Computing conditional probability with proxy variable Suppose $P(X | Y)$.
If I only know $P(X | Z)$ and $P(Z | Y)$, can I do 
$$       
       P(X|Y) = P(X|Z)P(Z|Y)? 
$$

We toss a fair coin independently three times. Let $X$ be the event head on the first toss, $Y$ the event head on the second, and $Z$ the event head on the third.
Then $\Pr(X\mid Y)=1/2$ and $\Pr(X\mid Z)\Pr(Z\mid Y)=(1/2)(1/2)$.
