Birnbaum’s proof missing one bar?!

Michael Evans just posted a new paper on arXiv yesterday about Birnbaum’s proof of his likelihood principle theorem. There has recently been a lot of activity around this theorem (some of which reported on the ‘Og!) and the flurry of proofs, disproofs, arguments, counterarguments, and counter-counterarguments, mostly by major figures in the field, is rather overwhelming! This paper  is however highly readable as it sets everything in terms of set theory and relations. While I am not completely convinced that the conclusion holds, the steps in the paper seem correct. The starting point is that the likelihood relation, L, the invariance relation, G, and the sufficiency relation, S, all are equivalence relations (on the set of inference bases/parametric families). The conditionality relation,C, however fails to be transitive and hence an equivalence relation. Furthermore, the smallest equivalence relation containing the conditionality relation is the likelihood relation. Then Evans proves that the conjunction of the sufficiency and the conditionality relations is strictly included in the likelihood relation, which is the smallest equivalence relation containing the union. Furthermore, the fact that the smallest equivalence relation containing the conditionality relation is the likelihood relation means that sufficiency is irrelevant (in this sense, and in this sense only!).

This is a highly interesting and well-written document. I just do not know what to think of it in correspondence with my understanding of the likelihood principle. That

rather than

makes a difference from a mathematical point of view, however I cannot relate it to the statistical interpretation. Like, why would we have to insist upon equivalence? why does invariance appear in some lemmas? why is a maximal ancillary statistics relevant at this stage when it does not appear in the original proof of Birbaum (1962)? why is there no mention made of weak versus strong conditionality principle?