  The Mapper algorithm does not include a check for whether the cover produced
conforms to the requirements of the nerve lemma. To perform a check for
obstructions to the nerve lemma, statistical considerations of multiple testing
quickly arise.
  In this paper, we propose several statistical approaches to finding
obstructions: through a persistent nerve lemma, through simulation testing, and
using a parametric refinement of simulation tests.
  We suggest Certified Mapper -- a method built from these approaches to
generate certificates of non-obstruction, or identify specific obstructions to
the nerve lemma -- and we give recommendations for which statistical approaches
are most appropriate for the task.
