While I was reading about Bayesian networks, I run into "Markov blanket" term and got severely confused with its independency in a Bayesian network graph.
Markov blanket briefly says that every node is only dependent on its parents, children and children's parents [it is gray area for node A in the picture].
What is the joint probability of this BN, $P(M,S,G,I,B,R)$?
(source: aiqus.com)
If I follow the step parent only independency rule, it is:
$$ P(M | S)P(S | G,I)P(I | B)P(R | B)P(G)P(B)$$
However, if I follow the Markov Blanket independency, I end up with this (notice $P(I|\mathbf{G},B)$ is different):
$$P(M | S)P(S | G,I)P(I | \mathbf{G},B)P(R | B)P(G)P(B)$$
So which is the correct joint probability of this BN?
Update: Crosslink of this question in AIQUS
and
Respective chapter and diagrams are below:
alt text http://img828.imageshack.us/img828/9783/img0103s.png
alt text http://img406.imageshack.us/img406/3788/img0104l.png