SeparableCMLSelector

Chooses each group's family independently, by the largest penalized log-likelihood on that group alone.

With the assignment fixed, the classification log-likelihood is a sum of independent per-group terms, so the family that maximizes it for one group can be chosen without reference to any other group. Selection therefore costs the catalog size times the number of groups, and one mixture is assembled at the end. The exhaustive alternative costs the catalog size raised to the number of groups in assembled mixtures.

The penalty is each group's own Bayesian information criterion term. Without it the selection would favour whichever family has the most parameters, since more parameters can only raise a maximized likelihood; with it, separability is preserved because the penalty is itself a sum of per-group terms.

What this does not do is optimize the observed-data criterion. That criterion is not separable, so the family vector chosen here can differ from the one an exhaustive search would choose. How often, and by how much, is precisely the quantity the experiments measure.