  This paper provides a systematic account of the hidden variable models (HVMs)
formulated to describe systems of random variables with mutually exclusive
contexts. Any such system can be described either by a model with free choice
but generally context-dependent mapping of the hidden variables into observable
ones, or by a model with context-independent mapping but generally compromised
free choice. These two types of HVMs are equivalent, one can always be
translated into another. They are also unfalsifiable, applicable to all
possible systems. These facts, the equivalence and unfalsifiability, imply that
freedom of choice and context-independent mapping are no assumptions at all,
and they tell us nothing about freedom of choice or physical influences exerted
by contexts as these notions would be understood in science and philosophy. The
conjunction of these two notions, however, defines a falsifiable HVM that
describes noncontextuality when applied to systems with no disturbance or to
consistifications of arbitrary systems. This HVM is most adequately captured by
the term "context-irrelevance," meaning that no distribution in the model
changes with context.
