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In Metropolis - Hasting sampling, the proposal distribution does not necessarily have to have a form similar to that of the target distribution from which attempts are made to sample from. For one, I understand that a symmetrical choice of proposal kernel can vastly simplify the conditional proposal which yield a simple ratio of target distribution but this says more about the simplification of the acceptance ratio rather than the goodness of a proposal distribution to a target distribution.

What motivates our choice of proposal distribution for efficient sampling?

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    $\begingroup$ Tools for proposal comparison are efficient sample size and expected (square) jumping distance. $\endgroup$
    – Xi'an
    Commented Feb 23, 2023 at 20:04
  • $\begingroup$ @Xi'an Am I correct to interpret your comment as "we cannot know a priori what proposal distribution is best for some target distribution"? $\endgroup$
    – Physkid
    Commented Feb 24, 2023 at 14:17
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    $\begingroup$ The ideal proposal is the target itself, with no burn-in and independence between the draws. $\endgroup$
    – Xi'an
    Commented Feb 24, 2023 at 15:00

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