Well, the answer to what would be the ideal choice of distribution is pretty obvious: a deterministic one, always yielding exactly E[f;p]. It’s absolutely perfect: it’s easy to sample from, you only need one sample from it, and estimating the expected value with it gives you the exact value. The only trouble, of course, is that value of E[f;p] is precisely what you don’t know, that’s a bummer.
Anyway, it seems we’re going to go the statistics route, looking for a generic way to conjure up unbiased estimators with not-too-large variance?

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