  Designs, and in particular symmetric, informationally complete (SIC)
structures, play an important role in the quantum tomographic reconstruction
process and, by extension, in certain interpretations of quantum theory
focusing on such a process. This fact is due to the symmetry of the
reconstruction formula that designs lead to. However, it is also known that the
same tomographic task, albeit with a less symmetric formula, can be
accomplished by any informationally complete (non necessarily symmetric)
structure. Here we show that, if the tomographic task is replaced by a
data-driven inferential approach, the reconstruction, while possible with
designs, cannot by accomplished anymore by an arbitrary informationally
complete structure. Hence, we propose the data-driven inference as the arena in
which the role of designs naturally emerges. Our inferential approach is based
on a minimality principle according to which, among all the possible inferences
consistent with the data, the weakest should be preferred, in the sense of
majorization theory and statistical comparison.
