  Variational Bayes (VB) is a popular estimation method for Bayesian inference.
However, most existing VB algorithms are restricted to cases where the
likelihood is tractable, which precludes their use in many important
situations. Tran et al. (2017) 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 as Big Data and Big Panel Data
models, where unbiased estimators of the gradient of the log-likelihood are
available. We demonstrate that such estimators can be easily obtained in many
Big Data applications. The proposed method is exact 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 likelihood, or full-data likelihood, is used. In
particular, we develop a computationally efficient approach, based on data
subsampling and the MapReduce programming technique, for analyzing massive
datasets which cannot fit into the memory of a single desktop PC. We illustrate
the method using several simulated datasets and a big real dataset based on the
arrival time status of U. S. airlines.
