In his late piece 'La nouvelle cuisine' (Bell 1990), John Bell describes the steps from an intuitive, informal principle of locality to a mathematical rule called Factorizability. This rule stipulates that when possible past causes are held fixed, the joint probabilities of outcomes of spacelike separated measurements, conditional on measurement settings, be the product of the local conditional probabilities individually. Bell shows that Factorizability conflicts with predictions of QM, predictions since confirmed in many experiments. However, Bell warns his readers that the steps leading to Factorizability should 'be viewed with the utmost suspicion'. He says that 'it is precisely in cleaning up intuitive ideas for mathematics that one is likely to throw the baby out with the bathwater' (1990, 239). Bell's suspicions were well-founded, for he himself misses an important baby. Here we retrieve and identify it: it is selection bias. We explain how failure of Factorizability may be regarded as a selection artefact, requiring no violation of locality in the intuitive, conceptual sense with which Bell begins his analysis. The argument begins with a central principle of causal discovery, Reichenbach's Principle of Common Cause (PCC). It is well known that correlations due to selection bias are not subject to PCC. Several writers have proposed that EPR-Bell correlations are also an exception to PCC, but it has not been noticed that they fall under this well-known exclusion. The point is relevant not only to the status of Bell nonlocality, but also for statistics and causal modeling. For these fields, the news is that selection effects play a ubiquitous role in quantum phenomena, in a form akin to collider bias.
In his late piece 'La nouvelle cuisine' (Bell 1990), John Bell describes the steps from an intuitive, informal principle of causal locality to a mathematical rule called Factorizability. This rule stipulates that when possible past causes are held fixed, the joint probabilities of outcomes of spacelike separated measurements, conditional on measurement settings, be the product of the local conditional probabilities individually. Bell shows that Factorizability conflicts with predictions of quantum mechanics (QM), predictions since confirmed in many experiments. So, modulo the assumptions of Bell's argument, Factorizability fails. This is the basis for claims of nonlocality in QM, insofar as they stem from Bell's work. However, Bell warns his readers that the steps leading to Factorizability should 'be viewed with the utmost suspicion'. He says that 'it is precisely in cleaning up intuitive ideas for mathematics that one is likely to throw the baby out with the bathwater ' (2004, 239). It turns out that Bell's suspicions were well-founded, for he himself discards an important baby. Here we retrieve and identify it: it is selection bias. We explain how failures of Factorizability in EPR-Bell experiments may be represented as selection artefacts, requiring no violation of locality in the intuitive, conceptual sense with which Bell begins his analysis.
Our argument exploits a well-known loophole, or exclusion, to a familiar principle of causal inference. We begin by introducing this principle and the issue of its limits. We then turn to an outline of selection bias in general, before returning to its application in the quantum case.
Correlation is not causation, but they keep close company. A central principle of causal discovery tells us that where we find correlation we should expect causation, too -not necessarily direct causation between the correlated variables, but otherwise mutual links to other variables, causing both. This is called Reichenbach’s Principle, or the Principle of Common Cause (PCC). David Papineau formulates it like this: 1 If A and B are correlated, then one must be causing the other or they must have one or more common causes, and in the latter case controlling for the common causes will screen off the correlation. (Papineau 2025, 9) This principle is well known in science, though not always under this name. As (Myrvold et al 2024) note, Bell himself relies on a version of it, in formulating his notions of local causality. Writing about Bell’s discussion in (Bell 1976), for example, these authors say that implicit in it is the assumption that correlations between two variables be susceptible to causal explanation, either via a causal connection between the variables, or via a common cause. This assumption was stated explicitly in a later article (Bell 1981), in which he says that “the scientific attitude is that correlations cry out for explanation” [Bell 1981, 152]. (Myrvold et al 2024, §3.1.1) Myrvold et al identify this assumption with PCC.
Useful though it is, PCC comes with some caveats and exclusions. One exception, according to some writers, concerns the EPR-Bell correlations themselves. David Papineau again:
EPR correlations certainly pose a challenge to Reichenbach’s Principle. After all, they are genuine correlations, by anybody’s counting. So, by Reichenbach’s Principle, either one measurement is causing the other, or they have a common cause. But in the former case special relativity would require the two measurements not to be spacelike separated, which they aren’t, and in the latter case Reichenbach’s Principle says that the common cause should screen off the correlation, which Bell’s inequality tells us it can’t. So, whichever way we turn it, Reichenbach’s Principle is in trouble. (2025, 9-10) Similarly from (Brown & Timpson 2016), describing an option that they take PCC to miss:
[I]n a non-separable theory there is a further way in which correlations can be explained which Reichenbach’s stipulations miss out: correlations between systems (e.g., the fact that certain correlations between measurement outcomes will be found to obtain in the future) can be explained directly by irreducible relational properties holding between the systems, relational properties themselves which can be further explained in dynamical terms as arising under local dynamics from a previous non-separable state for the total system. Which is precisely what happens in the Everettian context, for example. (Brown & Timpson 2016) Not everyone agrees. Some think that there are direct causal connections in EPR-Bell experiments, despite relativity (e.g., Maudlin 2011, 2014) Is nonlocality real, according to this proposal? That depends on what we mean by the term. We’ll return to this in §12, but looking ahead, the upshot will be that if nonlocality means a failure of Bell’s condition of Factorizability, the proposal confirms and explains it. If it means direct spacelike influence, the pro
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