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The criteria reported alongside the ranking criterion when none are named.
A factory, because the extended criterion is constructed against a catalog size and because a criterion is free to carry state. All five are penalised likelihoods differing only in penalty, so they are reported side by side rather than combined: averaging them would manufacture an agreement the evidence does not support.
Ordered by penalty, lightest first, because that is what makes their disagreement readable. AIC charges two per parameter and over-selects; BIC charges the log of the sample size and under-selects; Hannan-Quinn sits between them and was measured the most accurate of the five on this method's component count. A reader who sees AIC choose five and BIC choose two is looking at the two ends of that scale rather than at a contradiction.
Parameters
the number of families the fitter may choose from, which the extended criterion charges for