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Center for Nonlinear Studies |
The talk will develop an infinite-dimensional Hilbert manifold of probability measures. The manifold, M, retains the first and second order features of finite-dimensional information geometry: the α-divergences admit first derivatives and mixed second derivatives, enabling the definition of the Fisher metric as a pseudo-Riemannian metric. M was constructed with the Fenchel-Legendre transform between Kullback-Leibler divergences, and its role in Bayesian estimation, in mind. This transform retains, on M, the symmetry of the finite-dimensional case. Many of the manifolds of finite-dimensional information geometry are shown to be C∞-embedded submanifolds of M. The recursive equations of nonlinear filtering are usually expressed in terms of the Ito stochastic calculus, in which the so-called L2 theory is particularly simple and elegant. The Hilbert nature of M lends itself to this theory. By expressing the equations of nonlinear filtering for Markov processes in terms of stochastic processes on M, we show that the quadratic variation of a filter, in the Fisher metric, bears a simple relation to its rate of information supply. The filter representation can also be used as a basis for projective approximations of the type proposed by Brigo, Hanzon and Le Gland.
Host: Frank Alexander, 665-4518, Institutes Office