## The likelihood principle as a relation

29 January 2015

Evans, Michael

What does the proof of Birnbaum’s theorem prove? (English summary)

Electron. J. Stat. 7 (2013), 2645–2655.

62A01 (62F99)

According to Birnbaum’s theorem [A. D. Birnbaum, J. Amer. Statist. Assoc. 57 (1962), 269–326; MR0138176 (25 #1623)], compliance with the sufficiency principle and the conditionality principle of statistics would require compliance with the likelihood principle as well. The result appears paradoxical: whereas the first two principles seem reasonable in light of simple examples, the third is routinely violated in statistical practice. Although the theorem has provided ammunition for assaults on frequentist statistics [see, e.g., J. K. Ghosh, M. Delampady and T. K. Samanta, An introduction to Bayesian analysis, Springer Texts Statist., Springer, New York, 2006 (Section 2.4); MR2247439 (2007g:62003)], most Bayesian statisticians do not comply with it at all costs, as attested by current procedures of checking priors and assessing models more generally.

The author formalizes the theorem in terms of set theory to say that the likelihood relation is the equivalence relation generated by the union of the sufficiency relation and the conditionality relation. He finds the result trivial because it relies on extending the conditionality relation, itself intuitively appealing, to the equivalence relation it generates, which conflicts with usual frequentist reasoning and which may even be meaningless for statistical practice. This viewpoint is supported with a counterexample.

While some would regard the irrelevance of the theorem as repelling an attack on frequentist inference, emboldening the advancement of novel methods rooted in fiducial probability [R. Martin and C. Liu, Statist. Sci. 29 (2014), no. 2, 247–251; MR3264537; cf. J. Hannig, Statist. Sci. 29 (2014), no. 2, 254–258; MR3264539; S. Nadarajah, S. Bityukov and N. Krasnikov, Stat. Methodol. 22 (2015), 23–46; MR3261595], the author criticizes the conditionality principle as formalized by the conditionality relation. The problem he sees is that the equivalence relation generated by the conditionality relation and needed for the applicability of the theorem “is essentially equivalent to saying that it doesn’t matter which maximal ancillary we condition on and it is unlikely that this is acceptable to most frequentist statisticians”.

The author concludes by challenging frequentists to resolve the problems arising from the plurality of maximal ancillary statistics in light of the “intuitive appeal” of the conditionality relation. From the perspective of O. E. Barndorff-Nielsen [Scand. J. Statist. 22(1995), no. 4, 513–522; MR1363227 (96k:62010)], that might be accomplished by developing methods for summarizing and weighing “diverse pieces of evidence”, with some of that diversity stemming from the lack of a unique maximal ancillary statistic for conditional inference.

The author formalizes the theorem in terms of set theory to say that the likelihood relation is the equivalence relation generated by the union of the sufficiency relation and the conditionality relation. He finds the result trivial because it relies on extending the conditionality relation, itself intuitively appealing, to the equivalence relation it generates, which conflicts with usual frequentist reasoning and which may even be meaningless for statistical practice. This viewpoint is supported with a counterexample.

While some would regard the irrelevance of the theorem as repelling an attack on frequentist inference, emboldening the advancement of novel methods rooted in fiducial probability [R. Martin and C. Liu, Statist. Sci. 29 (2014), no. 2, 247–251; MR3264537; cf. J. Hannig, Statist. Sci. 29 (2014), no. 2, 254–258; MR3264539; S. Nadarajah, S. Bityukov and N. Krasnikov, Stat. Methodol. 22 (2015), 23–46; MR3261595], the author criticizes the conditionality principle as formalized by the conditionality relation. The problem he sees is that the equivalence relation generated by the conditionality relation and needed for the applicability of the theorem “is essentially equivalent to saying that it doesn’t matter which maximal ancillary we condition on and it is unlikely that this is acceptable to most frequentist statisticians”.

The author concludes by challenging frequentists to resolve the problems arising from the plurality of maximal ancillary statistics in light of the “intuitive appeal” of the conditionality relation. From the perspective of O. E. Barndorff-Nielsen [Scand. J. Statist. 22(1995), no. 4, 513–522; MR1363227 (96k:62010)], that might be accomplished by developing methods for summarizing and weighing “diverse pieces of evidence”, with some of that diversity stemming from the lack of a unique maximal ancillary statistic for conditional inference.

Reviewed by David R. Bickel

**References**

- Barndorff-Nielsen, O. E. (1995) Diversity of evidence and Birnbaum’s theorem (with discussion). Scand. J. Statist., 22(4), 513–522. MR1363227 MR1363227 (96k:62010)
- Birnbaum, A. (1962) On the foundations of statistical inference (with discussion). J. Amer. Stat. Assoc., 57, 269–332. MR0138176 MR0138176 (25 #1623)
- Cox, D. R. and Hinkley, D. V. (1974) Theoretical Statistics. Chapman and Hall. MR0370837 MR0370837 (51 #7060)
- Durbin, J. (1970) On Birnbaum’s theorem on the relation between sufficiency, conditionality and likelihood. J. Amer. Stat. Assoc., 654, 395–398.
- 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. MR0859631 MR0859631 (87m:62017)
- Gandenberger, G. (2012) A new proof of the likelihood principle. To appear in the British Journal for the Philosophy of Science.
- Halmos, P. (1960) Naive Set Theory. Van Nostrand Reinhold Co. MR0114756 MR0114756 (22 #5575)
- Helland, I. S. (1995) Simple counterexamples against the conditionality principle. Amer. Statist., 49, 4, 351–356. MR1368487 MR1368487 (96h:62003)
- 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. MR0795054 MR0795054 (86k:62002)
- Jang, G. H. (2011) The conditionality principle implies the sufficiency principle. Working paper.
- Kalbfleisch, J. D. (1975) Sufficiency and conditionality. Biometrika, 62, 251–259. MR0386075 MR0386075 (52 #6934)
- 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. MR2640508 MR2640508
- Robins, J. and Wasserman, L. (2000) Conditioning, likelihood, and coherence: A review of some foundational concepts. J. Amer. Stat. Assoc., 95, 452, 1340–1346. MR1825290 MR1825290

This review first appeared at “What does the proof of Birnbaum’s theorem prove?” (Mathematical Reviews) and is used with permission from the American Mathematical Society.

Categories: reviews, statistical evidence

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