 
Summary: Neural network uncertainty assessment using Bayesian
statistics with application to remote sensing:
1. Network weights
F. Aires
Department of Applied Physics and Applied Mathematics, Columbia University/NASA Goddard Institute for Space Studies,
New York, USA
CNRS/IPSL/Laboratoire de Me´te´orologie Dynamique, E´cole Polytechnique, Palaiseau, France
Received 22 September 2003; revised 17 February 2004; accepted 15 March 2004; published 21 May 2004.
[1] Neural network techniques have proved successful for many inversion problems in
remote sensing; however, uncertainty estimates are rarely provided. This study has
three parts. In this article, we present an approach to evaluate uncertainties (i.e., error
bars and the correlation structure of these errors) of the neural network parameters, the
socalled ``synaptic weights'' on the basis of a Bayesian technique. In contrast to more
traditional approaches based on ``point estimation'' of the neural network weights (i.e.,
only one set of weights is determined by the learning process), we assess uncertainties on
such estimates to monitor the quality of the neural network model. Uncertainties of the
network parameters are used in the following two papers to estimate uncertainties of the
network output [Aires et al., 2004a] and of the network Jacobians [Aires et al., 2004b].
These new theoretical developments are illustrated by applying them to the problem of
retrieving surface skin temperature, microwave surface emissivities, and integrated water
