  Bell's inequality was originally derived under the assumption that
experimenters are free to select detector settings independently of any local
"hidden variables" that might affect the outcomes of measurements on entangled
particles. This assumption has come to be known as "measurement independence"
(also referred to as "freedom of choice" or "settings independence"). For a
two-setting, two-outcome Bell test, we derive modified Bell inequalities that
relax measurement independence, for either or both observers, while remaining
locally causal. We describe the loss of measurement independence for each
observer using the parameters $M_1$ and $M_2$, as defined by Hall in 2010, and
also by a more complete description that adds two new parameters, which we call
$\hat{M}_1$ and $\hat{M}_2$, deriving a modified Bell inequality for each
description. These "relaxed" inequalities subsume those considered in previous
work as special cases, and quantify how much the assumption of measurement
independence needs to be relaxed in order for a locally causal model to produce
a given violation of the standard Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH)
inequality. We show that both relaxed Bell inequalities are tight bounds on the
CHSH parameter by constructing locally causal models that saturate them. For
any given Bell inequality violation, the new two-parameter and four-parameter
models each require significantly less mutual information between the hidden
variables and measurement settings than previous models. We conjecture that the
new models, with optimal parameters, require the minimum possible mutual
information for a given Bell violation. We further argue that, contrary to
various claims in the literature, relaxing freedom of choice need not imply
superdeterminism.
