A variational approach at uncertainty estimation in data-driven rainfall-runoff modeling

Manuel Álvarez Chaves, Eduardo Acuña Espinoza, Daniel Klotz, Hoshin Gupta, Uwe Ehret, Anneli Guthke

Reliable uncertainty estimation is essential for decision making, evaluating model performance, and defining the limits of what can be inferred from data. While uncertainty estimation typically requires specifying prior assumptions about distributional form, we introduce an approach to learn the structure of uncertainty directly from data. Specifically, we introduce a variational long short-term memory network (vLSTM) that uses variational inference to enable flexible, non-parametric probabilistic predictions. The vLSTM is assessed against deep learning baseline models for probabilistic rainfall–runoff prediction. We discuss training dynamics of probabilistic models, including concerns of overfitting, and compare predictive strategies that emphasize coverage versus point-wise accuracy. Results demonstrate that the vLSTM achieves state-of-the-art performance when evaluated using log-likelihood, while offering a distinct approach to uncertainty estimation that lets uncertainty patterns emerge instead of prescribing them. In our case study, the learned predictive distributions closely resemble that of the current baseline approach, which prescribes a mixture of asymmetric Laplacian distributions. This finding validates our approach, but also points to its fundamental strength: our variational approach to learning uncertainty structure has the potential to provide a more fundamental understanding of predictive uncertainty in arbitrary types of dynamic models and applications across scientific disciplines, enabling progress especially in fields where a priori assumptions seem hard to justify. In general, the vLSTM serves as a valuable approach for exploring uncertainty structures before transitioning to more computationally efficient models once the emerging patterns of uncertainty are better understood.