# Joint state-parameter estimation of a nonlinear stochastic energy balance model from sparse noisy data

## TL;DR

This paper develops a regularized Bayesian approach for joint state-parameter estimation in nonlinear stochastic energy balance models, effectively handling sparse noisy data and ill-posedness to improve inference accuracy in paleoclimate studies.

## Contribution

It introduces a strongly regularized posterior with physical priors and a particle Gibbs sampler tailored for high-dimensional SEBM, enhancing estimation robustness and uncertainty quantification.

## Key findings

- Regularized posteriors overcome ill-posedness and stay within physical ranges.
- Posterior of states concentrates near the true states, reducing noise impact.
- Parameter posteriors exhibit large uncertainty due to ill-posedness and regularization.

## Abstract

While nonlinear stochastic partial differential equations arise naturally in spatiotemporal modeling, inference for such systems often faces two major challenges: sparse noisy data and ill-posedness of the inverse problem of parameter estimation. To overcome the challenges, we introduce a strongly regularized posterior by normalizing the likelihood and by imposing physical constraints through priors of the parameters and states. We investigate joint parameter-state estimation by the regularized posterior in a physically motivated nonlinear stochastic energy balance model (SEBM) for paleoclimate reconstruction. The high-dimensional posterior is sampled by a particle Gibbs sampler that combines MCMC with an optimal particle filter exploiting the structure of the SEBM. In tests using either Gaussian or uniform priors based on the physical range of parameters, the regularized posteriors overcome the ill-posedness and lead to samples within physical ranges, quantifying the uncertainty in estimation. Due to the ill-posedness and the regularization, the posterior of parameters presents a relatively large uncertainty, and consequently, the maximum of the posterior, which is the minimizer in a variational approach, can have a large variation. In contrast, the posterior of states generally concentrates near the truth, substantially filtering out observation noise and reducing uncertainty in the unconstrained SEBM.