I have a stupid question about the [Metropolis-Hastings sampling algorithm](http://en.wikipedia.org/wiki/Metropolis%25E2%2580%2593Hastings_algorithm). 


If I got this right, for every variable $X$ in turn, which currently has value $x\_{old}$, you generate a new sample $x\_{new}$. To do that, you draw $x\_{new}$ from the proposal distribution $Q(x\_{new}\mid x\_{old})$, then you draw a number $\alpha$ uniformly at random from the range between $0$ and $1$. Then, accept $x\_{new}$ if 
$\alpha < \min{1,\frac{P(x\_{new})}{P(x\_{old})}\frac{Q(x\_{old}\mid x\_{new})}{Q(x\_{new}\mid x\_{old})}}$


The second ratio does not really make sense to me: Why are we more likely to accept if $Q(x\_{new}\mid x\_{old})$ is low? 



Question about this ratio in Metropolis-Hastings MCMC algorithm
From what you're saying, I'm not sure if you want a proof or intuition. As the proof is written up in many places, I'll just guess that you want intuition.


Very informally: the algorithm allows you to, in effect, sample from distribution P using samples from distribution Q. So in a sense we want to take the samples from Q and "remove" statistical properties of these samples that reveal that they come from Q, replacing them with the properties of P. The thing that "gives away" that they came from Q is that they're more likely to come from areas where Q is high. So we want our acceptance probability to be reduced when our samples come from such an area. That's exactly what dividing by $Q(x\_{new}|x\_{old})$ does.


(BTW The $min$ in your expression is redundant.)


