Hierarchical Variational Models (Appendix)

Relationship to empirical Bayes and RL. The augmentation with a variational prior has strong ties to empirical Bayesian methods, which use data to estimate hyperparameters of a prior distribution (Robbins, 1964; Efron & Morris, 1973). In general, empirical Bayes considers the fully Bayesian treatment of a hyperprior on the original prior—here, the variational prior on the original meanfield—and proceeds to integrate it out. As this is analytically intractable, much work has been on parametric estimation, which seek point estimates rather than the whole distribution encoded by the hyperprior. We avoid this at the level of the hyperprior (variational prior) via the hierarchical ELBO; however, our procedure can be viewed in this framework at one higher level. That is, we seek a point estimate of the "variational hyperprior" which governs the parameters on the variational prior.