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arXiv · 1910.14511

Backward Nonlinear Smoothing Diffusions

Brian D. O. Anderson·Adrian N. Bishop·Pierre Del Moral·Camille Palmier
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Abstract

We present a backward diffusion flow (i.e. a backward-in-time stochastic differential equation) whose marginal distribution at any (earlier) time is equal to the smoothing distribution when the terminal state (at a latter time) is distributed according to the filtering distribution. This is a novel interpretation of the smoothing solution in terms of a nonlinear diffusion (stochastic) flow. This solution contrasts with, and complements, the (backward) deterministic flow of probability distributions (viz. a type of Kushner smoothing equation) studied in a number of prior works. A number of corollaries of our main result are given including a derivation of the time-reversal of a stochastic differential equation, and an immediate derivation of the classical Rauch-Tung-Striebel smoothing equations in the linear setting.

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Brian D. O. Anderson, Adrian N. Bishop, Pierre Del Moral, Camille Palmier. 2019-10-31. Backward Nonlinear Smoothing Diffusions. https://doi.org/10.1137/s0040585x97t99037x

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