  Variational Bayesian neural networks (BNNs) perform variational inference
over weights, but it is difficult to specify meaningful priors and approximate
posteriors in a high-dimensional weight space. We introduce functional
variational Bayesian neural networks (fBNNs), which maximize an Evidence Lower
BOund (ELBO) defined directly on stochastic processes, i.e. distributions over
functions. We prove that the KL divergence between stochastic processes equals
the supremum of marginal KL divergences over all finite sets of inputs. Based
on this, we introduce a practical training objective which approximates the
functional ELBO using finite measurement sets and the spectral Stein gradient
estimator. With fBNNs, we can specify priors entailing rich structures,
including Gaussian processes and implicit stochastic processes. Empirically, we
find fBNNs extrapolate well using various structured priors, provide reliable
uncertainty estimates, and scale to large datasets.
