Understanding Variational Inference in Function-Space

David R. Burt, Sebastian W. Ober, Adrià Garriga-Alonso, Mark van der Wilk

Keywords: Functional variational inference, Bayesian neural networks, Gaussian processes, Bayesian linear regression

TL;DR: We provide theoretical results surrounding the objective function in functional variational inference as well as a method of benchmarking function-space inference methods.

Abstract: Recent work has attempted to directly approximate the 'function-space' or predictive posterior distribution of Bayesian models, without approximating the posterior distribution over the parameters. This is appealing in e.g. Bayesian neural networks, where we only need the former, and the latter is hard to represent. In this work, we highlight some advantages and limitations of employing the Kullback-Leibler divergence in this setting. For example, we show that minimizing the KL divergence between a wide class of parametric distributions and the posterior induced by a (non-degenerate) Gaussian process prior leads to an ill-defined objective function. Then, we propose (featurized) Bayesian linear regression as a benchmark for 'function-space' inference methods that directly measures approximation quality. We apply this methodology to assess aspects of the objective function and inference scheme considered in Sun et al. (2018), emphasizing the quality of approximation to Bayesian inference as opposed to predictive performance.