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From Kruschke's Doing Bayesian Data Analysis, he states that with Metropolis-Hastings, "each position will be visited proportionally to its target value". By 'target value' he means the calculation of $P(\theta_{proposed})$.

If the visits are proportional to the target values, why not use the target values directly?

Instead, it seems as though MH discards the target values after using them to calculate acceptance. At least in the discrete case, this leaves open the possibility of visiting the same $\theta_{proposed}$ multiple times and re-calculating the same target value and the acceptance ratio over and over again.

I suppose what I'm proposing is more like a grid search of the posterior distribution that uses the proposal distribution (candidate generation) of MH. In my suggestion, one would still visit the same unique values of $\theta$ but ignore the visit counts and build a table mapping $\theta$ to $P(\theta)$ instead. Where does this idea break down?

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  • $\begingroup$ It is unclear what you plan to do with the visited positions, once you have computed $p(\theta)$, and how you propose to visit positions if the computing budget does not allow for visiting them all. $\endgroup$ – Xi'an Apr 27 '20 at 14:06
  • $\begingroup$ Can you supply a reference for what MH (or any MCMC method) does with the visit counts to build the simulated density function? That part isn't clear to me, but I thought that calculating $P(\theta_{proposed})$ provides more information than the visits counts and could be substituted. $\endgroup$ – Adam Ribaudo Apr 27 '20 at 14:32
  • $\begingroup$ The computation of $p(\theta)$ is used to decide whether or not the proposed value is accepted, relative to the previous value in the case of MCMC. It can be stored for computing quantities like normalising constants, but the value is usually discarded. The reason is that it is not particularly useful to approximate integrals and other functionals of $p(\cdot)$, while Monte Carlo samples are. Especially in large dimensions. If it was not self-promotion I would point out to my MCMC book! $\endgroup$ – Xi'an Apr 27 '20 at 16:47

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