Finite-sample inference for the compatibility polytope

Develop methods to construct valid finite-sample confidence regions for identified sets defined by the response-function/marginal-compatibility polytope, and characterize how polytope geometry interacts with statistical uncertainty; investigate whether entropic Bell inequalities can aid this extension.

Background

Classical partial identification results assume known population probabilities; in practice, analysts estimate these from data, introducing sampling uncertainty.

The authors call for a finite-sample treatment of polytope-based bounds and suggest that entropic approaches may provide suitable tools.

References

The unified framework raises several open questions that span the boundaries of quantum information, causal inference, and statistical computation. The Balke--Pearl bounds and Manski bounds assume the observed distribution P(Y,X \mid Z) is known exactly. In practice, it is estimated from data. Constructing valid confidence regions for the identified set—and understanding how the polytope geometry interacts with statistical uncertainty—requires extending the framework to finite-sample settings (see for the parallel problem in Bell testing). The entropic Bell inequalities (Section~\ref{sec:kolmogorov}) may provide a natural route, since entropy estimation has well-understood finite-sample properties.

— Bell's Inequality, Causal Bounds, and Quantum Bayesian Computation: A Unified Framework  (2603.28973 - Polson et al., 30 Mar 2026) in Section 7, Open Problems