\section{Discussion}
\label{sec:discussion}
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We proposed a simple inference method with a well-defined variational objective function for BNNs with GP priors in function-space.
As standard VI with functions-space priors suffers from an infinite KL divergence problem, we propose to follow the generalized VI framework.
Specifically, we substitute the conventional KL divergence in the ELBO by the regularized KL divergence, which is always finite, and which can be estimated consistently within the linearized BNN approximation.
We demonstrated that our method allows to incorporate interpretable structural properties via a GP prior, accurately approximates the true GP posterior on synthetic and small real-world data sets, and provides competitive uncertainty estimates for regression, classification and out-of-distribution detection compared to BNNs with both function-space and weight-space priors.
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Future work should investigate the use of more expressive variational distributions, such as Gaussian with low-rank plus diagonal covariance proposed by \citet{tomczak2020lowrankGausVI}.
