  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.
