Timeline for Acceptance ratio in Metropolis–Hastings algorithm

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May 7, 2012 at 3:17	comment	added	Drew		Another way to think about it is as follows: when $\sigma^2$ is large, most of your proposals ($x_j$) will have low density under the target distribution (for the reasons described above -- is that part okay?). Very rarely you will propose a value with high density under the proposal, and when this happens you will almost certainly accept it. Once there, you continue proposing unlikely values; since you rarely accept one of them, you just "stay" at your current, high-density sample for many iterations.
May 7, 2012 at 2:46	comment	added	Tim		+1. Thanks! When $\sigma^2$ is large, I am still not sure why $\pi(x_i)$ is usually big while $\pi(x_j)$ usually small? Can your reason that $\pi(x_j)$ is small apply to $\pi(x_i)$ and your reason that $\pi(x_i)$ big apply to $\pi(x_j)$?
May 7, 2012 at 1:57	history	answered	Drew	CC BY-SA 3.0