Ergodicity and Accuracy of Optimal Particle Filters for Bayesian Data Assimilation

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