From Least Action to Latent Forcing Tests for Root-Cause Attribution

Root-cause attribution in latent dynamical systems asks which hidden process was perturbed and when. We study sparse additive forcing in latent stochastic differential equations using the classical innovation likelihood-ratio statistic from model-based fault detection and isolation. On an Euler–Maruyama grid with diagonal diffusion, the diffusion-normalised action evaluated at the profile-maximising latent forcing is exactly the per-atom generalised log-likelihood ratio. The oracle detection laws follow from standard Gaussian testing. Our focus is the identification, reconstruction and reporting conditions needed to use this statistic in learned latent coordinates. We derive a deterministic condition under which representation, drift and diffusion errors preserve the oracle attribution ranking. On synthetic latent-target benchmarks, the resulting one-pass estimator achieves comparable atom accuracy to an iterative solver while being substantially cheaper to evaluate. Physical-system simulations and real-data experiments also expose the importance of observability and the limits of transferring latent forcing attribution to observed-variable diagnosis. These results support latent forcing tests as an efficient and theoretically grounded approach to root-cause attribution when latent coordinates are meaningful and sufficiently observable.