Journal Article

The correlated pseudomarginal method

Deligiannidis, George, Doucet, Arnaud

2018

Overview

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 number T of data points increases, it is typically necessary for the number N of Monte Carlo samples to be proportional to T to 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 when N increases sublinearly with T and 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 with T and exceeds two orders of magnitude in some examples.

• Publisher: Wiley,Oxford University Press
• Subject: Algorithms / Asymptotic posterior normality / Bayesian analysis / Bayesian theory / Computer simulation / Computing time / Convergence / Correlated random numbers / Correlation / Data points / Density / equations / Inference / Intractable likelihood / Likelihood ratio / Metropolis–Hastings algorithm / Particle filter / Probability / probability distribution / Random‐effects model / Regression analysis / Regularity / Simulation / State of the art / Statistical inference / Statistical methods / Statistics / Variance / Weak convergence