What does the proof of Birnbaum's theorem prove?

TL;DR

This paper critically examines Birnbaum's theorem, revealing that its original statement omits a key hypothesis, which, when included, shows the theorem's conclusion is almost vacuous due to flaws in the conditionality principle.

Contribution

The paper formalizes Birnbaum's theorem using set theory and identifies a missing hypothesis, challenging the validity of the original result.

Findings

Birnbaum's theorem is incorrectly stated due to a missing hypothesis.

When the hypothesis is included, sufficiency becomes irrelevant.

The flaw in the conditionality principle makes the theorem's conclusion nearly vacuous.

Abstract

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.