Bayesian network

Question

I have this following question and i would like to know if anyone can answer it. For a given Bayesian network where $P(a) =.6, P(b|a) =.8, P(b|-a)=.4, P(c|a)=.4$ and $P(c|-a) = .3$, compute $P(c|b)$. Note that $a, -a, b$, etc. are propositions: e.g.) $a \leftrightarrow A = true , -a \leftrightarrow A = false$.

I know that $P(a\land b\land c) = P(b|a)P(c|a)P(a)$ but i don't know how to solve the $P(c|b)$?

Thank you for help in advance.

Answer 1

Begin with the definition of conditional probability.

$\hspace{25ex}\mathsf P(c\mid b) = \dfrac{\mathsf P(b\cap c)}{\mathsf P(b)}$

Now apply the law of total probability,

$\hspace{25ex}\mathsf P(c\mid b) = \dfrac{\mathsf P(a\cap b\cap c)+\mathsf P(\neg a\cap b\cap c)}{\mathsf P(a\cap b)+\mathsf P(\neg a\cap b)}$

Then use the relations from the DAG (the directed acyclic graph).