I have the following probability model:
the joint probability p(A,B,D) is to be evaluated. I think there are 2 ways how to think about the problem:
- Marginalization over C of p(A,B,C,D) ignoring the conditional probabilities within the model $$p(A,B,D) = \sum_{C}{p(A,B,C,D)} = p(A,B,C,D) + p(A,B,\overline{C},D)$$
Assuming the $C = [C, \overline{C}]$, that is, it has only 2 values.
- Marginalization over C of p(A,B,C,D) taking the conditional links p(C|A,B) and p(D|C) into account
It can be shown, that according to the graph, the $p(A,B,C,D)$ can be marginalized as
$$p(A,B,D) = \sum_{C}{p(A,B,C,D)} \\ = \sum_{C}{p(A)p(B)p(C|A,B)p(D|C)} \\ = p(A)p(B)\sum_{C}{p(C|A,B)p(D|C)} \\ = p(A)p(B)\left(p(C|A,B)p(D|C) + p(\overline{C}|A,B)p(D|\overline{C})\right)$$
Question
Is the marginalization of the joint distribution a summation over conditioned observations ? (that is, the marginalization over C is the same as summation of conditioned cases on C)
Yes, it is an application of the Law of Total Probability.
When discussing Bayesian DAG, it is common to use $\sum_C$ to mean "sum over the evaluations of $C$", which is typically just $C$ itself and its complement, $\overline C$. So it is exactly as you had.
$$\begin{align}\mathsf p(A,B,D)&=\mathsf p(A,B,C,D)+\mathsf p(A,B,\overline C, D)\\&=\sum_{C} \mathsf p(A,B,C,D)\\&=\mathsf p(A)\mathsf p(B)\sum_C\mathsf p(C\mid B,A)\mathsf p(D\mid C)\\&=\mathsf p(A)\mathsf p(B)\left(\mathsf p(C\mid B,A)\mathsf p(D\mid C)+\mathsf p(\overline C\mid B,A)\mathsf p(D\mid \overline C)\right)\end{align}$$