  Whilst deep neural networks have shown great empirical success, there is
still much work to be done to understand their theoretical properties. In this
paper, we study the relationship between random, wide, fully connected,
feedforward networks with more than one hidden layer and Gaussian processes
with a recursive kernel definition. We show that, under broad conditions, as we
make the architecture increasingly wide, the implied random function converges
in distribution to a Gaussian process, formalising and extending existing
results by Neal (1996) to deep networks. To evaluate convergence rates
empirically, we use maximum mean discrepancy. We then compare finite Bayesian
deep networks from the literature to Gaussian processes in terms of the key
predictive quantities of interest, finding that in some cases the agreement can
be very close. We discuss the desirability of Gaussian process behaviour and
review non-Gaussian alternative models from the literature.
