On the Robustness of Optimal Scaling for Random Walk Metropolis Algorithms

In this thesis, we study the optimal scaling problem for sampling from a target distribution of interest using a random walk Metropolis (RWM) algorithm. In order to implement this method, the selection of a proposal distribution is required, which is assumed to be a multivariate normal distribution with independent components. We investigate how the proposal scaling (i.e. the variance of the normal distribution) should be selected for best performance of the algorithm. The d-dimensional target distribution we consider is formed of independent components, each of which has its own scaling term θ−2 j (d) (j = 1, . . . , d). This constitutes an extension of the d-dimensional iid target considered by Roberts, Gelman & Gilks (1997) who showed that for large d, the acceptance rate should be tuned to 0.234 for optimal performance of the algorithm. In a similar fashion, we show that for the aforementioned framework, the relative efficiency of the algorithm can be characterized by its overall acceptance rate. We first propose a method to determine the optimal form for the proposal