Question

2 part question:

A. Give an example of an application where a Metropolis-Hastings algorithm with an acceptable probability less than 1 might be preferred?

B. What is a possible issue with setting an acceptable probability too low for a Metropolis-Hastings algorithm?

Answer 1

A.

I assume that you don't want a real-life example but a practical example. So, I will give a Statistical example.

Suppose, you want to draw sample from a posterior/prior distribution whose closed form is unknown. So , we cantc run Gibb's Sampler. However, we know a distribution that can approximate the posterior/prior. At that point, even if the acceptance rate is low, you will go for the Metropolis Hastings Sampling.

B.

The main problem with low acceptance rate is its time. If too many samples are rejected, it takes huge time to achieve required nunber of samples. For example, if the acceptance rate is 0.3, then only 30% of the samples are accepted. This makes the whole simulation process very slow.