I want to calculate the probability given in the red in image attached. Can I apply simple Bayesian rule? If yes, what can be the first step?
So I tried to solve using Bayesian Rule. But, I don't know if I have taken the first step right. If so, How can evaluate the individual terms.. Thank You.
My Try:
No.
Use Bayes' Rule and the Law of Total Probability
It is not that hard.
The first step is: $\mathsf P(A\mid B) = \dfrac{\mathsf P(B\mid A)\mathsf P(A)}{\mathsf P(B)}$ where $B=X_1\cap X_2\cap \neg X_3$.
The next step will be: $\mathsf P(B)={\mathsf P(B\mid A)\,\mathsf P(A)+\mathsf P(B\mid \neg A)\,\mathsf P(\neg A)}$ .
Notice also, from the DAG, that $X_1,X_2,X_3$ are mutually conditionally independent when given $A$.
Put it together. $\phantom{\mathsf P(A\mid X_1,X_2,X_3^\mathsf c) = \tfrac{\mathsf P(X_1\mid A)\,\mathsf P(X_2\mid A)\,\mathsf P(X_3^\mathsf c\mid A)\,\mathsf P(A)}{\mathsf P(X_1\mid A)\,\mathsf P(X_2\mid A)\,\mathsf P(X_3^\mathsf c\mid A)\,\mathsf P(A)~+~\mathsf P(X_1\mid A^\mathsf c)\,\mathsf P(X_2\mid A^\mathsf c)\,\mathsf P(X_3^\mathsf c\mid A^\mathsf c)\,\mathsf P(A^\mathsf c)}}$