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We know the Metropolis Hastings (MH) in MCMC:

  • target distribution: $\pi(x)$
  • proposal distribution: $p(y|x)$
  • acceptance: $\alpha(x,y) = \min \Big(1, \dfrac{\pi(y)p(y|x)}{\pi(x)p(x|y)}\Big)$

Here are some examples of a proposal distribution:

  • $p(y|x)$ density of $U(x-\sigma, x+\sigma)$
  • $p(y|x)$ density of $N(x,\sigma^2).$

Then how to understand the statements that:

  1. large $\alpha$ or $\sigma^2$ will lower the acceptance;

  2. small $\alpha$ or $\sigma^2$ will increase the correlation.

I cannot mathematically deduce the result from the acceptance formula.

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    $\begingroup$ The intuition is that if your chain at iteration $i$ is $x_{i}$ and you propose a candidate that it is near $x^{'}_{cand}=x_{i}+0.1$ ($a$ small and $\sigma^{2}$ small) it is more likely to be accepted than a candidate $x_{cand}^{'}=x_{i}+1000$ ($a$ big and $\sigma^{2}$ big). So, obviously in the first case if you accept your candidate you will have higher correlation because it is almost the same with your $x_{i}$ $\endgroup$
    – Fiodor1234
    Jul 19, 2021 at 12:21
  • $\begingroup$ @Fiodor1234 agree on the intuition, actually I want to see the mathematical proof, especially for the first one. $\endgroup$ Jul 19, 2021 at 12:22
  • $\begingroup$ Sure, I think I can demonstrate one $\endgroup$
    – Fiodor1234
    Jul 19, 2021 at 12:25

1 Answer 1

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Assuming a target distribution with unrestricted support $\mathfrak X$, consider the simple random walk proposal written as $$y=x+\sigma\epsilon\,,$$ where $\epsilon$ is a unit-variance symmetric white noise. Then $$\dfrac{\pi(y)p(y|x)}{\pi(x)p(x|y)}=\dfrac{\pi(y)}{\pi(x)}=\dfrac{\pi(x+\sigma\epsilon)}{\pi(x)}$$ which a.s. converges to $1$ when $\sigma$ goes to zero and to $0$ when $\sigma$ goes to infinity.

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  • $\begingroup$ Sorry I cannot convince by above deduction, you should proof $P\Big(U\leq \min(1,\dfrac{\pi(x+\sigma\epsilon)}{\pi(x)})\Big)$ is an decreasing function of $\sigma.$ $\endgroup$ Jul 19, 2021 at 12:41
  • $\begingroup$ "to 0 when $\sigma$ goes to infinity" why? $\endgroup$ Jul 19, 2021 at 12:44
  • $\begingroup$ The property does not hold for all $\sigma$'s but only for small enough and large enough $\sigma$'s. $\endgroup$
    – Xi'an
    Jul 19, 2021 at 12:54
  • $\begingroup$ even for the large enough $\sigma,$ why goes to 0? Or everything here is "almost surely"? $\endgroup$ Jul 19, 2021 at 13:06
  • $\begingroup$ That the density goes to zero at infinity is a necessary condition for the density to be integrable. $\endgroup$
    – Xi'an
    Jul 19, 2021 at 13:50

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