Practical posterior error bounds from variational objectives

Variational inference has become an increasingly attractive fast alternative to Markov chain Monte Carlo methods for approximate Bayesian inference. However, a major obstacle to the widespread use of variational methods is the lack of post-hoc accuracy measures that are both theoretically justified and computationally efficient. In this paper, we provide rigorous bounds on the error of posterior mean and uncertainty estimates that arise from full-distribution approximations, as in variational inference. Our bounds are widely applicable, as they require only that the approximating and exact posteriors have polynomial moments. Our bounds are also computationally efficient for variational inference because they require only standard values from variational objectives, straightforward analytic calculations, and simple Monte Carlo estimates. We show that our analysis naturally leads to a new and improved workflow for validated variational inference. Finally, we demonstrate the utility of our proposed workflow and error bounds on a robust regression problem and on a real-data example with a widely used multilevel hierarchical model.Published versio

Negative Dependence Tightens Variational Bounds

International audienceImportance weighted variational inference (IWVI) is a promising strategy for learning latent variable models. IWVI uses new variational bounds, known as Monte Carlo objectives (MCOs), obtained by replacing intractable integrals by Monte Carlo estimates-usually simply obtained via importance sampling. Burda et al. (2016) showed that increasing the number of importance samples provably tightens the gap between the bound and the likelihood. We show that, in a somewhat similar fashion, increasing the negative dependence of importance weights monotonically increases the bound. To this end, we use the supermodular order as a measure of dependence. Our simple result provides theoretical support to several different approaches that leveraged negative dependence to perform efficient variational inference of deep generative models