Exact marginal prior distributions of finite Bayesian neural networks

Bayesian neural networks are theoretically well-understood only in the infinite-width limit, where Gaussian priors over network weights yield Gaussian priors over network outputs. Recent work has suggested that finite Bayesian networks may outperform their infinite counterparts, but their non-Gaussian output priors have been characterized only though perturbative approaches. Here, we derive exact solutions for the function space priors for individual input examples of a class of finite fully-connected feedforward Bayesian neural networks. For deep linear networks, the prior has a simple expression in terms of the Meijer G -function. The prior of a finite ReLU network is a mixture of the priors of linear networks of smaller widths, corresponding to different numbers of active units in each layer.