On the efficiency of pseudo-marginal random walk Metropolis algorithms

By Chris Sherlock, Alexandre Hoang Thiery, Gareth O. Roberts, Jeffrey S. Rosenthal. Annals of Statistics, 2013

202 citations recorded by the source corpus.

We examine the behaviour of the pseudo-marginal random walk Metropolis algorithm, where evaluations of the target density for the accept/reject probability are estimated rather than computed precisely. Under relatively general conditions on the target distribution, we obtain limiting formulae for the acceptance rate and for the expected squared jump distance, as the dimension of the target approaches infinity, under the assumption that the noise in the estimate of the log-target is additive and is independent of the position. For targets with independent and identically distributed components, we also obtain a limiting diffusion for the first component. We then consider the overall efficiency of the algorithm, in terms of both speed of mixing and computational time. Assuming the additive noise is Gaussian and is inversely proportional to the number of unbiased estimates that are used, we prove that the algorithm is optimally efficient when the variance of the noise is approximately 3.3 and the acceptance rate is approximately 7.0%. We also find that the optimal scaling is insensitive to the noise and that the optimal variance of the noise is insensitive to the scaling. The theory is illustrated with a simulation study using the particle random walk Metropolis.

Subfields: Mathematical Physics, Molecular Biology, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e99abc3672ea16847c90c0f6fcd3a67da1769a42

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Citations in the corpus

Uniform ergodicity of the iterated conditional SMC and geometric ergodicity of particle Gibbs samplers (2013, methodology): “While PMMH methods have been studied in a series of papers [4, 5, 16, 28, 11], a theoretical study of the PGibbs is still missing.”
Bayesian computation: a summary of the current state, and samples backwards and forwards (2015, background): “Sherlock et al. (2014), Doucet et al. (2012) and Sherlock (2014), on the other hand, investigate the efﬁciency as a function of the acceptance rate and variance of the noise, deriving the optimal scaling, as discussed in Sect.”
Exploiting Multi-Core Architectures for Reduced-Variance Estimation with Intractable Likelihoods (2014, background): “The pseudo-marginal MCMC typically leads to reduced efficiency relative to the (unavailable) marginal algorithm, but improved efficiency relative to a Markov chain constructed on the extended space (θ,x) (Sherlock et al., 2014).”
Bayesian computation: a perspective on the current state, and sampling backwards and forwards (2015, background): “Interestingly, the optimal scaling of the discussed in Section 2.5 pseudo-marginal algorithms as obtained in Sherlock et al. (2014) suggests the acceptance rate of just 0 .”