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Given the following Bayes network:

BayesNet

with

  • $p(k=t)=.2$
  • $p(o=t)=.1$
  • $p(s=t|k=f,o=f)=.0$
  • $p(s=t|k=f,o=t)=.2$
  • $p(s=t|k=t,o=f)=.5$
  • $p(s=t|k=t,o=t)=.95$

how would I calculate $p(s=t|o=t)$ and $p(o=t|s=t)$?

I tried the following way:

$p(s=t|o=t) = \dfrac{p(s=t \land o=t)}{p(o=t)} = \dfrac{p(s=t)p( o=t | s=t)}{p(o=t)}$, which obviously doesn't make sense, since I have to caluclate $ p( o=t | s=t)$ with Bayes rule.

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From the graph you know $P(s|k,o)$, so to get $P(s|o)$ you need to sum over all possible values of $k$, it will be then:

$$ P(s=t|o=t) = \sum_{i \in {t,f}} P(s=t|o=t,k=i) $$

which will be equal to :

$$ P(s=t|o=t) = P(s=t|o=t,k=f) + P(s=t|o=t,k=t) $$ In your case it doesn't sum to one your distribution, I think you should change the values.

Hope it helps

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