Say I have a simple Bayes Net that appears like that in the picture and am giving the following probabilities:
$P(y|x) = 0.5$
$P(z|x)=0.4$
$P(y|\bar{x})=0.8$
$P(z|\bar{x}) = 0.9 $
How would I calculate the following, or is it not possible to calculate them? I think I need to know $P(x)$ to be able to calculate them:
$P(y)$
$P(x|y \land z)$
$P(x|y)$
Yes.
$P(y)=P(x)P(y\mid x)+P(\bar x)P(y\mid\bar x)$
$P(x\mid y,z) = \dfrac{P(x)P(y\mid x)P(z\mid x)}{P(x)P(y\mid x)P(z\mid x)+P(\bar x)P(y\mid\bar x)P(z\mid\bar x)}$
$P(x\mid y)=\dfrac{P(x)P(y\mid x)}{P(y)}$