Conditional probability and instantiation

Question

This questions consists of two parts that are quite similar and concern conditional probability.

Firstly, I would like to confirm the calculation of the conditional probability when we know the value of a random variable. Let's assume that we have a conditional matrix which gives us the values for $P(D|A\&B)$ and that B can take the values $(b0, b1)$. If we know that $B = b0$, then if we have to calculate $P(D|A)$, it would just be $P(D|A\&b0)$, right?

Let's now assume that we have a conditional probability matrix giving us $P(F|G\&H)$. If we assume that $F$ is independent from $H$, how can we calculate $P(F|G)$ only using $P(F|G\&H)$?

Answer 1

According to the law of total probability,

$P(F) = \sum_{h} P(F|H=h)$, and $P(F|G) = \sum_{h} P(F|G, H=h)$.

The dependency structure does not matter here.

Here is an example to illustrate marginal vs. conditional independence.