Tightest conditions for violating the Bell inequality when measurement independence is relaxed

It has been believed that statistical inequalities such as the Bell inequality should be modified once measurement independence (MI), the assumption that observers can freely choose measurement settings without changing the probability distribution of hidden variables, is relaxed. However, we show that there exists the possibility that the Bell inequality is still valid even if MI is relaxed. MI is only a sufficient condition to derive the Bell inequality when both determinism and setting independence, usually called local realism, are satisfied. We thus propose a condition necessary and sufficient for deriving the Bell inequality, called concealed measurement dependence.