Learning Statistical Manifolds for Subsequent Inference

• Activity Number: 547
• Type: Contributed
• Date/Time: Wednesday, August 12, 2015 : 10:30 AM to 12:20 PM
• Sponsor: Section on Statistical Learning and Data Mining
• Author(s): Michael Trosset* and Carey E. Priebe
• Companies: Indiana University and The Johns Hopkins University
• Keywords: Fisher information ; Hellinger distance ; Riemannian geometry ; Isomap

Abstract

We describe 1-sample tests that exploit the Riemannian structure of parametric statistical manifolds. This structure is induced by Fisher information, or, equivalently, by Hellinger distance. Thus, the information distance between two distributions is the geodesic distance between them, and our test statistic is the information distance between the null distribution and the distribution of minimal Hellinger distance from the empirical distribution. We describe some asymptotic properties of this test, then consider problems for which the parametric statistical manifold is unknown. If the manifold can be sampled, then it may be possible to learn about its Riemannian structure. We use a variant of Isomap to obtain regularized estimates of the information distance. Examples demonstrate that one can increase power by learning an unknown low-dimensional manifold instead of relying on a known manifold of higher dimension.