Causal Fingerprints Reveal Hidden Mechanisms in Partially-Observed Dynamical Systems
In partially-observed dynamical systems, distinct hidden mechanisms can produce similar observable summary statistics while explaining individual events differently. Matching statistics therefore does not guarantee matching mechanisms. We introduce causal fingerprints: realization-dependent patterns that track how information about a hidden state is revealed through subsequent evolution. Within a prescribed dynamical model, we compare the filtering posterior, based on observations up to a given time, with the smoothing posterior, obtained after adding an expanding window of future observations. We quantify this retrospective information gain using Kullback-Leibler divergence and track its evolution as the future window of observational data expands. The fingerprint reveals both the strength and temporal reach of hidden influence, which can vary across events and their stages. Because the observed trajectory is held fixed while hidden-state inference depends on the candidate model, fing