  The demonstration and use of Bell-nonlocality, a concept that is
fundamentally striking and is at the core of applications in device independent
quantum information processing, relies heavily on the assumption of measurement
independence, also called the assumption of free choice. The latter cannot be
verified or guaranteed. In this paper, we consider a relaxation of the
measurement independence assumption. We briefly review the results of Phys.
Rev. Lett. 113, 190402 (2014), which show that with our relaxation, the set of
so-called measurement dependent local (MDL) correlations is a polytope, i.e. it
can be fully described using a finite set of linear inequalities. Here we
analyze this polytope, first in the simplest case of 2 parties with binary
inputs and outputs, for which we give a full characterization. We show that
partially entangled states are preferable to the maximally entangled state when
dealing with measurement dependence in this scenario. We further present a
method which transforms any Bell-inequality into an MDL inequality and give
valid inequalities for the case of arbitrary number of parties as well as one
for arbitrary number of inputs. We introduce the assumption of independent
sources in the measurement dependence scenario and give a full analysis for the
bipartite scenario with binary inputs and outputs. Finally, we establish a link
between measurement dependence and another strong hindrance in certifying
nonlocal correlations: nondetection events.
