A Bayesian network with 4 nodes question

Question

Hello. I need to find the following probabilities:

$ \qquad a. P(\neg A, B, \neg C, D) \\ \qquad b. P(A|B,C,D) $

This is my solution:

I would like to get an opinion on my solution.

Answer 1

Your answer looks correct to me. The Bayesian network encodes some independence relationships between events. In particular, your network encodes the fact that $A$ and $B$ are independent, but may be dependent given $D$. This is the so-called "collider" pattern. $A$ and $B$ are marginally independent. The relationship between $A$, $C$ and $D$ is of a fork. $C$ and $D$ are conditionally independent, given $A$.

In your answer, you use the above facts to write

$P(A,B,C,D)=P(C,D|A,B)P(A)P(B)=P(C|A,B)P(D|A,B)P(A)P(B)$

the first equality follows from the marginal independence between $A$ and $B$ and the second from the conditional indepedence between $C$ and $D$.

Finally,

$P(A,B,C,D)=P(C|A)P(A,B)P(D|A,B)=P(C|A)P(A)P(B)P(D|A,B)$

This is the typical result, that the joint distribution is the product of the conditional distributions of events given their parents.

https://en.wikipedia.org/wiki/Bayesian_network