Counterfactual Sufficiency
State Representation and Response Geometry
Research question
Which distinctions in a structural state must a representation preserve for a declared family of counterfactuals?
Abstract
Which additional state coordinates preserve a specified policy calculation? I study this question relative to a supplied representation, target loss, and admissible feature class. Restricted response derivatives give a local benchmark: discarded eigenvalues determine leading decoder risk under shrinking perturbations. Global coordinate choice also depends on the variation left within feature level sets. A conditional-generator certificate bounds this risk, and its spectral slack enters the comparison between estimated certificate minimization and a risk-optimal coordinate map. The bound separates that slack from score estimation, sampling, and optimization error. A finite-graph counterpart evaluates fitted-decoder residuals. Occupational-choice and heterogeneous-agent exercises illustrate the consequences for model-generated policy responses. The analysis keeps representation loss, fitted prediction error, and identification of the underlying response map distinct.
In one sentence
A compressed state is useful only if it preserves the distinctions that matter for the counterfactuals you actually want to answer.