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I've made this Bayes net based on a problem and I'm trying to find the probability of W but I'm stuck. I know I probably have to use Bayes theorem backwards through to find $P(W)$, but I'm not sure how to go about it. Any hints would be greatly appreciated.

Bayes Net

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    $\begingroup$ You have not provided enough information, such as several conditional probabilities. $\endgroup$
    – David G. Stork
    Commented Oct 4, 2016 at 1:18
  • $\begingroup$ Start by writing out the joint distribution over all variables and then begin eliminating or marginalizing out variables according to your bayes net. Then you'll likely realize that you're missing some conditional probabilities. $\endgroup$
    – ilanman
    Commented Oct 12, 2016 at 16:58

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Let's help get you started. First, you should write out the conditional probabilities associated with your bayes net. Note that in general we can write a joint probability:

$$P(x_1,x_2, ..., x_n) = P(X_1 = x_1, \ldots, X_n = x_n) = \prod_{i=1}^{n}P(x_i|parents(X_i))$$

In your second bayes net, for example, $S$ has two parents, $F$ and $O$, and neither $F$ nor $O$ have any parents. So we can write $$P(s, f, o) = P(S=s, F=f, O=o) = \sum_{FO}P(s|F,O)P(f)P(o)$$

Note that we need to sum over all possible values of $F,O$. That is:

F=true, O=true
F=true, O=false
F=false, O=true
F=false, O=false

For example, $P(s|F=false, O=true) = 0.2$, $P(f) = P(F=true) = 0.2$ and $P(o) = P(O=true) = 1$.

Can you figure out the rest? Are you missing any conditional probabilities?

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