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Let$ p$ and $ q$ be two probability densities a subset of $\mathbb{R}^d $. Assume I know how to generate samples from $ p $ and how to generate samples from $ q $, but I do so without actually being able to evaluate the densities.

How do I algorithmically generate samples from the distribution whose density is $\propto pq $?

Bananach
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I think I'm getting closer to understanding what you are asking. Is it helpful to reference this previous post for the general approach and then for your problem note that if you can sample from the distributions, you can create empirical pdfs for them e.g. take a sample of 10^6 from distribution q to form q_hat and substitute q_hat for q?

Robert Alan Greevy Jr PhD
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  • I am given densities, which in your discrete example would correspond to $ p=q=(1/6,...,1/6) $. The corresponding pointwise product density would correspond again to $(1/6,...,1/6) $ as this is the unique distribution on $\{1,...,6\}$ (!, not on a larger state space) that is proportional to $ pq=(1/36,...,1/36)$. I don't understand how your answer connects to this – Bananach Jul 17 '17 at 19:34
  • Still not there :) The post you linked concerns the PDF of the product of random variables. Note that if $P$ and $Q$ are random variables with densities $p$ and $q$, the density of the variables $PQ$ is not $pq$. Also, PQ doesn't even make sense if the domain of $p$ and $q$ is multi-dimensional – Bananach Jul 17 '17 at 21:20