  Different variants of a Bell inequality, such as CHSH and CH, are known to be
equivalent when evaluated on nonsignaling outcome probability distributions.
However, in experimental setups, the outcome probability distributions are
estimated using a finite number of samples. Therefore the nonsignaling
conditions are only approximately satisfied and the robustness of the violation
depends on the chosen inequality variant. We explain that phenomenon using the
decomposition of the space of outcome probability distributions under the
action of the symmetry group of the scenario, and propose a method to optimize
the statistical robustness of a Bell inequality. In the process, we describe
the finite group composed of relabeling of parties, measurement settings and
outcomes, and identify correspondences between the irreducible representations
of this group and properties of outcome probability distributions such as
normalization, signaling or having uniform marginals.
