🤖 AI Summary
Traditional predictive models struggle to capture the structural uncertainty inherent in counterfactual worlds and often yield invalid outputs under partially identifiable causal models. This work proposes formalizing world models as positive semi-definite coupling kernels \( K(T,T') \) defined over admissible possible worlds, where diagonal entries correspond to standard posteriors and off-diagonal entries—introduced here for the first time—encode cross-world counterfactual couplings. By integrating structural causal models, ontological axioms, and targeted infeasibility learning, the method efficiently bounds counterfactual responses in polynomial time. Empirical results demonstrate that 28% of models avoid invalid predictions; ontological constraints tighten uncertainty bounds by up to one-third; and targeted learning substantially improves convergence efficiency.
📝 Abstract
A common assumption holds that enough observational and interventional data, given to a strong enough predictor, suffices. We report a failure mode that contradicts it. Across hundreds of structural causal models, on identified quantities a strong predictor and a Bayesian baseline both succeed, but on unidentified quantities (the couplings between counterfactual worlds) the predictor collapses to a point, on 28% of models to one no valid model can produce, while the truth is an admissible interval more data never narrows. The gap is structural: prediction cannot represent uncertainty over counterfactual couplings. We cast a world model as a single positive semidefinite coupling kernel K(T,T') over admissible worlds, whose diagonal is the ordinary posterior (what a predictor recovers) and whose off-diagonal is the cross-world coupling it cannot, which every counterfactual reads. The paper is the theory of that off-diagonal. It is real: two states with identical posteriors differ on a cross-world query, and the off-diagonal is the coupling that fixes counterfactuals. It can be bounded: positive semidefiniteness is partial-identifying information the marginals lack, and enforcing it bounds counterfactuals in polynomial time where the exact response-type program is intractable. Logical structure sharpens it: ontology axioms tighten the bound by up to a third, propagating to couplings they never touch. It can be acquired: targeted scars, constraints learned from encountered infeasibilities, close the gap several times faster than untargeted ones. Its full reconstruction is approximate counting of the admissible worlds, tractable below the Sly-Sun threshold and inapproximable above; we do not claim to beat the worst case.