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, $d\rightarrow \infty$. 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 distributions for which we rigorously prove it. Finally we conduct a simulation study on a target and family of noise distributions which together satisfy neither of the two key conditions of the limit result on which our work is based: the target has $d=1$ and the noise distribution depends heavily on the position in the state-space. Our key conclusions are found to hold in this example also.