Thursday 02 Feb 2017: Statistical Science Seminar: The Correlated Pseudo-Marginal Method for Inference in Latent Variable Models
Arnaud Doucet - University of Oxford
The pseudo-marginal algorithm is a Metropolis--Hastings-type scheme which samples asymptotically from a target probability density when we are only able to estimate unbiasedly an unnormalised 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 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 to use a number N of Monte Carlo samples proportional to T. This linear scaling allows us to control the variance of the log-likelihood ratio estimator appearing in the acceptance probability of this algorithm. The correlated pseudo-marginal algorithm is a modification of the pseudo-marginal method using a log-likelihood ratio estimator computed using two correlated log-likelihood estimators. For random effects models, we show under regularity conditions that the parameters of this scheme can be selected such that the variance of this log-likelihood ratio estimator is controlled when N increases sublinearly with T and we provide guidelines to optimise the scheme based on a non-standard weak convergence analysis. In our examples, the efficiency of computations for Bayesian inference in random effects and state-space models relative to the pseudo-marginal method increases with T and is higher than two orders of magnitude when T is a few thousands.
This is joint work George Deligiannidis and Michael K. Pitt: http://arxiv.org/abs/1511.04992