Density estimation for the Metropolis-Hastings algorithm

Kernel density estimation is an important tool in visualizing posterior densities from Markov chain Monte Carlo output. It is well known that when smooth transition densities exist, the asymptotic properties of the estimator agree with those for independent data. In this paper, we show that because of the rejection step of the Metropolis-Hastings algorithm, this is no longer true and the asymptotic variance will depend on the probability of accepting a proposed move. We find an expression for this variance and apply the result to algorithms for automatic bandwidth selection.