  Almost all Bell-inequality experiments to date have used postselection, and
therefore relied on the fair sampling assumption for their interpretation. The
standard form of the fair sampling assumption is that the loss is independent
of the measurement settings, so the ensemble of detected systems provides a
fair statistical sample of the total ensemble. This is often assumed to be
needed to interpret Bell inequality experiments as ruling out hidden-variable
theories. Here we show that it is not necessary; the loss can depend on
measurement settings, provided the detection efficiency factorises as a
function of the measurement settings and any hidden variable. This condition
implies that Tsirelson's bound must be satisfied for entangled states. On the
other hand, we show that it is possible for Tsirelson's bound to be violated
while the CHSH-Bell inequality still holds for unentangled states, and present
an experimentally feasible example.
