Why does Markov blanket contain the children's parents?

Question

I am quite confused about why Markov blanket contains children's parents. Wikipedia says

its children's parents also have to be included, because they can be used to explain away the node in question.

But what's the node in question? I also found another info from BayesiaLab. It says

The Spouses (or co-parents, dark green) are used for cutting the information coming from the ascendants of the Children (blue nodes). The Target node is marginally independent of the Spouses, but becomes conditionally dependent, i.e. when some evidence is available on the Children.

Why do the Target node and the Spouse become conditionally dependent when some evidence is available on the Children? What does it mean that some evidence is available on the Children?

I hope someone could help clarify it, esp. with a concrete example. Thanks:)

Answer 1

The reason is that the blanket should make the rest of the network conditionally independent of A.

Say the nodes are named S(pouse) and C(hild).

S    A
 \  /
  C

then, the output of C is dependent on S and A. To get the the rest of the network independent of A, you need to control both S and A. In other words, the output at the boundary of the blanket from other parents of the child are directly influenced by A. If you include the parents of the children of a you got everything that A can influence.

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Subquestions:

The 'node in question' is A.

They become conditionally dependent because you can know something about A if you know the spouse and the output of the child.

'Evidence available from the children' means you know some output of the child nodes.

Answer 2

Let Mrk(A) be Markov Blanket for a node A. Then, Pr(A | (Mrk(A), any other node) = Pr(A | Mrk(A))

For this to hold, children's parents needs to be included as well. Consider this example (taken from "Artificial Intelligence: A Modern Approach, 3rd Edition, by Stuart Russell and Peter Norvig. "

Suppose we have a graphical structure given as, Nodes: B: Burglary E: Earthquake A: Alarm J: John Call M: Marry Calls

Links are: B and E are parents of A: It means if burglary or earthquake happens then the alarm will ring.

Further A is parent of J and M: In case alarm rings, John and Marry (Good Neighbors) will call.

Thus structure is like (all arrow are downwards):

B   E
 \ /
  A
 / \
J  M

Here, Pr(B|A, E) != Pr(B|A), because Alarm could be explained by an Earth quack as well. Hence E need to be included to Mrk(B) as well.

Mathematically, Consider the probabilities as:

P(B) = 0.001, P(E) = 0.002

P(A|B,E) as: (please note P(A) will be affected by it's parent so needs to be given this way:

B E | Pr(A|B, E)
T T   0.95
T F   0.94
F T   0.29
F F   0.001

Then Pr(A|B=T) = Pr(E=T)*Pr(A|B=T, E=T) + Pr(E=F)*Pr(A|B=T, E=F) = 0.002*0.95 + 0.998*0.94 = 0.94002

while Pr(A|B=T, E=T) = 0.95 and Pr(A|B=T, E=F) = 0.94, both different then the above probability.