The correlated pseudomarginal method
George Deligiannidis, Arnaud Doucet, Michael K. Pitt
Journal of the Royal Statistical Society: Series B (Statistical Methodology) · 2018 · Volume 80 · Issue 5
Abstract
The pseudomarginal algorithm is a Metropolis–Hastings-type scheme which samples asymptotically from a target probability density when we can only estimate unbiasedly an unnormalized version of it. In a Bayesian context, it is a state of the art posterior simulation technique when the likelihood function is intractable but can be estimated unbiasedly by using Monte Carlo samples. However, for the performance of this scheme not to degrade as the numberTof data points increases, it is typically necessary for the numberNof Monte Carlo samples to be proportional toTto control the relative variance of the likelihood ratio estimator appearing in the acceptance probability of this algorithm. The correlated pseudomarginal method is a modification of the pseudomarginal method using a likelihood ratio estimator computed by using two correlated likelihood estimators. For random-effects models, we show under regularity conditions that the parameters of this scheme can be selected such that the relative variance of this likelihood ratio estimator is controlled whenNincreases sublinearly withTand we provide guidelines on how to optimize the algorithm on the basis of a non-standard weak convergence analysis. The efficiency of computations for Bayesian inference relative to the pseudomarginal method empirically increases withTand exceeds two orders of magnitude in some examples.
Keywords
Asymptotic posterior normality, Correlated random numbers, Intractable likelihood, Metropolis–Hastings algorithm, Particle filter, Random-effects model, Weak convergence
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