Rejoinder: “On the Birnbaum Argument for the Strong Likelihood Principle”

I am honored and grateful to have so many interesting and challenging comments on my paper. I want to thank the discussants for their willingness to jump back into the thorny quagmire of Birnbaum’s argument. To a question raised in the paper “Does it matter?”, these discussions show the answer is yes. The enlightening connections to contemporary projects are especially valuable in galvanizing future efforts to address foundational issues in statistics. As long-standing as Birnbaum’s result has been, Birnbaum himself went through dramatic shifts in a short period of time following his famous (1962) result. More than of historical interest, these shifts provide a unique perspective on the current problem. Already in the rejoinder to Birnbaum (1962), he is worried about criticisms (by Pratt, 1962) pertaining to applying WCP to his constructed mathematical mixtures (what I call Birnbaumization), and hints at replacing WCP with another principle (Irrelevant Censoring). Then there is a gap until around 1968 at which point Birnbaum declares the SLP plausible “only in the simplest case, where the parameter space has but two” predesignated points [Birnbaum (1968), page 301]. He tells us in Birnbaum (1970a, page 1033) that he has pursued the matter thoroughly, leading to “rejection of both the likelihood concept and various proposed formalizations of prior information.” The basis for this shift is that the SLP permits interpretations that “can be seriously misleading with high probability” [Birnbaum (1968), page 301]. He puts forward the “confidence concept” (Conf) which takes from the Neyman–Pearson (N–P) approach “techniques for systematically appraising and bounding the probabilities (under respective hypotheses) of seriously misleading interpretations of data” while supplying it an evidential interpretation [Birnbaum (1970a), page 1033].