Posts Tagged With: Birnbaum

Midnight With Birnbaum (Happy New Year)

 Just as in the past 2 years since I’ve been blogging, I revisit that spot in the road, get into a strange-looking taxi, and head to “Midnight With Birnbaum”. There are a couple of brief (12/31/13) updates at the end.  

You know how in that (not-so) recent movie, “Midnight in Paris,” the main character (I forget who plays it, I saw it on a plane) is a writer finishing a novel, and he steps into a cab that mysteriously picks him up at midnight and transports him back in time where he gets to run his work by such famous authors as Hemingway and Virginia Wolf?  He is impressed when his work earns their approval and he comes back each night in the same mysterious cab…Well, imagine an error statistical philosopher is picked up in a mysterious taxi at midnight (New Year’s Eve 2011 2012, 2013) and is taken back fifty years and, lo and behold, finds herself in the company of Allan Birnbaum.[i]

ERROR STATISTICIAN: It’s wonderful to meet you Professor Birnbaum; I’ve always been extremely impressed with the important impact your work has had on philosophical foundations of statistics.  I happen to be writing on your famous argument about the likelihood principle (LP).  (whispers: I can’t believe this!)

BIRNBAUM: Ultimately you know I rejected the LP as failing to control the error probabilities needed for my Confidence concept.

ERROR STATISTICIAN: Yes, but I actually don’t think your argument shows that the LP follows from such frequentist concepts as sufficiency S and the weak conditionality principle WLP.[ii]  Sorry,…I know it’s famous…

BIRNBAUM:  Well, I shall happily invite you to take any case that violates the LP and allow me to demonstrate that the frequentist is led to inconsistency, provided she also wishes to adhere to the WLP and sufficiency (although less than S is needed).

ERROR STATISTICIAN: Well I happen to be a frequentist (error statistical) philosopher; I have recently (2006) found a hole in your proof,..er…well I hope we can discuss it.

BIRNBAUM: Well, well, well: I’ll bet you a bottle of Elba Grease champagne that I can demonstrate it!

ERROR STATISTICAL PHILOSOPHER:  It is a great drink, I must admit that: I love lemons.

BIRNBAUM: OK.  (A waiter brings a bottle, they each pour a glass and resume talking).  Whoever wins this little argument pays for this whole bottle of vintage Ebar or Elbow or whatever it is Grease.

ERROR STATISTICAL PHILOSOPHER:  I really don’t mind paying for the bottle.

BIRNBAUM: Good, you will have to. Take any LP violation. Let  x’ be 2-standard deviation difference from the null (asserting m = 0) in testing a normal mean from the fixed sample size experiment E’, say n = 100; and let x” be a 2-standard deviation difference from an optional stopping experiment E”, which happens to stop at 100.  Do you agree that:

(0) For a frequentist, outcome x’ from E’ (fixed sample size) is NOT evidentially equivalent to x” from E” (optional stopping that stops at n)

ERROR STATISTICAL PHILOSOPHER: Yes, that’s a clear case where we reject the strong LP, and it makes perfect sense to distinguish their corresponding p-values (which we can write as p’ and p”, respectively).  The searching in the optional stopping experiment makes the p-value quite a bit higher than with the fixed sample size.  For n = 100, data x’ yields p’= ~.05; while p”  is ~.3.  Clearly, p’ is not equal to p”, I don’t see how you can make them equal. Continue reading

Categories: Birnbaum Brakes, strong likelihood principle | Tags: , , , | 2 Comments

A.Birnbaum: Statistical Methods in Scientific Inference

Birnbaum: born May 27, 1923

Today is (statistician) Allan Birnbaum’s birthday. He lived to be only 53 [i]. From the perspective of philosophy of statistics and philosophy of science, Birnbaum is best known for his work on likelihood, the Likelihood Principle [ii], and for his attempts to blend concepts of likelihood with error probability ideas to obtain what he called “concepts of statistical evidence”. Failing to find adequate concepts of statistical evidence, Birnbaum called for joining the work of “interested statisticians, scientific workers and philosophers and historians of science”–an idea I would heartily endorse!  While known for attempts to argue that the (strong) Likelihood Principle followed from sufficiency and conditionality principles, a few years after publishing this result, he seems to have turned away from it, perhaps discovering gaps in his argument.

NATURE VOL. 225 MARCH 14, 1970 (1033)

LETTERS TO THE EDITOR

Statistical methods in Scientific Inference

 It is regrettable that Edwards’s interesting article[1], supporting the likelihood and prior likelihood concepts, did not point out the specific criticisms of likelihood (and Bayesian) concepts that seem to dissuade most theoretical and applied statisticians from adopting them. As one whom Edwards particularly credits with having ‘analysed in depth…some attractive properties” of the likelihood concept, I must point out that I am not now among the ‘modern exponents” of the likelihood concept. Further, after suggesting that the notion of prior likelihood was plausible as an extension or analogue of the usual likelihood concept (ref.2, p. 200)[2], I have pursued the matter through further consideration and rejection of both the likelihood concept and various proposed formalizations of prior information and opinion (including prior likelihood).  I regret not having expressed my developing views in any formal publication between 1962 and late 1969 (just after ref. 1 appeared). My present views have now, however, been published in an expository but critical article (ref. 3, see also ref. 4)[3] [4], and so my comments here will be restricted to several specific points that Edwards raised.

 If there has been ‘one rock in a shifting scene’ or general statistical thinking and practice in recent decades, it has not been the likelihood concept, as Edwards suggests, but rather the concept by which confidence limits and hypothesis tests are usually interpreted, which we may call the confidence concept of statistical evidence. This concept is not part of the Neyman-Pearson theory of tests and confidence region estimation, which denies any role to concepts of statistical evidence, as Neyman consistently insists. The confidence concept takes from the Neyman-Pearson approach techniques for systematically appraising and bounding the probabilities (under respective hypotheses) of seriously misleading interpretations of data. (The absence of a comparable property in the likelihood and Bayesian approaches is widely regarded as a decisive inadequacy.) The confidence concept also incorporates important but limited aspects of the likelihood concept: the sufficiency concept, expressed in the general refusal to use randomized tests and confidence limits when they are recommended by the Neyman-Pearson approach; and some applications of the conditionality concept. It is remarkable that this concept, an incompletely formalized synthesis of ingredients borrowed from mutually incompatible theoretical approaches, is evidently useful continuously in much critically informed statistical thinking and practice [emphasis mine].

While inferences of many sorts are evident everywhere in scientific work, the existence of precise, general and accurate schemas of scientific inference remains a problem. Mendelian examples like those of Edwards and my 1969 paper seem particularly appropriate as case-study material for clarifying issues and facilitating effective communication among interested statisticians, scientific workers and philosophers and historians of science.

Allan Birnbaum
New York University
Courant Institute of Mathematical Sciences,
251 Mercer Street,
New York, NY 10012

Birnbaum’s confidence concept, sometimes written (Conf), was his attempt to find in error statistical ideas a concept of statistical evidence–a term that he invented and popularized. In Birnbaum 1977 (24), he states it as follows:

(Conf): A concept of statistical evidence is not plausible unless it finds ‘strong evidence for J as against H with small probability (α) when H is true, and with much larger probability (1 – β) when J is true.

Birnbaum questioned whether Neyman-Pearson methods had “concepts of evidence”  simply because Neyman talked of “inductive behavior” and Wald and others cauched statistical methods in decision-theoretic terms. I have been urging that we consider instead how the tools may actually be used, and not be restricted by the statistical philosophies of founders (not to mention that so many of their statements are tied up with personality disputes, and problems of “anger management”). Recall, as well, E. Pearson’s insistence on an evidential construal of N-P methods, and the fact that Neyman, in practice, spoke of drawing inferences and reaching conclusions (e.g., Neyman’s nursery posts, links in [iii] below). Continue reading

Categories: Likelihood Principle, phil/history of stat, Statistics | Tags: | 3 Comments

Midnight With Birnbaum-reblog

 Reblogging Dec. 31, 2011:

You know how in that recent movie, “Midnight in Paris,” the main character (I forget who plays it, I saw it on a plane) is a writer finishing a novel, and he steps into a cab that mysteriously picks him up at midnight and transports him back in time where he gets to run his work by such famous authors as Hemingway and Virginia Wolf?  He is impressed when his work earns their approval and he comes back each night in the same mysterious cab…Well, imagine an error statistical philosopher is picked up in a mysterious taxi at midnight (New Year’s Eve 2011 2012) and is taken back fifty years and, lo and behold, finds herself in the company of Allan Birnbaum.[i]

ERROR STATISTICIAN: It’s wonderful to meet you Professor Birnbaum; I’ve always been extremely impressed with the important impact your work has had on philosophical foundations of statistics.  I happen to be writing on your famous argument about the likelihood principle (LP).  (whispers: I can’t believe this!)

BIRNBAUM: Ultimately you know I rejected the LP as failing to control the error probabilities needed for my Confidence concept.

ERROR STATISTICIAN: Yes, but I actually don’t think your argument shows that the LP follows from such frequentist concepts as sufficiency S and the weak conditionality principle WLP.[ii]  Sorry,…I know it’s famous… Continue reading

Categories: Birnbaum Brakes, strong likelihood principle | Tags: , , , | 2 Comments

P-values as Frequentist Measures

Working on the last two chapters of my book on philosophy of statistical inference, I’m revisiting such topics as weak conditioning, Birnbaum, likelihood principle, etc., and was reading from the Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer (1985)[i]. In a paper I had not seen (or had forgotten), Jim Berger “The Frequentist Viewpoint and Conditioning,” writes that the quoting of a P-value “may be felt to be a frequentist procedure by some, since it involves an averaging over the sample space. The reporting of P-values can be given no long-run frequency interpretation [in any of the set-ups generally considered].  A P-value actually lies closer to conditional (Bayesian) measures than to frequentist measures.” (Berger 1985, 23). These views are echoed in Berger’s more recent “Could Fisher,Jeffreys and Neyman Have Agreed on Testing?”(2003). This is at odds with what Fisher, N-P, Cox, Lehmann, etc. have held, and if true, would also seem to entail that a severity assessment had no frequentist interpretation!  The flaw lies in that all-too-common behavioristic, predesignated conception…

Among related posts:

http://errorstatistics.com/2012/04/28/3671/
http://errorstatistics.com/2012/05/10/excerpts-from-s-senns-letter-on-replication-p-values-and-evidence/

 


[i] Also because of Peter Gruenwald’s recent mention of Kiefer’s work, read long ago.

Categories: Statistics | Tags: , , , | Leave a comment

Don’t Birnbaumize that Experiment my Friend*

(A)  “It is not uncommon to see statistics texts argue that in frequentist theory one is faced with the following dilemma: either to deny the appropriateness of conditioning on the precision of the tool chosen by the toss of a coin[i], or else to embrace the strong likelihood principle which entails that frequentist sampling distributions are irrelevant to inference once the data are obtained.  This is a false dilemma … The ‘dilemma’ argument is therefore an illusion”. (Cox and Mayo 2010, p. 298)
Continue reading

Categories: Statistics | Tags: , , , | 16 Comments

Midnight With Birnbaum

You know how in that recent movie, “Midnight in Paris,” the main character (I forget who plays it, I saw it on a plane) is a writer finishing a novel, and he steps into a cab that mysteriously picks him up at midnight and transports him back in time where he gets to run his work by such famous authors as Hemingway and Virginia Wolf?  He is impressed when his work earns their approval and he comes back each night in the same mysterious cab…Well, imagine an error statistical philosopher is picked up in a mysterious taxi at midnight (new Year’s Eve 2011) and is taken back fifty years and, lo and behold, finds herself in the company of Allan Birnbaum.[i] Continue reading

Categories: Birnbaum Brakes | Tags: , , , | 2 Comments

The 3 stages of the acceptance of novel truths

There is an often-heard slogan about the stages of the acceptance of novel truths:

First people deny a thing.

Then they belittle it.

Then they say they knew it all along.

I don’t know who was first to state it in one form or another.  Here’s Schopenhauer with a slightly different variant:

“All truth passes through three stages: First, it is ridiculed; Second, it is violently opposed; and Third, it is accepted as self-evident.” – Arthur Schopenhauer

After recently presenting my paper criticizing the Birnbaum result on the likelihood principle (LP)[1] the reception of my analysis seems somewhere around stage two, in some cases, moving into stage three (see my blogposts of December 6 and 7, 2011). Continue reading

Categories: Statistics | Tags: , , | 2 Comments

ReBlogging the Likelihood Principle #2: Solitary Fishing:SLP Violations

Reblogging from a year ago. The Appendix of the “Cox/Mayo Conversation” (linked below [i]) is an attempt to quickly sketch Birnbaum’s argument for the strong likelihood principle (SLP), and its sins.  Couple of notes: Firstly, I am a philosopher (of science and statistics) not a statistician.  That means, my treatment will show all of the typical (and perhaps annoying) signs of being a trained philosopher-logician.  I’ve no doubt statisticians would want to use different language, which is welcome.  Second, this is just a blog (although perhaps my published version is still too informal for some). Continue reading

Categories: Likelihood Principle | Tags: , , | 9 Comments

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