Stability of Optimal Filter Higher-Order Derivatives

2018/06/25 by Vladislav Z. B. Tadic, Arnaud Doucet, Tadic, Vladislav Z. B. +1
Mathematics · #FOS: Mathematics #Optimization and Control (math.OC) #Probability (math.PR) #Statistics Theory (math.ST) #math.OC #math.PR #math.ST #stat.TH

paper · pdf · doi:10.48550/arxiv.1806.09595

arxiv created 2019/12/20 · arxiv updated 2019/12/23

Abstract

In many scenarios, a state-space model depends on a parameter which needs to be inferred from data. Using stochastic gradient search and the optimal filter (first-order) derivative, the parameter can be estimated online. To analyze the asymptotic behavior of online methods for parameter estimation in non-linear state-space models, it is necessary to establish results on the existence and stability of the optimal filter higher-order derivatives. The existence and stability properties of these derivatives are studied here. We show that the optimal filter higher-order derivatives exist and forget initial conditions exponentially fast. We also show that the optimal filter higher-order derivatives are geometrically ergodic. The obtained results hold under (relatively) mild conditions and apply to state-space models met in practice.

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