First of all, what is conditional independence in a Bayes Net?

In a Bayes Net, each node represents an event (or variable). When we want to know if two nodes A and B are conditionally independent given a node C, we're looking to see if P(A,B|C) = P(A|C)P(B|C), which is equivalent to P(A|C)=P(A|B,C).

In other words, if we know C, and we want to know the probability of A happening, knowing about B does not give us any additional information that influences our belief of when A can happen. Likewise, if we know C, and we want to know the probability of B happening, knowing about A doesn't give us any additional information that influences our belief of when B can happen. Knowing that A and B are conditionally independent given C tells us that if we want to know P(A,B|C), we can multiply P(A|C) and P(B|C) to get P(A,B|C).

Theorem (from lecture): P(X,Y|E) = P(X|E)P(Y|E) iif every undirected path from a node in X to a node in Y is "d-separated" (or "blocked") by E.

Definition of d-separation:
Path p is d-separated if there exists a node Z in path p where at least 1 of the following 3 conditions is true:

1. Observed event in causation, where Z is a member of E, the evidence set.

2. Observed common explanation, where Z is a member of E, the evidence set.

3. No observed common effect, where Z and the descendants of Z are NOT members of E, the evidence set.

To find conditional independence:
1. Identify the nodes in sets X, Y, and E
2. List all undirected paths between X and Y
3. Put the edge directions back into consideration of those paths in 2, and mark the evidence nodes.
4. Check each path for d-separation.

Examples

Are X and Y conditionally independent given E?

List of paths that are d-separated with their condition:

Practice Problems