Optimal Scaling for the Pseudo-Marginal Random Walk Metropolis: Insensitivity to the Noise Generating Mechanism

We examine the optimal scaling and the efficiency of the pseudo-marginal random walk Metropolis algorithm using a recently-derived result on the limiting efficiency as the dimension, (drightarrow infty ). We prove that the optimal scaling for a given target varies by less than 20 % across a wide range of distributions for the noise in the estimate of the target, and that any scaling that is within 20 % of the optimal one will be at least 70 % efficient. We demonstrate that this phenomenon occurs even outside the range of noise distributions for which we rigorously prove it. We then conduct a simulation study on an example with d = 10 where importance sampling is used to estimate the target density; we also examine results available from an existing simulation study with d = 5 and where a particle filter was used. Our key conclusions are found to hold in these examples also.