A technique to estimate the uncertainties of the parameters of a neural network model, i.e., the synaptic weights, was described in the work of Aires <2004>. Using these weight uncertainty estimates, we compute the uncertainties in the network outputs (i.e., error bars and correlation structure of these errors). Such quantities are very important for evaluating any application of the neural network technique. The theory is applied to the same remote sensing problem as in the work of Aires <2004> concerning the retrieval of surface skin temperature, microwave surface emissivities and integrated water vapor content from a combined analysis of microwave and infrared observations over land.