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How does the sampling procedure work if the posterior that is being sampled from is unknown?

If the proposed jump has a higher probability than the current one the random walk jumps to the proposed site, but how is the probability assessed at any time?

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You need to be able to evaluate the target density pointwise to run MH.

Suppose you want to obtain samples $x^{(s)} \sim \pi(x)$ and propose from a local random walk $q(x) = \mathcal{N}(x, \sigma)$. Then in order to accept/reject a step $x \to x^\star$ in a Metropolis-Hastings manner you need to evaluate:

$$1 \wedge \frac{\pi(x^\star)}{\pi(x)}$$

up to a multiplicative constant (since that'd be present in both the numerator and denumerator and thus drop out).

If you cannot evaluate $\pi(x)$ you cannot apply many MCMC methods. There are ways to do away with this requirement, but they carry no statistical guarantees and are rather advanced.

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  • $\begingroup$ Is there any chance you could expand upon your answer perhaps using an example? Thanks. $\endgroup$ – adkane Apr 15 '17 at 19:44
  • $\begingroup$ @ManassaMauler added a bit more detail $\endgroup$ – Oxonon Apr 15 '17 at 20:26

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