Decision Network Expected Utility

Question

Suppose I am given the following decision network, with $t\in\{T_n,T_t,T_s\}$ being the decision:

$f\in\{F,\bar{F}\}$, $h\in\{H,\bar{H}\}$, $s\in\{S,\bar{S}\}$.

Suppose I am given $P(f)$, $P(s|f)$, $P(h|t,f)$, and $Utility(t,h)$. My goal is to find the expected utility of the optimal policy. I'm stuck because I'm not sure how to compute $P(h,s,f|t)$. How do I calculate this?

Answer 1

From your network I deduce that $h$ is conditionally independent of $s$ given $f$ and $t$, i.e. $$P(h|f,s,t) = P(h|f,t), \tag{1}$$ and from the directionality of the arrows I guess that $$P(s,f|t) = P(s,f)\tag{2}.$$ Otherwise, I do not see how to answer the question. Assuming these two relations and the definition of conditional probability in the form $$ P(a|b) = \frac{P(a,b)}{P(b)},$$ we have \begin{align*} P(f,h,s|t) &= \frac{P(f,h,s,t)}{P(t)} \\ &=\frac{P(h|f,s,t)P(f,s,t)}{P(t)}\\ (1)&= \frac{P(h|f,t)P(f,s,t)}{P(t)}\\ &=\frac{P(h|f,t)P(f,s|t)P(t)}{P(t)}\\ (2)&=P(h|f,t)P(f,s)\\ &=P(h|f,t)P(s|f)P(f), \end{align*} which are all known.