It has long been known that a single-layer fully-connected neural network with an i.i.d. prior over its parameters is equivalent to a Gaussian process (GP), in the limit of infinite network width. This correspondence enables exact Bayesian inference for infinite width neural networks on regression tasks by means of evaluating the corresponding GP. Recently, kernel functions which mimic multi-layer random neural networks have been developed, but only outside of a Bayesian framework. As such, previous work has not identified that these kernels can be used as covariance functions for GPs and allow fully Bayesian prediction with a deep neural network.

In this work, we derive the exact equivalence between infinitely wide, deep, networks and GPs with a particular covariance function. We further develop a computationally efficient pipeline to compute this covariance function. We then use the resulting GP to perform Bayesian inference for deep neural networks on MNIST and CIFAR-10. We observe that the trained neural network accuracy approaches that of the corresponding GP with increasing layer width, and that the GP uncertainty is strongly correlated with trained network prediction error. We further find that test performance increases as finite-width trained networks are made wider and more similar to a GP, and that the GP-based predictions typically outperform those of finite-width networks. Finally we connect the prior distribution over weights and variances in our GP formulation to the recent development of signal propagation in random neural networks.