The confusion may come from mixing up my setup and Robins/Ritov's setup. There is no missing data in my setup.

I could write up my intuition for the hierarchical model. It's an almost trivial result if you don't assume smoothness, since for any x1,...,xn the parameters g(x1)...g(xn) are conditionally independent given p and distributed as F(p), where F is the maximum entropy Beta with mean p (I don't know the form of the parameters alpha(p) and beta(p) off-hand). Smoothness makes the proof much more difficult, but based on high-dimensional intuition one can be sure that it won't change the result substantially.

It is quite possible that estimating E[Y] and E[Y|event] are "equivalently hard", but they are both interesting problems with different quite different real-world applications. The reason I chose to write about estimating E[Y|event] is because I think it is easier to explain than importance sampling.