  Bayesian neural networks (BNN) promise to combine the predictive performance
of neural networks with principled uncertainty modeling important for
safety-critical systems and decision making. However, posterior uncertainty
estimates depend on the choice of prior, and finding informative priors in
weight-space has proven difficult. This has motivated variational inference
(VI) methods that pose priors directly on the function generated by the BNN
rather than on weights. In this paper, we address a fundamental issue with such
function-space VI approaches pointed out by Burt et al. (2020), who showed that
the objective function (ELBO) is negative infinite for most priors of interest.
Our solution builds on generalized VI (Knoblauch et al., 2019) with the
regularized KL divergence (Quang, 2019) and is, to the best of our knowledge,
the first well-defined variational objective for function-space inference in
BNNs with Gaussian process (GP) priors. Experiments show that our method
incorporates the properties specified by the GP prior on synthetic and small
real-world data sets, and provides competitive uncertainty estimates for
regression, classification and out-of-distribution detection compared to BNN
baselines with both function and weight-space priors.
