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 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.
Pseudo marginal Markov chain Monte Carlo Random walk Metropolis Optimal scaling Particle MCMC Robustness
Mathematics Subject Classification (2010)
This is a preview of subscription content, log in to check access.
Beaumont MA (2003) Estimation of population growth or decline in genetically monitored populations. Genetics 164:1139–1160Google Scholar
Bérard J, Del Moral P, Doucet A (2014) A lognormal central limit theorem for particle approximations of normalizing constants. Electron J Probab 19(94):28MathSciNetzbMATHGoogle Scholar
Bornn L, Pillai N, Smith A, Woodard D (2014) The use of a single pseudo-sample in approximate Bayesian computation. ArXiv e-printsGoogle Scholar
Carlin BP, Louis TA (2009) Bayesian methods for data analysis. Texts in statistical science series, 3rd edn. CRC Press, Boca RatonGoogle Scholar
Doucet A, Pitt MK, Deligiannidis G, Kohn R (2015) Efficient implementation of Markov chain Monte Carlo when using an unbiased likelihood estimator. Biometrika 102(2):295–313MathSciNetCrossRefzbMATHGoogle Scholar
Filippone M, Girolami M (2014) Pseudo-marginal Bayesian inference for Gaussian processes. IEEE Trans Pattern Anal Mach Intell 36(11):2214–2226CrossRefGoogle Scholar
Giorgi E, Sesay SSS, Terlouw DJ, Diggle PJ (2015) Combining data from multiple spatially referenced prevalence surveys using generalized linear geostatistical models. Stat Soc 178(2):445–464MathSciNetCrossRefGoogle Scholar
Golightly A, Wilkinson DJ (2011) Bayesian parameter inference for stochastic biochemical network models using particle Markov chain Monte Carlo. Interface Focus 1(6):807–820CrossRefGoogle Scholar
Haran M, Tierney L (2012) On automating Markov chain Monte Carlo for a class of spatial models. ArXiv e-printsGoogle Scholar
Knape J, de Valpine P (2012) Fitting complex population models by combining particle filters with Markov chain Monte Carlo. Ecology 93(2):256–263CrossRefGoogle Scholar
Lampaki I (2015) Markov chain Monte Carlo methodology for inference on generalised linear spatial models. Ph.D. thesis, Lancaster University, UK.Google Scholar
Pitt MK, dos Santos Silva R, Giordani P, Kohn R (2012) On some properties of Markov chain Monte Carlo simulation methods based on the particle filter. J Econ 171(2):134–151MathSciNetCrossRefGoogle Scholar