Optimized Predictive Coverage by Averaging Time‐Windowed Bayesian Distributions

Abstract

Hydrogeological models require reliable uncertainty intervals that honestly reflect the total uncertainties of model predictions. The operation of a conventional Bayesian framework only produces realistic (interpretable in the context of the natural system) inference results if the model structure matches the data‐generating process, that is, applying Bayes' theorem implicitly assumes the underlying model to be true. With an imperfect model, we may obtain a too‐narrow‐for‐its‐bias uncertainty interval when conditioning on a long time‐series of calibration data, because the assumption of a quasi‐true model becomes too strict. To overcome the problem of overconfident posteriors, we propose a non‐parametric Bayesian method, called Tau‐averaging method: it applies Bayesian analysis on sliding time windows along the data time series for calibration. Thus, it obtains so‐called transitional posteriors per time window. Then, we average these into a wider predictive posterior. With the proposed routine, we explicitly capture the time‐varying impact of model error on prediction uncertainty. The length of the calibration window is optimized to maximize goal‐oriented statistical skill scores for predictive coverage. Our method loosens the perfect‐model‐assumption by conditioning only on small windows of the data set at a time, that is, it assumes that “the model is sufficient to follow the system dynamics for a smaller duration.” We test our method on two cases of soil moisture modeling and show how it improves predictive coverage as compared to the conventional Bayesian approach. Our findings demonstrate that the proposed method convincingly overcomes the overconfidence drawback of Bayesian inference under model misspecification and long calibration time‐series.