Mathematics equivalent to Bell's derivation of the inequalities, also allows a local hidden variables explanation for the correlation between distant measurements.
Bell inequalities exclude a broad class of local hidden-variable explanations of quantum correlations. A recurring objection is that the usual Bell form is static, whereas real measuring devices may contain local memory, stochastic dynamics, and measurement-induced disturbances of their hidden variables. We formulate this objection as a general transition-kernel model for dynamical hidden variables. The only locality assumption is imposed at the ontic level: conditional on the pre-measurement ontic state, the transition kernel and response function in each wing depend on the local setting but not on the distant setting or on distant post-measurement variables. Under measurement independence, all such dynamics can be absorbed into effective local response functions. The resulting probabilities have exactly the static Bell-local form and therefore obey the CHSH inequality. The result identifies the available escape routes for reproducing quantum correlations: violating ontic dynamical locality, violating measurement independence, or abandoning a classical hidden-variable ontology. As a consequence, local classical dynamical complexity cannot by itself spoof Bell-nonlocal statistics in device-independent protocols.
Bell inequalities characterize the boundary of the local-realist correlation polytope -- the set of joint probability distributions achievable by classical hidden-variable models. Quantum mechanics exceeds this boundary through non-commutativity, reaching the Tsirelson bound 22 for CHSH. We show that this polytope structure is not specific to quantum foundations: it appears identically in the causal inference literature, where the instrumental inequality, the Balke--Pearl linear programming bounds, and the Tian--Pearl probabilities of causation all arise as facets of the same marginal compatibility polytope. Fine's theorem -- that CHSH inequalities hold if and only if a joint distribution exists -- is precisely the pivot: the instrumental variable model in causal inference is structurally equivalent to the Bell local hidden-variable model, with the instrument playing the role of the measurement setting and the latent confounder playing the role of the hidden variable λ. We develop this correspondence in detail, extending it to algorithmic (Kolmogorov complexity) and entropic formulations of Bell inequalities, the NPA semidefinite programming hierarchy, and the MIP∗=RE undecidability result. We further show that the Born-rule / Bayes-rule duality underlying quantum Bayesian computation exploits the same non-commutativity that enables Bell violation, providing polynomial speedups for posterior inference. The framework yields a concrete dictionary between quantum information theory, causal econometrics, and Bayesian computation, and suggests new directions including NPA-based quantum causal inference algorithms and quantum architectures for function approximation.
Bell's inequality was originally derived under the assumption that experimenters are free to select detector settings independently of any local ``hidden variables'' that might affect the outcomes of measurements on entangled particles. This assumption has come to be known as ``measurement independence'' (also referred to as ``freedom of choice'' or ``settings independence''). For a two-setting, two-outcome Bell test, we derive modified Bell inequalities that relax measurement independence, for either or both observers, while remaining locally causal. We describe the loss of measurement independence for each observer using the parameters M1 and M2, as defined by Hall in 2010, and also by a more complete description that adds two new parameters, which we call $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{{M}_{1}}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{{M}_{2}}$, deriving a modified Bell inequality for each description. These ``relaxed'' inequalities subsume those considered in previous work as special cases, and quantify how much the assumption of measurement independence needs to be relaxed in order for a locally causal model to produce a given violation of the standard Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality. We show that both relaxed Bell inequalities are tight bounds on the CHSH parameter by constructing locally causal models that saturate them. For any given Bell inequality violation, the new two-parameter and four-parameter models each require significantly less mutual information between the hidden variables and measurement settings than previous models. We conjecture that the new models, with optimal parameters, require the minimum possible mutual information for a given Bell violation. We further argue that, contrary to various claims in the literature, relaxing freedom of choice need not imply superdeterminism.