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Ergodicity and Accuracy of Optimal Particle Filters for Bayesian Data Assimilation

Kelly, David and Stuart, Andrew M. (2016) Ergodicity and Accuracy of Optimal Particle Filters for Bayesian Data Assimilation. . (Submitted)

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Data assimilation refers to the methodology of combining dynamical models and observed data with the objective of improving state estimation. Most data assimilation algorithms are viewed as approximations of the Bayesian posterior (filtering distribution) on the signal given the observations. Some of these approximations are controlled, such as particle filters which may be refined to produce the true filtering distribution in the large particle number limit, and some are uncontrolled, such as ensemble Kalman filter methods which do not recover the true filtering distribution in the large ensemble limit. Other data assimilation algorithms, such as cycled 3DVAR methods, may be thought of as approximating the mean of the posterior, but are also uncontrolled in general. For particle filters and ensemble Kalman filters it is of practical importance to understand how and why data assimilation methods can be effective when used with a fixed small number of particles, since for many large-scale applications it is not practical to deploy algorithms close to the large particle limit asymptotic. In this paper we address this question for particle filters and, in particular, study their accuracy (in the small noise limit) and ergodicity (for noisy signal and observation) without appealing to the large particle number limit. We first prove the accuracy and ergodicity properties for the true filtering distribution, working in the setting of conditional Gaussianity for the dynamics-observation model. We then show that these properties are inherited by optimal particle filters for any fixed number of particles. For completeness we also prove large particle number consistency results for the optimal particle filters, by writing the update equations for the underlying distributions as recursions.

Item Type:Report or Paper (Discussion Paper)
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Additional Information:DK is supported as a Courant instructor. The work of AMS is supported by EPSRC, DARPA and ONR.
Funding AgencyGrant Number
Courant Institute, New York UniversityUNSPECIFIED
Engineering and Physical Sciences Research Council (EPSRC)UNSPECIFIED
Defense Advanced Research Projects Agency (DARPA)UNSPECIFIED
Office of Naval Research (ONR)UNSPECIFIED
Record Number:CaltechAUTHORS:20161221-161911353
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:73108
Deposited By: Linda Taddeo
Deposited On:22 Dec 2016 00:26
Last Modified:22 Dec 2016 01:18

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