Abstract: Sketches of an alternative family of indices, for the set of probability measures of a statistical experiment, that allows for rigorous numerical decision-making, even in the presence of finite precision in measurement of the empirical phenomenon of interest, will be given. The often overlooked "Empirical indiscernibility" of elements of the index set, usually induced by physical limits on measurement precision, as well as, any nonidentifiability inherent in the mathematical model itself around the optimal decision will be rigorously accounted for, in addition to all the usual errors associated with shadows of real computations confined to a finite screen of floating-points embedded in the reals. Applications from phylogenetics and finite mixture problems will be covered. Key words: rigorous global optimization, interval analysis, automatic differentiation, Jukes and Cantor model of DNA evolution, efficient M-H sampling