Estimation of three-class ideal observer decision functions with a Bayesian artificial neural network

We are using Bayesian artificial neural networks (BANNs) to eliminate false-positive detections in our computer-aided diagnosis schemes. In the present work, we investigated whether BANNs can be used to estimate likelihood ratio, or ideal observer, decision functions for distinguishing observations which are drawn from three classes. Three univariate normal distributions were chosen representing three classes. We sampled 3,000 values of x for each of 10 training datasets, and 3,000 values of x for a single testing dataset. A BANN was trained on each training dataset, and the two outputs from each trained BANN, which estimate p(class 1\x) and p(class 2\x), were recorded for each value of x in the testing dataset. The mean BANN output and its standard error were calculated using the ten sets of BANN output. We repeated the above procedure to estimate the means and standard errors of the two likelihood ratio decision functions p(x\class 1)/p(x\class 3) and p(x\class 2)/p(x\class 3). We found that the BANN can estimate the a posteriori class probabilities quite accurately, except in regions of data space where outcomes are unlikely. Estimation of the Likelihood ratios is more problematic, which we attribute to error amplification caused by taking the ratio of two imprecise estimates. We hope to improve these estimates by constraining the BANN training procedure.