The Ridgelet Prior: A Covariance Function Approach to Prior Specification for Bayesian Neural Networks

Bayesian neural networks attempt to combine the strong predictive performance of neural networks with formal quantification of uncertainty associated with the predictive output in the Bayesian framework. However, it remains unclear how to endow the parameters of the network with a prior distribution that is meaningful when lifted into the output space of the network. A possible solution is proposed that enables the user to posit an appropriate Gaussian process covariance function for the task at hand. Our approach constructs a prior distribution for the parameters of the network, called a ridgelet prior, that approximates the posited Gaussian process in the output space of the network. In contrast to existing work on the connection between neural networks and Gaussian processes, our analysis is non-asymptotic, with finite sample-size error bounds provided. This establishes the universality property that a Bayesian neural network can approximate any Gaussian process whose covariance function is sufficiently regular. Our experimental assessment is limited to a proof-of-concept, where we demonstrate that the ridgelet prior can out-perform an unstructured prior on regression problems for which a suitable Gaussian process prior can be provided.

Automated summary

Bayesian neural networks aim to combine predictive performance with uncertainty quantification, proposing a way to approximate Gaussian processes for parameter priors. Non-asymptotic analysis with finite error bounds shows the ability of Bayesian neural networks to approximate any sufficiently regular covariance Gaussian process. Experimental assessment demonstrates the superiority of the proposed ridgelet prior in regression tasks.