On Statistical Optimality of Variational Bayes

Variational inference [25, 7, 40] is now a well-established tool to approximate intractable posterior distributions in hierarchical multi-layered Bayesian models. The traditional Markov chain Monte Carlo (MCMC; [17]) approach of approximating distributions with intractable normalizing constants draws (correlated) samples according to a discrete-time Markov chain whose stationary distribution is the target distribution. Despite their success and popularity, MCMC methods can be slow to converge and lack scalability in big data problems and/or problems involving very many latent variables, which has fueled search for alternatives. In contrast to the sampling approach of MCMC, variational inference approaches the problem from an optimization viewpoint. First, a class of analytically tractable distributions, referred to as the variational family, is identified for the problem at hand. For example, in mean-field approximation, the set of parameters and latent variables is divided into blocks and the variational distribution is assumed to be independent across blocks.