conditional probability of joints in bayesian net

Question

I have been staring at a bayesian net for an hour and can't understand why this is correct to write:

$$P(W|B,E)\cdot P(E)\cdot P(R|E)= P(W,R,E|B)$$

Note that the joint probability of $P(A,B,E,W,R)$ can be decomposed as follows according to the bayesian net structure:

$$P(A,B,E,W,R) = P(B)\cdot P(E)\cdot P(A|B,E)\cdot P(R|E)\cdot P(A|B,E)\cdot P(W|A)$$

Answer 1

By the Chain Rule, we have:

\begin{eqnarray*} P(W,R,E\mid B) &=& P(W\mid R,E,B)\;P(R\mid E,B)\;P(E\mid B) \\ &=& P(W\mid E,B)\;P(R\mid E)\;P(E). \end{eqnarray*}

This last equality because:

• Given $E,B$ then $W$ is conditionally independent of $R$, so $P(W\mid R,E,B) = P(W\mid E,B)$
• Given $E$ then $R$ is conditionally independent of $B$, so $P(R\mid E,B) = P(R\mid E)$
• $B,E$ are independent events, so $P(E\mid B) = P(E)$.