  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.
