  Recent works on Bayesian neural networks (BNNs) have highlighted the need to
better understand the implications of using Gaussian priors in combination with
the compositional structure of the network architecture. Similar in spirit to
the kind of analysis that has been developed to devise better initialization
schemes for neural networks (cf. He- or Xavier initialization), we derive a
precise characterization of the prior predictive distribution of finite-width
ReLU networks with Gaussian weights. While theoretical results have been
obtained for their heavy-tailedness, the full characterization of the prior
predictive distribution (i.e. its density, CDF and moments), remained unknown
prior to this work. Our analysis, based on the Meijer-G function, allows us to
quantify the influence of architectural choices such as the width or depth of
the network on the resulting shape of the prior predictive distribution. We
also formally connect our results to previous work in the infinite width
setting, demonstrating that the moments of the distribution converge to those
of a normal log-normal mixture in the infinite depth limit. Finally, our
results provide valuable guidance on prior design: for instance, controlling
the predictive variance with depth- and width-informed priors on the weights of
the network.
