Consistency of Posterior Distributions for Neural Networks

Herbert Lee


In this paper we show that the posterior distribution for feedforward neural networks is asymptotically consistent. This paper extends earlier results on universal approximation properties of neural networks to the Bayesian setting. The proof of consistency embeds the problem in a density estimation problem, then uses bounds on the bracketing entropy to show that the posterior is consistent over Hellinger neighborhoods. It then relates this result back to the regression setting. We show consistency in both the setting of the number of hidden nodes growing with the sample size, and in the case where the number of hidden nodes is treated as a parameter. Thus we provide a theoretical justification for using neural networks for nonparametric regression in a Bayesian framework.

Keywords: Bayesian statistics, Asymptotic consistency, Posterior approximation, Nonparametric regression, Sieve Asymptotics, Hellinger distance, Bracketing entropy

Here is the full postscript text for this technical report. It is 623652 bytes long.