In a recent paper [Phys. Rev. A 105, 042220 (2022)], Daley et al. claim that some superdeterministic models are disfavored relative to standard quantum mechanics because such models overfit the statistics of a Bell-type experiment which the authors conducted. We add to the discussion by providing additional context about how few superdeterministic models fall into the category they analyze, and by emphasizing that overfitting, while better as a measure of fine-tuning than other measures given in the literature, does not necessarily indicate a model is universally bad.
In a recent paper [Phys. Rev. A 105, 042220 (2022)], Daley et al. claim that some superdeterministic models are disfavored relative to standard quantum mechanics because such models overfit the statistics of a Bell-type experiment which the authors conducted. We add to the discussion by providing additional context about how few superdeterministic models fall into the category they analyze, and by emphasizing that overfitting, while better as a measure of fine-tuning than other measures given in the literature, does not necessarily indicate a model is universally bad.
Our article described an experiment that adjudicates between different causal accounts of Bell inequality violations by a comparison of their predictive power, finding that certain types of models that are structurally radical but parametrically conservative---of which a class of superdeterministic models are an example---overfit the data relative to models that are structurally conservative but parametrically radical in the sense of endorsing an intrinsically quantum generalization of the framework of causal modeling. In their comment [Phys. Rev. A 109, 026201 (2024)], Hance and Hossenfelder argue that we have misrepresented the purpose of superdeterministic models. We here dispute this claim by recalling the different classes of superdeterministic models we defined in our article and our conclusions regarding which of these are disfavored by our experimental results. Their confusion on this point seems to have arisen in part from the fact that we characterized superdeterministic models within a causal modeling framework and from the fact that we referred to this framework as ``classical'' in order to contrast it with an intrinsically quantum alternative. In this Reply, therefore, we take the opportunity to clarify these points. They also claim that if one is adjudicating between a pair of models, where one model can account for strictly more operational statistics than the other, the first model will tend to overfit the data relative to the second. Because this model inclusion relation can arise for pairs of models in a reductionist hierarchy, they conclude that overfitting should not be taken as evidence against the first model. We point out here that, contrary to this claim, one does not expect overfitting to arise generically in cases of model inclusion, so that it is indeed sometimes appropriate to consider overfitting as a criterion for adjudicating between such models.
Our article described an experiment that adjudicates between different causal accounts of Bell inequality violations by a comparison of their predictive power, finding that certain types of models that are structurally radical but parametrically conservative, of which a class of superdeterministic models are an example, overfit the data relative to models that are structurally conservative but parametrically radical in the sense of endorsing an intrinsically quantum generalization of the framework of causal modelling. In their comment (arXiv:2206.10619), Hance and Hossenfelder argue that we have misrepresented the purpose of superdeterministic models. We here dispute this claim by recalling the different classes of superdeterministic models we defined in our article and our conclusions regarding which of these are disfavoured by our experimental results. Their confusion on this point seems to have arisen in part from the fact that we characterized superdeterministic models within a causal modelling framework and from the fact that we referred to this framework as "classical" in order to contrast it with an intrinsically quantum alternative. In this reply, therefore, we take the opportunity to clarify these points. They also claim that if one is adjudicating between a pair of models, where one model can account for strictly more operational statistics than the other, the first model will tend to overfit the data relative to the second. Because this model inclusion relation can arise for pairs of models in a reductionist heirarchy, they conclude that overfitting should not be taken as evidence against the first model. We point out here that, contrary to this claim, one does not expect overfitting to arise generically in cases of model inclusion, so that it is indeed sometimes appropriate to consider overfitting as a criterion for adjudicating between such models.
Bell famously showed that it was possible to adjudicate between quantum theory and classical hidden-variable theories on experimental grounds rather than merely philosophical ones. These days, there is a new dispute, regarding the question of how to causally account for Bell correlations, with some suggesting a quantum common cause, others a classical superluminal influence. Here, the authors propose a way to move beyond philosophical disputes by comparing the predictive power of different models. They show that the classical models actually tend to overfit Bell test data.
This is the first of two papers that attempt to comprehensively analyse superdeterministic hidden-variables models of Bell correlations. We first give an overview of superdeterminism and discuss various criticisms of it raised in the literature. We argue that the most common criticism, the violation of ‘free-will’, is incorrect. We take up Bell’s intuitive criticism that these models are ‘conspiratorial’. To develop this further, we introduce non-equilibrium extensions of superdeterministic models. We show that the measurement statistics of these extended models depend on the physical system used to determine the measurement settings. This suggests a fine-tuning in order to eliminate this dependence from experimental observation. We also study the signalling properties of these extended models. We show that although they generally violate the formal no-signalling constraints, this violation cannot be equated to an actual signal. We therefore suggest that the so-called no-signalling constraints be more appropriately named the marginal-independence constraints. We discuss the mechanism by which marginal-independence is violated in superdeterministic models. Lastly, we consider a hypothetical scenario where two experimenters use the apparent-signalling of a superdeterministic model to communicate with each other. This scenario suggests another conspiratorial feature peculiar to superdeterminism. These suggestions are quantitatively developed in the second paper.
We prove that superdeterministic models of quantum mechanics are conspiratorial in a mathematically well-defined sense, by further development of the ideas presented in a previous article A . We consider a Bell scenario where, in each run and at each wing, the experimenter chooses one of N devices to determine the local measurement setting. We prove, without assuming any features of quantum statistics, that superdeterministic models of this scenario must have a finely tuned distribution of hidden variables. Specifically, fine-tuning is required so that the measurement statistics depend on the measurement settings but not on the details of how the settings are chosen. We quantify this as the overhead fine-tuning F of the model, and show that F > 0 (corresponding to ‘fine-tuned’) for any N > 1. The notion of fine-tuning assumes that arbitrary (‘non-equilibrium’) hidden-variables distributions are possible in principle. We also show how to quantify superdeterministic conspiracy without using non-equilibrium. This second approach is based on the fact that superdeterministic correlations can mimic actual signalling. We argue that an analogous situation occurs in equilibrium where, for every run, the devices that the hidden variables are correlated with are coincidentally the same as the devices in fact used. This results in extremely large superdeterministic correlations, which we quantify as a drop of an appropriately defined formal entropy. Non-local and retrocausal models turn out to be non-conspiratorial according to both approaches.
An active area of research in the fields of machine learning and statistics is the development of causal discovery algorithms , the purpose of which is to infer the causal relations that hold among a set of variables from the correlations that these exhibit . We apply some of these algorithms to the correlations that arise for entangled quantum systems. We show that they cannot distinguish correlations that satisfy Bell inequalities from correlations that violate Bell inequalities, and consequently that they cannot do justice to the challenges of explaining certain quantum correlations causally. Nonetheless, by adapting the conceptual tools of causal inference, we can show that any attempt to provide a causal explanation of nonsignalling correlations that violate a Bell inequality must contradict a core principle of these algorithms, namely, that an observed statistical independence between variables should not be explained by fine-tuning of the causal parameters. In particular, we demonstrate the need for such fine-tuning for most of the causal mechanisms that have been proposed to underlie Bell correlations, including superluminal causal influences, superdeterminism (that is, a denial of freedom of choice of settings), and retrocausal influences which do not introduce causal cycles.