Stein-type Estimator Assisted by Dynamics

Estimating the equilibrium parameters of environmental systems is a fundamental task, yet it is often hampered by sparse and noisy sensor data. While the standard Maximum Likelihood Estimator (MLE) is intuitive, Stein’s paradox famously shows that it is statistically inefficient in high dimensions. To address this, we introduce STEADY, an estimator that generalises Stein’s paradox by integrating physical knowledge. We derive our estimator from a principled empirical Bayes model where the prior distribution over the equilibria is a direct consequence of the stationary properties of the system’s governing differential equations. This leads to a novel adaptive shrinkage mechanism, where the amount of shrinkage applied to each observation is naturally modulated by the physical stability of the measured system. We provide a rigorous frequentist analysis of our estimator, proving that STEADY not only dominates the MLE but is also minimax under certain conditions, offering the strongest possible guarantee of robustness. We validate our claims on synthetic data and demonstrate STEADY’s utility on the global Argo ocean float dataset, showing that it effectively filters noise to reveal the “North Atlantic Warming Hole”