express the equality of divergences, not divergences multiplied by probabilities (which wouldn’t make sense I think).

${ϵ}_{\text{blanket}}geq{D}_{KL}\left(P\right[X,Y,Z\left]||P\right[X|Y\left]P\right[Y\left]P\right[Z|Y\left]\right)$

$=D_{KL}\left(P\right[X,Y,Z\left]||P\right[X,Y\left]P\right[Z,Y\left]/P\right[Y\left]\right)$

Typo: $geq$ → $\ge$

[johnswentworth]

Man, that top one was a mess. Fixed now, thank you!