  Device-independent certifications employ Bell tests to guarantee the proper
functioning of an apparatus from the sole knowledge of observed measurement
statistics, i.e. without assumptions on the internal functioning of the
devices. When these Bell tests are implemented with devices having too low
efficiency, one has to post-select the events that lead to successful
detections and thus rely on a fair sampling assumption. The question that we
address in this paper is what remains of a device-independent certification
under fair sampling. We provide an intuitive description of post-selections in
terms of filters and define the fair sampling assumption as a property of these
filters, equivalent to the definition introduced in [Berry et. al., PRA 81(1),
012109 (2010)]. When this assumption is fulfilled, the post-selected data is
reproduced by an ideal experiment where lossless devices measure a filtered
state which can be obtained from the actual state via local probabilistic maps.
Trusted conclusions can thus be obtained on the quantum properties of this
filtered state and the corresponding measurement statistics can reliably be
used, e.g., for randomness generation or quantum key distribution. We also
explore a stronger notion of fair sampling leading to the conclusion that the
post-selected data is a fair representation of the data that would be obtained
with lossless detections. Furthermore, we show that our conclusions hold in
cases of small deviations from exact fair sampling. Finally, we describe setups
previously or potentially used in Bell-type experiments under fair sampling and
identify the underlying device-specific assumptions.
