Evaluating Bayesian ANN estimates of ideal observer decision variables by comparison with identity functions

Abstract

Bayesian artificial neural networks (BANNs) have proven useful in two-class classification tasks, and are claimed to provide good estimates of ideal-observer-related decision variables (the a posteriori class membership probabilities). We wish to apply the BANN methodology to three-class classification tasks for computer-aided diagnosis, but we currently lack a fully general extension of two-class receiver operating characteristic (ROC) analysis to objectively evaluate three-class BANN performance. It is well known that "the likelihood ratio of the likelihood ratio is the likelihood ratio." Based on this, we found that the decision variable which is the a posteriori class membership probability of an observational data vector is in fact equal to the a posteriori class membership probability of that decision variable. Under the assumption that a BANN can provide good estimates of these a posteriori probabilities, a second BANN trained on the output of such a BANN should perform very similarly to an identity function. We performed a two-class and a three-class simulation study to test this hypothesis. The mean squared error (deviation from an identity function) of a two-class BANN was found to be 2.5x10E-4. The mean squared error of the first component of the output of a three-class BANN was found to be 2.8x10^-4, and that of its second component was found to be 3.8x10^-4. Although we currently lack a fully general method to objectively evaluate performance in a three-class classification task, circumstantial evidence suggests that two- and three-class BANNs can provide good estimates of ideal-observer-related decision variables.