Implementing Gaussian process inference with neural networks

Gaussian processes compare favourably with backpropagation neural networks as a tool for regression, and Bayesian neural networks have Gaussian process behaviour when the number of hidden neurons tends to infinity. We describe a simple recurrent neural network with connection weights trained by one-shot Hebbian learning. This network amounts to a dynamical system which relaxes to a stable state in which it generates predictions identical to those of Gaussian process regression. In effect an infinite number of hidden units in a feed-forward architecture can be replaced by a merely finite number, together with recurrent connections.