I have two random variables, X and Y, and Y is a positive real number. I can sample from $p(y|x)$, but I need to sample from $p(x)$, which I know to be proportional to $\frac 1 {E[y|x]}$. I could estimate $p(x)$ from the mean of a lot of samples from $p(y|x)$ and then use a Metropolis algorithm, but sampling from y isn't cheap, so sampling a lot of them for each step is somewhere between prohibitively expensive and ain't gonna happen. Is there a better way to do this?


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