The optimal detector for a binary detection problem under a variety of criteria is the likelihood ratio test. Despite this simple characterization of the detector, analytic performance analysis in most cases is difficult because of the complexity of integrals involved in its computation. When the two hypotheses are signals in additive Gaussian noise, performance analysis leads to a geometric notion, whereby the signals are considered as elements of a Euclidean space and the distance between them being a measure of the performance. We extend this notion to non-Gaussian problems, assuming only that the nominal densities are mutually absolutely continuous. We adopt a completely non-parametric approach, considering the two hypotheses as points in the space of all the probability measures over the observation space. Employing the tools of differential geometry, we induce a manifold structure on this space of all probability measures, and enforce a detection theoretic specific geometry on it. The only Riemannian metric on this manifold is the Fisher information, whenever it exists. Because the detection theoretic covariant derivative is incompatible with this metric the manifold for the space of probability measures is non-Riemannian. We show that geodesics on this manifold are the exponential mixture densities. Despite not being able to define distance metric on our non-Riemannian manifold, we show that the Kullback information can play the role of "squared" distance. Because the Kullback information is asymptotically related to the performance of the optimal detector, geometry and performance are linked.
We apply this geometry to solve some classic problems in detection theory. Using the Kullback information to define the contamination neighborhoods around the nominals, the likelihood ratio of the nominals is shown to yield the robust detector. We obtain a density "halfway" between the nominals to employ as the importance sampling biasing density. Using this density, we demonstrate that the importance sampling gain is inversely related to probability of error. In an M-ary hypotheses testing problem in communications, the M signals constitute a "signal constellation" in the space of all probability measures and the underlying geometry can be employed in signal set design.