## Birnbaum’s proof missing one bar?!

Posted in Statistics with tags , , , , on March 4, 2013 by xi'an

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

$\overline{S \cup C} = L$

rather than

$S \cup C =L$

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?

Posted in Statistics, Travel, University life with tags , , , , , , , , , , , , on February 20, 2013 by xi'an

True randomness was the topic of the `Random numbers; fifty years later’ talk in DESY by Frederick James from CERN. I had discussed a while ago a puzzling book related to this topic. This talk went along a rather different route, focussing on random generators. James put this claim that there are computer based physical generators that are truly random. (He had this assertion that statisticians do not understand randomness because they do not know quantum mechanics.) He distinguished those from pseudo-random generators: “nobody understood why they were (almost) random”, “IBM did not know how to generate random numbers”… But then spent the whole talk discussing those pseudo-random generators. Among other pieces of trivia, James mentioned that George Marsaglia was the one exhibiting the hyperplane features of congruential generators. That Knuth achieved no successful definition of what randomness is in his otherwise wonderful books! James thus introduced Kolmogorov’s mixing (not Kolmogorov’s complexity, mind you!) as advocated by Soviet physicists to underlie randomness. Not producing anything useful for RNGs in the 60′s. He then moved to the famous paper by Ferrenberg, Landau and Wong (1992) that I remember reading more or less at the time. In connection with the phase transition critical slowing down phenomena in Ising model simulations. And connecting with the Wang-Landau algorithm of flipping many sites at once (which exhibited long-term dependences in the generators). Most interestingly, a central character in this story is Martin Lüscher, based in DESY, who expressed the standard generator of the time RCARRY into one studied by those Soviet mathematicians,

X’=AX

showing that it enjoyed Kolmogorov mixing, but with a very poor Lyapunov coefficient. I partly lost track there as RCARRY was not perfect. And on how this Kolmogorov mixing would relate to long-term dependencies. One explanation by James was that this property is only asymptotic. (I would even say statistical!) Also interestingly, the 1994 paper by Lüscher produces the number of steps necessary to attain complete mixing, namely 15 steps, which thus works as a cutoff point. (I wonder why a 15-step RCARRY is slower, since A15 can be computed at once… It may be due to the fact that A is sparse while A15 is not.) James mentioned that Marsaglia’s Die Hard battery of tests is now obsolete and superseded by Pierre Lecuyer’s TestU01.

In conclusion, I did very much like this presentation from an insider, but still do not feel it makes a contribution to the debate on randomness, as it stayed put on pseudorandom generators. To keep the connection with von Neumann, they all produce wrong answers from a randomness point of view, if not from a statistical one. (A final quote from the talk: “Among statisticians and number theorists who are supposed to be specialists, they do not know about Kolmogorov mixing.”) [Discussing with Fred James at the reception after the talk was obviously extremely pleasant, as he happened to know a lot of my Bayesian acquaintances!]

Posted in Books, Statistics, University life with tags , , , , , , , , , on December 26, 2012 by xi'an

(I received the following set of comments from Mark Chang after publishing a review of his book on the ‘Og. Here they are, verbatim, except for a few editing and spelling changes. It’s a huge post as Chang reproduces all of my comments as well.)

Professor Christian Robert reviewed my book: “Paradoxes in Scientific Inference”. I found that the majority of his criticisms had no foundation and were based on his truncated way of reading. I gave point-by-point responses below. For clarity, I kept his original comments.

Robert’s Comments: This CRC Press book was sent to me for review in CHANCE: Paradoxes in Scientific Inference is written by Mark Chang, vice-president of AMAG Pharmaceuticals. The topic of scientific paradoxes is one of my primary interests and I have learned a lot by looking at Lindley-Jeffreys and Savage-Dickey paradoxes. However, I did not find a renewed sense of excitement when reading the book. The very first (and maybe the best!) paradox with Paradoxes in Scientific Inference is that it is a book from the future! Indeed, its copyright year is 2013 (!), although I got it a few months ago. (Not mentioning here the cover mimicking Escher’s “paradoxical” pictures with dices. A sculpture due to Shigeo Fukuda and apparently not quoted in the book. As I do not want to get into another dice cover polemic, I will abstain from further comments!)

Thank you, Robert for reading and commenting on part of my book. I had the same question on the copyright year being 2013 when it was actually published in previous year. I believe the same thing had happened to my other books too. The incorrect year causes confusion for future citations. The cover was designed by the publisher. They gave me few options and I picked the one with dices. I was told that the publisher has the copyright for the art work. I am not aware of the original artist. Read more »

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , on November 23, 2012 by xi'an

This CRC Press book was sent to me for review in CHANCE: Paradoxes in Scientific Inference is written by Mark Chang, vice-president of AMAG Pharmaceuticals. The topic of scientific paradoxes is one of my primary interests and I have learned a lot by looking at Lindley-Jeffreys and Savage-Dickey paradoxes. However, I did not find a renewed sense of excitement when reading the book. The very first (and maybe the best!) paradox with Paradoxes in Scientific Inference is that it is a book from the future! Indeed, its copyright year is 2013 (!), although I got it a few months ago. (Not mentioning here the cover mimicking Escher’s “paradoxical” pictures with dices. A sculpture due to Shigeo Fukuda and apparently not quoted in the book. As I do not want to get into another dice cover polemic, I will abstain from further comments!)

Now, getting into a deeper level of criticism (!), I find the book very uneven and overall quite disappointing. (Even missing in its statistical foundations.) Esp. given my initial level of excitement about the topic!