# Relating Causal and Probabilistic Approaches to Contextuality

## TL;DR

This paper bridges causal and probabilistic frameworks to better understand contextuality, demonstrating their equivalence and formalizing the concept through hidden influences and signaling in probabilistic models.

## Contribution

It introduces a causal perspective on contextuality, proving the equivalence between model-based and CbD frameworks, and formalizes hidden influences as a core aspect of contextuality.

## Key findings

- M-contextuality aligns with standard contextuality in consistent systems
- M-contextuality is equivalent to CbD-contextuality in general systems
- Hidden influences or signaling are necessary for non-contextual models

## Abstract

A primary goal in recent research on contextuality has been to extend this concept to cases of inconsistent connectedness, where observables have different distributions in different contexts. This article proposes a solution within the framework of probabilistic causal models, which extend hidden-variables theories, and then demonstrates an equivalence to the contextuality-by-default (CbD) framework. CbD distinguishes contextuality from direct influences of context on observables, defining the latter purely in terms of probability distributions. Here we take a causal view of direct influences, defining direct influence within any causal model as the probability of all latent states of the system in which a change of context changes the outcome of a measurement. Model-based contextuality (M-contextuality) is then defined as the necessity of stronger direct influences to model a full system than when considered individually. For consistently connected systems, M-contextuality agrees with standard contextuality. For general systems, it is proved that M-contextuality is equivalent to the property that any model of a system must contain "hidden influences", meaning direct influences that go in opposite directions for different latent states, or equivalently signaling between observers that carries no information. This criterion can be taken as formalizing the "no-conspiracy" principle that has been proposed in connection with CbD. M-contextuality is then proved to be equivalent to CbD-contextuality, thus providing a new interpretation of CbD-contextuality as the non-existence of a model for a system without hidden direct influences.