  Free choice (or statistical independence) assumption in a hidden variable
model (HVM) means that the settings chosen by experimenters do not depend on
the values of the hidden variable. The assumption of context-independent (CI)
mapping in an HVM means that the results of a measurement do not depend on
settings for other measurements. If the measurements are spacelike separated,
this assumption is known as local causality. Both free choice and CI mapping
assumptions are considered necessary for derivation of the Bell-type criteria
of contextuality/nonlocality. It is known, however, for a variety of special
cases, that the two assumptions are not logically independent. We show here, in
complete generality, for any system of random variables with or without
disturbance/signaling, that an HVM that postulates CI mapping is equivalent to
an HVM that postulates free choice. If one denies the possibility that a given
empirical scenario can be described by an HVM in which measurements depend on
other measurements' settings, free choice violations should be denied too, and
vice versa.
  KEYWORDS: Contextuality; context-independent mapping; free choice; local
causality; nonlocality.
