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In implementations of the Metropolis-Hastings algorithm, how is the target distribution $\pi(\mathbf{x}) = P(\mathbf{x}|\mathbf{e})$ computed or estimated while computing the acceptance probability $\alpha(\mathbf{x'}|\mathbf{x}) = \min \left(1,\frac{\pi(\mathbf{x'})q(\mathbf{x}|\mathbf{x'})}{\pi(\mathbf{x})q(\mathbf{x'}|\mathbf{x})}\right)$?

For special cases of Metropolois-Hastings such as Gibbs Sampling, I understand that detailed balance of $q$ with $\pi$ simplifies the acceptance probably to $1$.

What I'm not understanding is how compute the target distribution when using something like a symmetric random walk proposal.

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  • $\begingroup$ For the moment, I've explicitly computed the joint distribution as the target. Something is terribly wrong here and I'm unsure how to fix it. $\endgroup$ – ehuang Sep 23 '14 at 6:44

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