Statistics, Causality and Bell's 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. 
Richard D. Gill
stat.AP physics.hist-ph quant-ph stat.ME