By Lawrence J Hubert; Howard Wainer
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Additional info for A statistical guide for the ethically perplexed
24 A Statistical Guide for the Ethically Perplexed ¯ which occurs when A does (1) For the complementary event, A, ¯ not, P (A) = 1 − P (A). (2) For events A and B that are not necessarily mutually exclusive, P (A or B) = P (A) + P (B) − P (A and B) . (3) The rule of total probability: Given a collection of mutually exclusive and exhaustive events, B1 , . . , BK (that is, all are pairwise mutually exclusive and their union gives the sure event), K P (A|Bk )P (Bk ) . P (A) = k=1 (4) Bayes’ theorem (or rule) for two events, A and B: P (A|B) = P (B|A)P (A) ¯ (A) ¯ .
Dick Cavett Although the assignment of probabilities to events consistent with the disjoint rule may lead to an internally valid system mathematically, there is still no assurance that this assignment is “meaningful,” or bears any empirical validity for observable long-run expected frequencies. There seems to be a never-ending string of misunderstandings in the way probabilities can be generated that are either blatantly wrong, or more subtly incorrect, irrespective of the internally consistent system they might lead to.
This is a mistake. A good fit reveals nothing about the flexibility of the theory (how much it cannot fit), the variability of the data (how firmly the data rule out what the theory cannot fit), or the likelihood of other outcomes (perhaps the theory could have fit any plausible result), and a reader needs all three pieces of information to decide how much the fit should increase belief in the theory. The use of good fits as evidence is not supported by philosophers of science nor by the history of psychology; there seem to be no examples of a theory supported mainly by good fits that has led to demonstrable progress.