Abstract
Nonlinear filtering is a branch of Bayesian estimation, in which a “signal” process is progressively estimated from the history of a related “observations” process. Nonlinear filters are typically represented in terms of stochastic differential equations for the posterior distribution of the signal. The natural “state space” for a filter is a sufficiently rich family of probability measures having a suitable topology, and the manifolds of infinite-dimensional Information Geometry are obvious candidates. After some discussion of these, the paper goes on to summarise recent results that “lift” the equations of nonlinear filtering to a Hilbert manifold, M. Apart from providing a starting point for the development of approximations, this gives insight into the information-theoretic properties of filters, which are related to their quadratic variation in the Fisher–Rao metric. A new result is proved on the regularity of a multi-objective measure of approximation errors on M.
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The author thanks the anonymous referee for carefully reading the paper and suggesting a number of improvements in its presentation.
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Newton, N.J. (2018). Nonlinear Filtering and Information Geometry: A Hilbert Manifold Approach. In: Ay, N., Gibilisco, P., Matúš, F. (eds) Information Geometry and Its Applications . IGAIA IV 2016. Springer Proceedings in Mathematics & Statistics, vol 252. Springer, Cham. https://doi.org/10.1007/978-3-319-97798-0_7
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