Asymptotic Consistency of $\alpha$-{R}\'enyi-Approximate Posteriors

Prateek Jaiswal, Vinayak Rao, Harsha Honnappa.

Year: 2020, Volume: 21, Issue: 156, Pages: 1−42


We study the asymptotic consistency properties of $\alpha$-{R}\'enyi approximate posteriors, a class of variational Bayesian methods that approximate an intractable Bayesian posterior with a member of a tractable family of distributions, the member chosen to minimize the $\alpha$-{R}\'enyi divergence from the true posterior. Unique to our work is that we consider settings with $\alpha > 1$, resulting in approximations that upperbound the log-likelihood, and consequently have wider spread than traditional variational approaches that minimize the Kullback-Liebler (KL) divergence from the posterior. Our primary result identifies sufficient conditions under which consistency holds, centering around the existence of a `good' sequence of distributions in the approximating family that possesses, among other properties, the right rate of convergence to a limit distribution. We further characterize the good sequence by demonstrating that a sequence of distributions that converges too quickly cannot be a good sequence. We also extend our analysis to the setting where $\alpha$ equals one, corresponding to the minimizer of the reverse KL divergence, and to models with local latent variables. We also illustrate the existence of a good sequence with a number of examples. Our results complement a growing body of work focused on the frequentist properties of variational Bayesian methods.