How to understand the scaling in Metropolis Hastings MCMC We know the Metropolis Hastings (MH) in MCMC:

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*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:

*

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


*small  $\alpha$ or $\sigma^2$ will increase the correlation.
I cannot mathematically deduce the result from the acceptance formula.

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.
