Distributional shift, especially temporal shift in which the underlying distribution or model evolves over time, is a common feature of climate, energy, and financial systems. This makes reliable modeling, forecasting, and decision-making fundamentally challenging, and is also closely related to broader concerns in AI alignment and AI safety, where systems must operate under changing and imperfectly understood environments.

This talk presents a foundational view of distributional shift through the M-structure, a nonlinear probabilistic framework for studying model uncertainty from a data perspective. Within this framework, notions such as independence, together with global asymptotic results including an M-law of large numbers and an M-central limit theorem, provide a language for describing uncertainty and long-run behavior when a single fixed probability law is no longer adequate. From this viewpoint, distributional shift is treated as a structural feature of the data-generating mechanism rather than a minor perturbation around a fixed model.

On the methodological side, the talk considers a central question under temporal shift: when and how should a model be updated? In time-series settings, this is closely related to the choice of window size. A suitable window helps better understand and approximate the underlying distributional uncertainty set induced by distributional shift, whereas windows that are too short or too long may fail to capture the relevant structure. Based on this perspective, a principled data-driven method for window-size selection under nonstationarity is introduced.

Data illustrations are given on an electricity dataset and financial market data, showing how the framework can help analyze evolving environments and regime-sensitive behavior. The talk also briefly discusses aspects beyond prediction, including robust decision-making and adaptive procedures.

Overall, the talk aims to connect foundational theory, practical methodology, and data illustrations in a unified framework for studying distributional shift, a natural concern in climate risk, renewable energy, and finance under model uncertainty.