  The Bell theorem is investigated as a trade-off relation between assumptions
for the underlying hidden variable model. Considering the introduction of a set
of hidden variables itself to be one of the essential assumptions, we introduce
a measure of hiddenness, a quantity that expresses the degree to which hidden
variables are needed. We derive novel relaxed Bell-Clauser-Horne-Shimony-Holt
(Bell-CHSH) inequalities for separable models, which are hidden variable models
that only satisfy the locality but not the measurement independence condition,
in terms of their measurement dependence and hiddenness. The derived relations
can be interpreted as trade-off relations between the measurement dependence
and hiddenness for separable models in the CHSH scenario. It is also revealed
that the relation gives a necessary and sufficient condition for the measures
to be realized by a separable model.
