Posterior contraction rates for non-parametric state and drift estimation

Sebastian Reich; Paul J. Rozdeba

doi:10.3934/fods.2020016

Article Contents

2020, Volume 2, Issue 3: 333-349. Doi: 10.3934/fods.2020016

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Posterior contraction rates for non-parametric state and drift estimation

Sebastian Reich^, and
Paul J. Rozdeba

Institute of Mathematics, University of Potsdam, Karl-Liebknecht-Str. 24/25, D-14476 Potsdam, Germany

^* Corresponding author: Sebastian Reich
^* Corresponding author: Sebastian Reich

Received: March 2020

Revised: August 2020

Early access: October 2020

Published: September 2020

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Abstract

We consider a combined state and drift estimation problem for the linear stochastic heat equation. The infinite-dimensional Bayesian inference problem is formulated in terms of the Kalman–Bucy filter over an extended state space, and its long-time asymptotic properties are studied. Asymptotic posterior contraction rates in the unknown drift function are the main contribution of this paper. Such rates have been studied before for stationary non-parametric Bayesian inverse problems, and here we demonstrate the consistency of our time-dependent formulation with these previous results building upon scale separation and a slow manifold approximation.

Keywords:

State and parameter estimation,
Bayesian inference,
posterior contraction rates,
stochastic partial differential equations,
Kalman–Bucy filter.

Mathematics Subject Classification: Primary: 62G20, 62G05, 62F15; Secondary: 60H15, 62M20.

Citation:

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Figure 1. We display the time evolution of the bias $ |m_t^f(k)-\mathcal{F}^\ast(k)| $ and the variances $ \sigma_t^f(k) $ and $ p_t^f(k) $ for increasing values of $ k $. We compare the results from the combined state and parameter estimation problem (labelled 'dynamic') with those from the associated direct inference problem in the parameter alone (labelled 'stationary'). The delay in the onset of the asymptotic $ t^{-1} $ regime as $ k $ increases can be clearly seen as well as that $ p_t^f(k) < \sigma_t^f(k) $ for all $ t>0 $. Furthermore, as theoretically predicted, the dynamic and stationary problem formulations behave almost identically in terms of their drift estimates

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