Relationship between Counting and Sampling

This course focuses on Markov chain based algorithms for randomly sampling from probability distributions defined on a large state space. Typically we are interested in generating a sample in time polynomial of the logarithm of the size of the state space. In other words, we want to sample from the distribution after exploring only an exponentially small portion of it. We’ll begin with two classical examples to illustrate the type of sampling problem we are interested in. These two examples will arise several times during the course. The main topic of this lecture is to show the intimate relationship between random sampling and approximate counting. One consequence is that an efficient algorithm for random sampling yields an efficient randomized approximation algorithm to an associated counting problem. Our running examples will clarify the type of sampling and counting problems.