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## 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.

Reviewed by David R. Bickel

References

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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.