On the relationship between measures of discrimination and the performance of suboptimal detectors

The problem of designing and analyzing suboptimal detectors via statistical distance measures is considered. As a preliminary result, we show that only the minimum and maximum probability of error are valid measures of discrimination between the input statistics. This result would seem then to imply that the use of distance measures in this context can be inappropriate. However, to overcome this apparent obstacle, we demonstrate explicit relationships between various f-divergences and the loss in performance of an arbitrary detector relative to the optimal detector. In particular, we establish both upper and lower bounds on the performance loss of a suboptimal detector in terms of the “distance” between the pertinent statistics of both the optimal and suboptimal detectors. While designing detectors by minimizing these upper bounds can be an elusive task, in many practical cases, the lower bound presented herein holds with equality. In this case, minimizing the separation of the output statistics of the detector with respect to a particular f-divergence equivalently minimizes the resulting probability of error of the detector. To facilitate design, other researchers have established conditions under which one may design arbitrary detection strategies with respect to a specific f-divergence (Kullback-Leibler distance being a principal example). We extend this approach by deriving necessary and sufficient conditions under which one may design detection strategies with respect to an arbitrarily chosen f-divergence. Thus when these conditions are met, one may optimize a detector with respect to the most analytically tractable distance measure to obtain the minimum probability of error detector over a selected class of detection strategies. Examples demonstrating the utility of this theory for the problem of designing optimal linear detectors and optimal signal sets are presented