Ergodicity and Accuracy of Optimal Particle Filters for Bayesian Data Assimilation
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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 controlled estimators of the state, in the small observational noise scenario, but are also uncontrolled in general in relation to the true filtering distribution. 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, the authors 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. The authors first overview the accuracy and minorization properties for the true filtering distribution, working in the setting of conditional Gaussianity for the dynamics-observation model. They then show that these properties are inherited by optimal particle filters for any fixed number of particles, and use the minorization to establish ergodicity of the filters. 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. In addition to looking at the optimal particle filter with standard resampling, they derive all the above results for (what they term) the Gaussianized optimal particle filter and show that the theoretical properties are favorable for this method, when compared to the standard optimal particle filter.
KeywordsParticle filters Data assimilation Ergodic theory
2000 MR Subject Classification60G35 62M20 37H99 60F99
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DK was supported as an NYU-Courant instructor. The authors are grateful to an anonymous referee for comments on an earlier version of this paper which have led to a correction of the presentation in Subsection 2.2, and pointers to relevant literature that were previously omitted; they are also grateful to Jonathan Mattingly for discussions relating to ergodicity of nonlinear Markov processes, and to Arnaud Doucet and Adam Johansen for helpful pointers to the literature. The authors also highlight, on the occasion of his 70th birthday, the inspirational work of Andy Majda, in applied mathematics in general, and in the area of data assimilation and filtering (the subject of this paper) in particular.
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