In this paper, we present a new distribution metric for image segmentation that arises as a result in prediction theory. Forming a natural geodesic, our metric quantifies "distance" for two density functionals as the standard deviation of the difference between logarithms of those distributions. Using level set methods, we incorporate an energy model based on the metric into the Geometric Active Contour framework. Moreover, we briefly provide a theoretical comparison between the popular Fisher Information metric, from which the Bhattacharyya distance originates, with the newly proposed similarity metric. In doing so, we demonstrate that segmentation results are directly impacted by the type of metric used. Specifically, we qualitatively compare the Bhattacharyya distance and our algorithm on the Kaposi Sarcoma, a pathology that infects the skin. We also demonstrate the algorithm on several challenging medical images, which further ensure the viability of the metric in the context of image segmentation.