Optimal scaling for the pseudo-marginal random walk Metropolis: insensitivity to the noise generating mechanism

TL;DR

This paper investigates the optimal scaling of the pseudo-marginal random walk Metropolis algorithm, showing it is robust to noise distribution variations and remains efficient within a 20% scaling margin across high-dimensional targets.

Contribution

It proves that the optimal scaling varies little across different noise distributions and that near-optimal scalings maintain high efficiency, supported by theoretical and simulation results.

Findings

Optimal scaling varies less than 20% across noise distributions.

Scaling within 20% of the optimum achieves at least 70% efficiency.

Results hold in high-dimensional and practical importance sampling scenarios.

Abstract

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. 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 simulations study with $d=5$ and where a particle filter was used. Our key conclusions are found to hold in these examples also.