An Analysis of the Diaconis-Holmes-Neal Markov Chain Sampler Under Generalized Unimodal Underlying Probabilities

Abstract

Upon the introduction of the Metropolis algorithm, the question of how many steps in the Markov chain were needed to achieve convergence to stationarity became apparent. The convergence was rather slow, i.e. for a process on $n$ states the number of steps needed to achieve convergence to stationarity was found to be on the order of $n^2$ if the underlying distribution is uniform. The obvious problem with Metropolis et. al is that the Markov chain is reversible. In other words, for any state $j$ we can move from $j$ to $j + 1$ and back to $j$ in two steps. To correct for this, Diaconis, Holmes, and Neal improved Metropolis et. al by introducing a non-reversible Markov chain. The Diaconis-Holmes-Neal sampler, as it is known, is a Markov chain on two copies of $n$ states, a $+1$ copy and a $-1$ copy. Applications of the Diaconis-Holmes-Neal sampler include Markov chain sampling and situations in statistical physics, among others. However, an answer to the question of how many steps are needed to achieve convergence to stationarity was required. Hildebrand showed that if the underlying probabilities are log-concave then the sampler achieves convergence to stationarity in at least a constant multiple of $n$ steps. Nonetheless, the question of whether a similar convergence exists when the underlying probabilities are instead unimodal was posed in Hildebrand. While Lange answered the question in the three simplest cases - the simple case, the function of $n$ case, and the asymmetric function of $n$ case - and Lange answered the question in the general symmetric unimodal case, the general unimodal case is left to this paper.