Recursive nonlinear estimation: Geometry of a space of posterior densities

The paper establishes a geometric formulation for nonlinear parameter estimation using reduced statistics. If a reduced, rather than sufficient statistic is used in estimation, an equivalence class of densities, [p(t)], rather than the true posterior density p(t) is determined. Two questions arise in this connection: (1) What kind of a reduced statistic allows recursive computations? (2) What is the appropriate representative p(t) of the equivalence class [p(t)]? Typically, the first question is not posed at all and the second one is resolved by heuristic considerations. The present paper attempts to cast the problem into a solid mathematical structure. It closely follows the differential-geometric approach suggested in Kulhavý (Automatica, 26, 545–555, 1990) but goes into more detail. Roughly, admissible statistics are characterized here as homomorphisms of a group containing all possible likelihoods. A representative of the pertinent equivalence class is constructed by a projection along this class onto an orthogonal submanifold going through a prespecified density p∗ and imbedded in the manifold of possible posteriors. The obtained projection p(t) has attractive extremal properties: it minimizes the Kullback-Leibler distance from the “reference” point p∗ and, at the same time, the dual Kullback-Leibler distance from the true point p(t).