Statistics, Causality and Bells Theorem

Bell’s [Physics 1 (1964) 195-200] theorem is popularly supposed to establish the nonlocality of quantum physics. Violation of Bell’s inequality in experiments such as that of Aspect, Dalibard and Roger [Phys. Rev. Lett. 49 (1982) 1804-1807] provides empirical proof of nonlocality in the real world. This paper reviews recent work on Bell’s theorem, linking it to issues in causality as understood by statisticians. The paper starts with a proof of a strong, finite sample, version of Bell’s inequality and thereby also of Bell’s theorem, which states that quantum theory is incompatible with the conjunction of three formerly uncontroversial physical principles, here referred to as locality, realism and freedom. Locality is the principle that the direction of causality matches the direction of time, and that causal influences need time to propagate spatially. Realism and freedom are directly connected to statistical thinking on causality: they relate to counterfactual reasoning, and to randomisation, respectively. Experimental loopholes in state-of-the-art Bell type experiments are related to statistical issues of post-selection in observational studies, and the missing at random assumption. They can be avoided by properly matching the statistical analysis to the actual experimental design, instead of by making untestable assumptions of independence between observed and unobserved variables. Methodological and statistical issues in the design of quantum Randi challenges (QRC) are discussed. The paper argues that Bell’s theorem (and its experimental confirmation) should lead us to relinquish not locality, but realism.

💡 Research Summary

The paper revisits Bell’s theorem from a statistical causality perspective, arguing that the experimental violations of Bell inequalities do not compel us to abandon locality but rather to relinquish realism. It begins by restating Bell’s 1964 result and clarifying three historically uncontroversial principles: locality (causal influences propagate forward in time and cannot exceed the speed of light), realism (physical properties have definite values prior to measurement), and freedom (experimenters can choose settings independently of hidden variables). The authors map realism onto the counterfactual “potential outcomes” framework used by statisticians, and freedom onto the notion of randomisation that underpins causal inference.

A central contribution is a finite‑sample proof of Bell’s inequality. By employing concentration inequalities and explicit sample‑size bounds, the authors show that any data set drawn from a locally causal, realist, and free model must satisfy a strong inequality. Empirical violations therefore indicate that at least one of the three assumptions fails.

The discussion of experimental loopholes (detection efficiency, freedom‑of‑choice, and locality timing) is reframed as familiar statistical problems: post‑selection bias and the “missing at random” (MAR) assumption in observational studies. When detectors miss events non‑randomly, the observed sample is no longer independent of the unobserved outcomes, mirroring the failure of MAR. The authors argue that careful alignment of experimental design with statistical analysis—pre‑registering selection criteria, using appropriate weighting, and modelling the detection process—can close these loopholes without invoking untestable independence assumptions.

The paper also proposes a “Quantum Randi Challenge” (QRC) as a statistical stress test for quantum‑mechanical claims. By specifying a priori hypothesis, required sample size, and error‑rate thresholds, a QRC would provide a transparent, reproducible framework for falsifying spurious quantum interpretations, analogous to classical Randi challenges in pseudoscience.

In the concluding philosophical section, the authors reinterpret Bell’s theorem: the empirical data force us to abandon the classical realist picture that hidden variables pre‑exist measurement, while preserving locality. This anti‑realist stance aligns with information‑centric and relational interpretations of quantum mechanics and suggests a shift in how future quantum technologies are conceptually grounded. The paper thus bridges foundational physics, statistical methodology, and experimental practice, offering a coherent roadmap for both interpreting Bell‑type experiments and designing more robust quantum tests.