*(Sneaking this up on “Rejected Posts” when no one’s looking; I took it off my regular blog in July after ….well, e-mail me if you want to know.) *

Four different people now have sent me a letter circulating on an ISBA e-mail list (by statistician Thomas Leonard) presumably because it mentions the (strong) likelihood principle (SLP). Even in exile, those ISBA e-mails reach me, maybe through some Elba-NSA retrieval or simply past connections. I had already written a note to Professor Leonard* about my new paper on the controversial Birnbaum argument. I’m not sure what to make of the letter (I know nothing about Leonard): I surmise it pertains to a recent interview of Dennis Lindley (of which I watched just the beginning). Anyway, the letter and follow-ups may be found at their website: http://bayesian.org/forums/news/5374.

Dear fellow Bayesians,

Peter Wakker is to be complimented on his deep understanding of the De Finetti and Lindley-Savage Axiom systems. Nevertheless

(1) The Likelihood Principle doesn’t need to be justified by any axiom systems at all. As so elegantly proved by Alan Birnbaum (JASA,1962) , it is an immediate consequence of the Sufficiency Principle, when applied to a mixed experiment, and the Conditionality Principle. The frequency arguments used to prove the Neyman-Fisher factorization theorem substantiiate this wonderful result

(2) The strong additivity assumptions in the appropriately extended De Finetti axiom system are, I think, virtually tautologous wih finite additivity of the prior measure..So why not just assume the latter, and forget the axioms altogether? The axioms are just window dressing, a sprinkling of holy water from Avignon, Rome or wherever..

(3) The Sure Thing Principle is an extremely strong assumption, since it helps to imply the Expected Utility Hypothesis, which has been long since refuted by the economists. See for example Maurice Allais’ famous 1953 paradox and the other paradoxes described in Ch.4 of my book Bayesian Methods (with John Hsu, C.U.P.,1999) where one of many reasonable extensions to the Expected Utility hypthesis is proposed..

When Dennis brought me up to be a Bayesian Boy, he emphasised the following normative philosophies::

If you want to be coherent you have to be a (proper) Bayesian

If you’re not a Bayesian, then you’re incoherent. and a sure loser to boot

Therefore all frequentists are criiminals

(After 1973) So are Bayesiabs who use improper priors

Sorry, Dennis, but I still don’t believe a word pf it

(Note that the counterexamples to improper priors described by Stone, Dawid and Zidek, 1973, relate to quite contrived, anomalous situations,. While some sampling models can only be analysed using proper priors, a judicious choice of improper prior distribution will produce a sensible posterior when analysing most standard parametrised models)

Yours sincerely

Thomas Leonard

Re: Interview with Dennis Lindley

Without wishing to generate any spam, could I possibly add that Michael Evans (University of Toronto) has advised me that Birnbaum’s 1962 justification of the LP is mathematical unsound, It should be more correctly stated as

Theorem: If we accept SP and accept CP, and we accept all the equivalences generated jointly by these principles, then we must accept LP

Michael also proves:

Theorem: If we accept CP and we accept all equivalences generated by CP then we must accept LP

Therefore all the counterexamples to LP published by Deborah Mayo (Virginia Tech) are presumably correct. Moreover the extra conditions may be very difficult to satisfy in practice. History has been made!

Gee whiz, Dennis! Where does that put the mathematical foundations of Bayesian statistics now? Both De Finetti and Birnbaum have misled us with their mathematically unsound proofs. I think that either you or Adrian should break cover and respond to this. And how about the highly misleading empirical claims in your 1972 paper on M-Group regression which I’ve long since refuted (e.g. Sun, Hsu, Guttman, and Leonard (1996), and the inaugural ISBA meeting in San Francisco in 1993)? I call upon you and Adrian to finally formally retract them in JRSSB..

And now back to my poetry—-

With best wishes to Bayesians and frequentists everywhere,

Thomas Leonard

Writer, Poet, and Statistician

Edinburgh, Scotland