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snarles comments on The trouble with Bayes (draft) - Less Wrong Discussion
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Comments (58)
Right. You could say the cases of Y1|D=1 you observe in the population are an importance sample from Y1, the hypothetical population that would result if everyone in the population were treated. E[Y1], the quantity to be estimated, is the mean of this hypothetical population. The importance sampling weights are q(x) = Pr[D=1|x]/p(x) where p(x) is the marginal distribution (ie you invert these weights to get the average), the importance sampling distribution is the conditional density of X|D=1.
Still slightly confused.
I think Robins and Ritov has a theorem (cited in your blog link) claiming to get E[Y] if Y is MAR you need to incorporate info about 1/p(x) somewhere into your procedure (?the prior?) or you don't get uniform consistency. Is your claim that you can get around this via some hierarchical model, e.g.:
Is this just intuition or did you write this up somewhere? That sounds very interesting.
Why did you start thinking about conditional sampling at all? If estimating E[Y] via importance sampling/inverse weights/covariate adjustment is already something of a difficulty for Bayesians, why think about E[Y | event]? Isn't that trivially at least as hard?
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.