Let's say there are random variables $A$ and $B$ being independent. And random variable $X$.
Are there any properties to simplify $\mathbb{E}(X \mid (A,B))$ : expectation of $X$ given $A$ and $B$ ?
In particular, do we have a "simplification", like $\mathbb{E}(X \mid (A,B)) = \mathbb{E}( \mathbb{E}(X \mid A) \mid B)$ ?
I think it's incorrect (take X = A*B) but it seems strange as intuitively this would seem to be correct, so I feel like there must be some formula linking $\mathbb{E}(X \mid (A,B))$ and the conditional expectations given a single variable
thanks !