## Birnbaum’s proof missing one bar?!

**M**ichael 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!).

**T**his 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?

November 11, 2013 at 10:56 pm

Evans got it published: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ejs/1382706342

P.S. It was my pleasure today to vote one answer of yours that scored you above 5K.

March 6, 2013 at 7:15 pm

I’ll read Evans paper. Note that we also stated everything set theoretically in our 2008 paper

http://link.aip.org/link/doi/10.1063/1.3039028

which is not one of Evans’s references. He probably doesn’t know about the existence of our little paper.

March 5, 2013 at 6:18 am

Without requiring equivalence for the WCP, Birnbaum’s “proof” does not go through. One may find a true equivalence, just not the one he needs. But I have argued all of this at too much length already….. I include some relevant background paper links for the interested reader.

http://errorstatistics.com/2012/11/25/likelihood-links-for-28-nov-seminar-and-current-u-phil/

http://errorstatistics.com/2012/11/23/announcement-28-november-my-seminar-at-the-lse-contemporary-philstat/

http://www.phil.vt.edu/dmayo/personal_website/ch 7 mayo birnbaum proof.pd