0
$\begingroup$

Given the joint probability and the factored term:

$ P(S,T,G,F,B) = P(G|B,F) \cdot P(S|T,F) \cdot P(T|B) \cdot P(B) \cdot P(F) $

I want to compute $P(S = s)$, thus I want to compute

$\sum_{T,G,F,B} P(S = s, T,G,F,B)$

Since I want to do the concrete calculation by hand, I would like to eliminate some terms in order to ease the process. Is there anything which could be done? I am thinking of making use of $\sum_X P(X|Y) = 1$

  • 0
    @RobertIsrael Exactly, sorry for my sloppy notation.2012-05-21

1 Answers 1

1

As per my comment, you are really saying $ P(S=s,T=t,G=g,F=f,B=b) = P(G=g|B=b,F=f)P(S=s|T=t,F=f)P(T=t|B=b)P(B=b)P(F=f)$ for discrete random variables $S,T,G,F,B$.
Sum both sides over $g$ and this says $ P(S=s,T=t,F=f,B=b) = P(S=s|T=t,F=f)P(T=t|B=b)P(B=b)P(F=f)$ Now $P(T=t) = \sum_b P(T=t|B=b) P(B=b)$, so calculate this for each $t$. Then summing over $b$, $P(S=s,T=t,F=f) = P(S=s|T=t,F=f) P(T=t) P(F=f)$ To calculate $P(S=s)$, you'll have to do a double sum of this over all $t$ and $f$.