Bayesian approach to neural-network modeling with input uncertainty

It is generally assumed when using Bayesian inference methods for neural networks that the input data contains no noise or corruption. For real-world (errors in variable) problems this is clearly an unsafe assumption. This paper presents a Bayesian neural-network framework which allows for input noise provided that some model of the noise process exists. In the limit where the noise process is small and symmetric it is shown, using the Laplace approximation, that this method gives an additional term to the usual Bayesian error bar which depends on the variance of the input noise process. Further by treating the true (noiseless) input as a hidden variable and sampling this jointly with the network's weights, using a Markov chain Monte Carlo method, it is demonstrated that it is possible to infer the regression over the noiseless input.