For very basic Metropolis Algorithm with one parameter, what happens when you are at the tail?

Question

I'm not sure how to phrase the question, but let's say you are running the Metropolis Algorithm and the distribution you are trying to produce is just a single distribution. Let's say the values of the distribution range between 0 and 100 and are discrete with step 1.

At each point, you have a 50% chance of making a decision of whether or not to move right or left. For example, if you are at 45, you have a 50% chance of making a decision on whether or not you want to go to point 46 and a 50% chance of making a decision on whether or not you want to go to point 44. The decision to actually go to an adjacent point is another probability decision but it's not important for this question.

My question is, what happens at point 0 or 100. At point 0, do you decide between point 100 and 1, or do you have a 100% chance of making a decision to go to point 1?

In other words, is the whole thing a circular loop?

Thanks!

Answer 1

You can do either you can reject the impossible points and simply stay at the current location: https://xianblog.wordpress.com/2011/07/05/bounded-target-support/ Do not generate a new proposal if you get an impossible move. You just want to stay where you are rather than making an impossible move.

Or you can make the whole thing into a ring and simply propose to go to the other end. This is because making it a ring preserves the symmetry of the proposal distribution. The probability of going from 0 to 100 is 0.5 and the probability of going from 100 to 0 is 0.5.

The thing you don't want to do is make it a 100% chance of leaving 0 because then you violate the symmetry of the proposal distribution. Probability of going from 1 to 0 is 0.5 but probability of 0 to 1 is 1.0.

This is the simplest version where the proposal distribution is symmetric. Asymmetric distributions are possible but need to be considered in the algorithm so for this answer we will pretend they don't exist. If you want to make it more complicated and use an asymmetric proposal distribution then please see this reference (or any book that includes the metropolis-hastings algorithm): http://www.mit.edu/~ilkery/papers/MetropolisHastingsSampling.pdf