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270 DS perspective on statistical inferenceby a third probability of “don’t know,” where all three probabilities are nonnegativeand sum to one. It needs to be emphasized that the small worldof possible true states is characterized by binary outcomes interpretable astrue/false assertions, while uncertainties about such two-valued outcomes arerepresented by three-valued probabilities.For a given targeted outcome, a convenient notation for a three-valuedprobability assessment is (p, q, r), where p represents personal probability “for”the truth of an assertion, while q represents personal probability “against,”and r represents personal probability of “don’t know.” Each of p, q, and r isnon-negative and together they sum to unity. The outcome complementary toa given target associated with (p, q, r) has the associated personal probabilitytriple (q, p, r). The “ordinary” calculus is recovered from the “extended” calculusby limiting (p, q, r) uncertaintyassessmentstotheform(p, q, 0), or (p, q)for short. The “ordinary” calculus permits “you” to be sure that the assertionis true through (p, q, 0) = (1, 0, 0), or false through (p, q, r) =(0, 1, 0), whilethe “extended” calculus additionally permits (p, q, r) =(0, 0, 1), representingtotal ignorance.Devotees of the “ordinary” calculus are sometimes inclined, when confrontedwith the introduction of r>0, to ask why the extra term is needed.Aren’t probabilities (p, q) withp + q =1sufficienttocharacterizescientificand operational uncertainties? Who needs probabilities of “don’t know”? Oneanswer is that every application of a Bayesian model is necessarily based ona limited state space structure (SSS) that does not assess associated (p, q)probabilities for more inclusive state space structures. Such extended statespace structures realistically always exist, and may be relevant to reportedinferences. In effect, every Bayesian analysis makes implicit assumptions thatevidence about true states of variables omitted from an SSS is “independent”of additional probabilistic knowledge, including ECP expressions thereof, thatshould accompany explicitly identifiable state spaces. DS methodology makesavailable a wide range of models and analyses whose differences from narroweranalyses can point to “biases” due to the limitations of state spaces associatedwith reported Bayesian analyses. Failure of narrower assumptions often accentuatesnon-reproducibility of findings from non-DS statistical studies, castingdoubts on the credibility of many statistical studies.DS methodology can go part way at least to fixing the problem throughbroadening of state space structures and indicating plausible assignments ofpersonal probabilities of “don’t know” to aspects of broadened state spaces,including the use of (p, q, r) =(0, 0, 1) when no empirical basis “whatever” toquote Keynes exists for the use of “a good Benthamite calculation of a series ofprospective advantages and disadvantages, each multiplied by its appropriateprobability, waiting to he summed” that can be brought to bear. DS allows awide range of possible probabilistic uncertainty assessments between completeignorance and the fully “Benthamite” (i.e., Bayesian) models that Keynesrejected for many applications.