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 so-called 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 vapor content from a combined analysis of microwave and infrared observations over land.