Bayesian Causal Experimental Design for CATE Estimation under Noncompliance

Causal experimental design often assumes direct control over treatment assignment, but many experiments can only assign encouragements that affect treatment receipt indirectly. We formulate this as adaptive instrumental variable (IV) design under noncompliance: the experimenter chooses unit--instrument queries, observes stochastic treatment realizations and outcomes, and seeks the treatment-level conditional average treatment effect (CATE). This creates an acquisition mismatch: uncertainty in prospective feedback does not necessarily correspond to uncertainty in the target causal estimand, so outcome-predictive or marginal variance-based criteria may select queries weakly informative for CATE. We introduce instrumental variable information gain (IVIG), a Bayesian acquisition principle that scores queries by the expected information their joint treatment-realization and outcome feedback provides about CATE values over a target population. Under the working Bayesian posterior, IVIG is Bayes-risk aligned under logarithmic loss; we also characterize the residual information discarded by outcome-only acquisition and connect greedy IVIG-style acquisition to target posterior-variance reduction under a fixed-covariance analysis. To approximate IVIG, we use an empirical-Bayes Gaussian Process IV posterior approximation with first-stage treatment-realization modeling and treatment-realization-specific fantasy updates. Experiments on synthetic and two semi-synthetic IV benchmarks show improved CATE sample efficiency over adaptive-design baselines; a real-label diagnostic shows improved recovery of full-data CATE estimates from observed IV data.