CRISAN, Dan; DEL MORAL, Pierre; JASRA, Ajay; RUZAYQAT, Hamza

doi:10.1017/apr.2021.62

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CRISAN, Dan
Department of Mathematics [Imperial College London]

DEL MORAL, Pierre
Méthodes avancées d’apprentissage statistique et de contrôle [ASTRAL]

JASRA, Ajay
Computer, Electrical and Mathematical Sciences and Engineering Division [Thuwal]

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Advances in Applied Probability. 2022-12, vol. 54, n° 4, p. 1139-1163

Applied Probability Trust

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Abstract In this article we consider the estimation of the log-normalization constant associated to a class of continuous-time filtering models. In particular, we consider ensemble Kalman–Bucy filter estimates based upon several nonlinear Kalman–Bucy diffusions. Using new conditional bias results for the mean of the aforementioned methods, we analyze the empirical log-scale normalization constants in terms of their $\mathbb{L}_n$ -errors and $\mathbb{L}_n$ -conditional bias. Depending on the type of nonlinear Kalman–Bucy diffusion, we show that these are bounded above by terms such as $\mathsf{C}(n)\left[t^{1/2}/N^{1/2} + t/N\right]$ or $\mathsf{C}(n)/N^{1/2}$ ( $\mathbb{L}_n$ -errors) and $\mathsf{C}(n)\left[t+t^{1/2}\right]/N$ or $\mathsf{C}(n)/N$ ( $\mathbb{L}_n$ -conditional bias), where t is the time horizon, N is the ensemble size, and $\mathsf{C}(n)$ is a constant that depends only on n , not on N or t . Finally, we use these results for online static parameter estimation for the above filtering models and implement the methodology for both linear and nonlinear models.Read less <

English Keywords

Kalman-Bucy filter

Riccati equations

nonlinear Markov processes

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Log-normalization constant estimation using the ensemble Kalman–Bucy filter with application to high-dimensional models

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