  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, Zhang, Shi, and Grosse (2018), emphasizing the quality of approximation to
Bayesian inference as opposed to predictive performance.
