Information-Theoretic Lower Bounds for Causal Inference under Credal Uncertainty

Causal effect estimation from observational data typically assumes a single fixed observational distribution. We study the setting in which the distribution is known only to belong to a credal set, a convex set of plausible observational laws. Our results characterize how this observational ambiguity is transformed by causal identification formulas. First, using standard two-point information-theoretic tools after composition with the interventional map, we derive lower bounds for causal effect estimation. The key message is a causal amplification phenomenon: observational discrepancies in low-propensity treatment strata can be hard to detect while producing separated interventional effects. Consequently, in the binary hard-pair constructions, sample complexity scales as $\Omega(1/(\pi_0\varepsilon^2))$ or $\Omega(1/(\beta_0\varepsilon^2))$, and the average treatment effect need not incur an additional dependence on the number of strata. Second, we prove Lipschitz bounds for the interventional mapping, with amplification factor $(1+1/\alpha_{\min})$, showing how positivity controls robust identification width. Third, we study minimax regret for causal decisions under credal uncertainty and show that randomization can halve worst-case regret in a symmetric action-separation construction. We also give matching upper bounds in special cases and an NP-hardness result for computing robust identification width. The experiments are diagnostic checks of the tight hard-instance scaling laws rather than broad empirical benchmarks.