Exercise 10.3 - Markov blanket for a DGM

The trick in the required reduction is to partition all variables into:

where $𝒴=\text{ch}(X_i)$, $𝒜$ contains the collection of all variable that are topological smaller than $X_i$, hence $\text{Pa}(X_i)\subset 𝒜$. Moreover, we have $𝒜$ incorporate:

So all elements of $𝒜$ can be safely evaluated before $X_i$. Then all $X_i$’s children can be computed according to a topological order. Finally, the rest variables $B$ can be evaluated.

We have:

Eliminating the terms $p(𝒜)$ and $p(B|𝒜,𝒴)$ from both the numerator and the denominator we have (10.58).