  The majority of recent works investigating the link between non-locality and
randomness, e.g. in the context of device-independent cryptography, do so with
respect to some specific Bell inequality, usually the CHSH inequality. However,
the joint probabilities characterizing the measurement outcomes of a Bell test
are richer than just the degree of violation of a single Bell inequality. In
this work we show how to take this extra information into account in a
systematic manner in order to optimally evaluate the randomness that can be
certified from non-local correlations. We further show that taking into account
the complete set of outcome probabilities is equivalent to optimizing over all
possible Bell inequalities, thereby allowing us to determine the optimal Bell
inequality for certifying the maximal amount of randomness from a given set of
non-local correlations.
