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

Published online by Cambridge University Press:  02 September 2022

Dan Crisan*
Affiliation:
Imperial College London
Pierre Del Moral*
Affiliation:
Institut de Mathématiques de Bordeaux
Ajay Jasra*
Affiliation:
King Abdullah University of Science and Technology
Hamza Ruzayqat*
Affiliation:
King Abdullah University of Science and Technology
*
*Postal address: SW7 2AZ, London, UK. Email address: d.crisan@ic.ac.uk
**Postal address: 33405, Bordeaux, France. Email address: pierre.del-moral@inria.fr
***Postal address: 23955, Thuwal, Kingdom of Saudi Arabia.
***Postal address: 23955, Thuwal, Kingdom of Saudi Arabia.

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.

Type
Original Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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