On the optimal Markov chain of IS simulation

We investigate the importance sampling (IS) simulation for the sample average of an output sequence from an irreducible Markov chain. The optimal Markov chain used in simulation is known to be a twisted Markov chain, however, the previous proofs are very complicated and do not give us a good perspective. We give a simple and natural proof for the optimality of the simulation Markov chain in terms of the Kullback-Leibler (KL) divergence of Markov chains. The performance degradation of the IS simulation by using a not optimal simulation Markov chain, i.e., the difference between the obtained variance and the minimum variance is shown to be represented by the KL divergence. Moreover, we show a geometric relationship between a simulation Markov chain and the optimal one.