2DkcTXceO

274 DS perspective on statistical inference“p-value” of contemporary applied statistics. The choice offered by such a DSsignificance test is not to either “accept” or “reject” the null hypothesis, butinstead is either to “not reject” or “reject.”In a similar vein a (p, q, r) triplecanbeassociatedwithanyspecifiedrange of P values, such as the interval (.25,.75), thus creating an “intervalestimate.” Similarly, if so desired, a “sharp” null hypothesis such as P = .25can be rendered “dull” using an interval such as (.24,.26). Finally, if theSSS is expanded to include a future toss or tosses, then (p, q, r) “predictions”concerning the future outcomes of such tosses can be computed given observedsample data.There is no space here to set forth details concerning how independentinput mass distributions on margins of an SSS are up-projected to mass distributionson the full SSS, and are combined there and used to derive inferencesas in the preceding paragraphs. Most details have been in place, albeitusing differing terminology, since the 1960s. The methods are remarkably simpleand mathematically elegant. It is surprising to me that research on thestandard protocol has not been taken up by any but an invisible sliver of themathematical statistics community.The inference system outlined in the preceding paragraphs can and shouldbe straightforwardly developed to cover many or most inference situationsfound in statistical textbooks. The result will not only be that traditionalBayesian models and analyses can be re-expressed in DS terms, but more significantlythat many “weakened” modifications of such inferences will becomeapparent, for example, by replacing Bayesian priors with DS mass distributionsthat demand less in terms of supporting evidence, including limiting“total ignorance” priors concerning “parameter” values. In the case of such(0, 0, 1)-based priors, traditional “likelihood functions” assume a restated DSform having a mass distribution implying stand-alone DS inferences. But whena “prior” includes limited probabilities of “don’t know,” the OCP “likelihoodprinciple” no longer holds, nor is it needed. It also becomes easy in principleto “weaken” parametric forms adopted in likelihood functions, for example,by exploring DS analyses that do not assume precise normality, but mightassume that cumulative distribution functions (CDFs) are within, say, .10 ofa Gaussian CDF. Such “robustness” research is in its infancy, and is withoutfinancial support, to my knowledge, at the present time.The concepts of DS “weakening,” or conversely “strengthening,” providebasic tools of model construction and revision for a user to consider in thecourse of arriving at final reports. In particular, claims about complex systemsmay be more appropriately represented in weakened forms with increasedprobabilities of “don’t know.”