### Abstract

It is generally assumed when using Bayesian inference methods for neural networks that the input data contains no noise. For real-world (errors in variable) problems this is clearly an unsafe assumption. This paper presents a Bayesian neural network framework which accounts for input noise provided that a 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 adds an extra 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. This leads to the possibility of training an accurate model of a system using less accurate, or more uncertain, data. This is demonstrated on both the, synthetic, noisy sine wave problem and a real problem of inferring the forward model for a satellite radar backscatter system used to predict sea surface wind vectors.

Original language | English |
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Pages (from-to) | 169-188 |

Number of pages | 20 |

Journal | Journal of VLSI Signal Processing Systems for Signal Image and Video Technology |

Volume | 26 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 2000 |

### Bibliographical note

The original publication is available at www.springerlink.com### Keywords

- Bayesian inference methods
- neural networks
- noise
- errors in variable
- Bayesian neural network framework
- input noise
- noise process exists
- noiseless input
- Markov chain Monte Carlo
- satellite radar backscatter system
- sea surface wind vectors

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## Cite this

Cornford, D., Wright, W. A., Ramage, G., & Nabney, I. T. (2000). Neural network modelling with input uncertainty: theory and application.

*Journal of VLSI Signal Processing Systems for Signal Image and Video Technology*,*26*(1-2), 169-188. https://doi.org/10.1023/A:1008111920791