## On principles and arguments to likelihood

### Abstract

Birnbaum (1962a) argued that the conditionality principle (C) and the sufficiency principle (S) implied the likelihood principle (L); he then argued (Birnbaum 1972) that C and a mathematical equivalence principle M implied L. Evans, Fraser, and Monette (1985a) gave reference details, and this paper gives proof that C alone implies L. The level of support by the profession for L is sharply less than that for S or even for C; thus the paradoxical nature of these results. In this regard, we elaborate on the Monette example (Fraser, Monette, and Ng 1984), which provides a strong case against L. We also examine closely the various proofs linking the principles and find that S and C can each be used operationally to suppress information otherwise deemed relevant. From another viewpoint this says that S and C can each be used in contexts that directly conflict with the original examples and motivations supporting them; the principles can thus be viewed as inappropriately used, or more strongly, as invalid. In either case, the result that C and S imply L or that C implies L can be regarded as noneffective in the context of discriminating applications. A resolution of the apparent anomalies can be obtained by allowing the statistical model to include ingredients additional to those usually present (particularly for subsequent use with conditionality), or alternatively by restricting the application of the principles to contexts where the conflicts would seem not to arise.

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