Uncertainty in the output of artificial neural networks.

Analysis of the performance of artificial neural networks (ANNs) is usually based on aggregate results on a population of cases. In this paper, we analyze ANN output corresponding to the individual case. We show variability in the outputs of multiple ANNs that are trained and "optimized" from a common set of training cases. We predict this variability from a theoretical standpoint on the basis that multiple ANNs can be optimized to achieve similar overall performance on a population of cases, but produce different outputs for the same individual case because the ANNs use different weights. We use simulations to show that the average standard deviation in the ANN output can be two orders of magnitude higher than the standard deviation in the ANN overall performance measured by the Az value. We further show this variability using an example in mammography where the ANNs are used to classify clustered microcalcifications as malignant or benign based on image features extracted from mammograms. This variability in the ANN output is generally not recognized because a trained individual ANN becomes a deterministic model. Recognition of this variability and the deterministic view of the ANN present a fundamental contradiction. The implication of this variability to the classification task warrants additional study.