Fast Inference for Intractable Likelihood Problems using Variational B ayes

Variational Bayes (VB) is a popular statistical method for Bayesian inference. The existing VB algorithms are restricted to cases where the likelihood is tractable, which precludes their use in many interesting models. Tran et al. (2015) extend the scope of application of VB to cases where the likelihood is intractable but can be estimated unbiasedly, and name the method "Variational Bayes with Intractable Likelihood (VBIL)". This paper presents a version of VBIL, named Variational Bayes with Intractable Log-Likelihood (VBILL), that is useful for cases, such as big data and big panel data models, where only unbiased estimators of the log-likelihood are available. In particular, we develop an estimation approach, based on subsampling and the MapReduce programming technique, for analysing massive datasets which cannot fit into a single desktop's memory. The proposed method is theoretically justified in the sense that, apart from an extra Monte Carlo error which can be controlled, it is able to produce estimators as if the true log-likelihood or full data were used. The proposed methodology is robust in the sense that it works well when only highly variable estimates of the log-likelihood are available. The method is illustrated empirically using several simulated datasets and a big real dataset based on the arrival time status of U. S. airlines. Keywords. Pseudo Marginal Metropolis-Hastings, Debiasing Approach, Big Data, Panel Data, Difference Estimator.