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What does the proof of Birnbaum’s theorem prove? (English) Zbl 1294.62002
Summary: Birnbaum’s theorem, that the sufficiency and conditionality principles entail the likelihood principle, has engendered a great deal of controversy and discussion since the publication of the result in 1962. In particular, many have raised doubts as to the validity of this result. Typically these doubts are concerned with the validity of the principles of sufficiency and conditionality as expressed by Birnbaum. Technically it would seem, however, that the proof itself is sound. In this paper we use set theory to formalize the context in which the result is proved and show that in fact Birnbaum’s theorem is incorrectly stated as a key hypothesis is left out of the statement. When this hypothesis is added, we see that sufficiency is irrelevant, and that the result is dependent on a well-known flaw in conditionality that renders the result almost vacuous.

##### MSC:
 62A01 Foundations and philosophical topics in statistics 62F99 Parametric inference
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##### References:
 [1] Barndorff-Nielsen, O. E. (1995) Diversity of evidence and Birnbaum’s theorem (with discussion). Scand. J. Statist. , 22(4), 513-522. · Zbl 0836.62003 [2] Birnbaum, A. (1962) On the foundations of statistical inference (with discussion). J. Amer. Stat. Assoc. , 57, 269-332. · Zbl 0107.36505 [3] Cox, D. R. and Hinkley, D. V. (1974) Theoretical Statistics. Chapman and Hall. · Zbl 0334.62003 [4] Durbin, J. (1970) On Birnbaum’s theorem on the relation between sufficiency, conditionality and likelihood. J. Amer. Stat. Assoc. , 654, 395-398. [5] Evans, M., Fraser, D. A. S. and Monette, G. (1986) On principles and arguments to likelihood (with discussion). Canad. J. of Statistics , 14, 3, 181-199. · Zbl 0607.62002 [6] Gandenberger, G. (2012) A new proof of the likelihood principle. To appear in the British Journal for the Philosophy of Science . · Zbl 1360.62021 [7] Halmos, P. (1960) Naive Set Theory. Van Nostrand Reinhold Co. · Zbl 0087.04403 [8] Helland, I. S. (1995) Simple counterexamples against the conditionality principle. Amer. Statist. , 49, 4, 351-356. [9] Holm, S. (1985) Implication and equivalence among statistical inference rules. In Contributions to Probability and Statistics in Honour of Gunnar Blom . Univ. Lund, Lund, 143-155. · Zbl 0573.62003 [10] Jang, G. H. (2011) The conditionality principle implies the sufficiency principle. Working paper. [11] Kalbfleisch, J. D. (1975) Sufficiency and conditionality. Biometrika , 62, 251-259. · Zbl 0313.62004 [12] Mayo, D. (2010) An error in the argument from conditionality and sufficiency to the likelihood principle. In Error and Inference: Recent Exchanges on Experimental Reasoning, Reliability and the Objectivity and Rationality of Science (D. Mayo and A. Spanos eds.). Cambridge University Press, Cambridge, 305-314. [13] Robins, J. and Wasserman, L. (2000) Conditioning, likelihood, and coherence: A review of some foundational concepts. J. Amer. Stat. Assoc. , 95, 452, 1340-1346. · Zbl 1072.62507
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