  Bell inequalities follow from a set of seemingly natural assumptions about
how to provide a causal model of a Bell experiment. In the face of their
violation, two types of causal models that modify some of these assumptions
have been proposed: (i) those that are parametrically conservative and
structurally radical, such as models where the parameters are conditional
probability distributions (termed 'classical causal models') but where one
posits inter-lab causal influences or superdeterminism, and (ii) those that are
parametrically radical and structurally conservative, such as models where the
labs are taken to be connected only by a common cause but where conditional
probabilities are replaced by conditional density operators (these are termed
'quantum causal models'). We here seek to adjudicate between these alternatives
based on their predictive power. The data from a Bell experiment is divided
into a training set and a test set, and for each causal model, the parameters
that yield the best fit for the training set are estimated and then used to
make predictions about the test set. Our main result is that the structurally
radical classical causal models are disfavoured relative to the structurally
conservative quantum causal model. Their lower predictive power seems to be due
to the fact that, unlike the quantum causal model, they are prone to a certain
type of overfitting wherein statistical fluctuations away from the
no-signalling condition are mistaken for real features. Our technique shows
that it is possible to witness quantumness even in a Bell experiment that does
not close the locality loophole. It also overturns the notion that it is
impossible to experimentally test the plausibility of superdeterminist models
of Bell inequality violations.
