Multi-Method Inference for Critical Thresholds from Sparse Observational Data

Critical thresholds are often latent functionals rather than directly observed breakpoints. Under sparse observations, structurally different estimators can return similar numerical locations while relying on incompatible inverse mappings or targeting different quantities. We introduce a qualification-first framework that fixes the scientific target before model choice, profiles each method's native inferential object and failure modes, and uses known-truth validation to assign one of four roles: target-responsive, calibrated within a stated validation range, native diagnostic, or abstention. In a compact mechanistically aligned but observation-model-mismatched experiment at truths 0.90, 0.94, and 0.97, a hierarchical Bayesian model (HBM) responds monotonically but compresses near the upper end and is positively biased at the two lower truths (RMSE 0.032, 0.026, and 0.006); change-point detection (CPD) does not track the mechanistic target and remains native-only. Applied to 10,153 U.S. county-years from 2,212 counties during 2018–2023, the HBM posterior median unconstrained coverage crossing is 0.991, with 31.8% of posterior draws implying no feasible crossing at coverage $\le 1$ under $\eta =0.97$. CPD returns a nonsignificant native breakpoint of 0.992 ($p=0.366$), while a biologically informed neural network (BINN) abstains because profile support spans 0.88–0.99. The framework therefore assigns different inferential roles and performs no fusion: numerical proximity alone is not evidence that heterogeneous methods recover the same latent threshold.