I am new to the Metropolis-Hastings algorithm and am trying to wrap my head around the key points of it. I understand that it uses a Markov Chain Monte Carlo simulation to sample points throughout a region and then calculates a value which is either greater or less than one, which in turn decides whether the chain moves on or not.
If one was to create a proposal density that is very small width-wise, would that lead to the Markov Chain converging quickly to the true pdf value or would it simply restrict the accepted values of the Markov Chain to continue moving to a small set of values?
Thanks for any help :)
The frequency of each values must be the same in the long run because they must coincide with the stationary distribution. You will get many accepted moves with little difference between successive accepted states. The opposite case would lead to large move tries but many refused moves, so, once a very probable state is reached, it will be picked many successive times.