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.

本文提出直接近似贝叶斯模型函数空间或预测后验分布的方法，并指出了使用Kullback-Leibler divergence方法的优劣，提出了基于Bayesian linear regression的benchmark方法来评估预测质量和后验近似质量。

函数空间中变分推断的理解