Unbiased estimation of the Hessian for partially observed diffusions

Abstract
In this article, we consider the development of unbiased estimators of the Hessian, of the log-likelihood function with respect to parameters, for partially observed diffusion processes. These processes arise in numerous applications, where such diffusions require derivative information, either through the Jacobian or Hessian matrix. As time-discretizations of diffusions induce a bias, we provide an unbiased estimator of the Hessian. This is based on using Girsanov’s Theorem and randomization schemes developed through Mcleish (2011 Monte Carlo Methods Appl.17, 301–315 (doi:10.1515/mcma.2011.013)) and Rhee & Glynn (2016 Op. Res.63, 1026–1043). We demonstrate our developed estimator of the Hessian is unbiased, and one of finite variance. We numerically test and verify this by comparing the methodology here to that of a newly proposed particle filtering methodology. We test this on a range of diffusion models, which include different Ornstein–Uhlenbeck processes and the Fitzhugh–Nagumo model, arising in neuroscience.

Citation
Chada, N. K., Jasra, A., & Yu, F. (2022). Unbiased estimation of the Hessian for partially observed diffusions. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 478(2262). https://doi.org/10.1098/rspa.2021.0710

Acknowledgements
This work was supported by KAUST baseline funding.

Publisher
The Royal Society

Journal
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

DOI
10.1098/rspa.2021.0710

arXiv
2109.02371

Additional Links
https://royalsocietypublishing.org/doi/10.1098/rspa.2021.0710

Relations
Is Supplemented By:
  • [Software]
    Title: fangyuan-ksgk/Hessian_Estimate: Inference of PODPDO model through MLE on the estimation of the Jacobian & Hessian of data likelihood with respect to the unknown parameter.. Publication Date: 2021-06-22. github: fangyuan-ksgk/Hessian_Estimate Handle: 10754/671186

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2022-06-23 05:55:25
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