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:
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.