Automatic step size selection in randon walk Metropolis algorithms

Practitioners of Markov chain Monte Carlo (MCMC) may hesitate to use random walk Metropolis-Hastings algorithms, especially variable-at-a-time algorithms with many parameters, because these algorithms require users to select values of tuning parameters (step sizes). These algorithms perform poorly if the step sizes are set to be too low or too high. We show in this paper that it is not difficult for an algorithm to tune these step sizes automatically to obtain a desired acceptance probability, since the logit of the acceptance probability is very nearly linear in the log of the step size, with known slope coefficient. These ideas work in most applications, including single parameter or block moves on the linear, log, or logit scales. We discuss the implementation of this algorithm in the software package YADAS.