Geometrical Aspects of Detection Theoryy

Using the tools of category theory and diierential geometry, we extend the geometric notions consequent of Gaussian detection problems to non-Gaussian ones. The nominal probability measures associated with the hypotheses in a binary detection problem form points on an innnite dimensional manifold. These measures may correspond to non-additive as well as additive noise situations and can express any dependence structure. The natural geometry for detection theory is not Riemannian, with the nominal measures characterizing the detection problem non-metrically connected by geodesic curves formed from the exponential mixtures of these measures. While no metric exists between nominals on this manifold, we show that the Kullback-Leibler information is related to squared intermeasure distance. We use this geometry to pose and solve classic robust detection problems and to nd biasing densities that guarantee signiicant importance sampling gain in detector performance simulations.