The recent post by Normal Deviate, and my comments on it, remind me of why/how I got back into the Bayesian-frequentist debates in 2006, as described in my first “deconstruction” (and “U-Phil”) on this blog (Dec 11, 2012):
Some time in 2006 (shortly after my ERROR06 conference), the trickle of irony and sometime flood of family feuds issuing from Bayesian forums drew me back into the Bayesian-frequentist debates.1 2 Suddenly sparks were flying, mostly kept shrouded within Bayesian walls, but nothing can long be kept secret even there. Spontaneous combustion is looming. The true-blue subjectivists were accusing the increasingly popular “objective” and “reference” Bayesians of practicing in bad faith; the new O-Bayesians (and frequentist-Bayesian unificationists) were taking pains to show they were not subjective; and some were calling the new Bayesian kids on the block “pseudo Bayesian.” Then there were the Bayesians somewhere in the middle (or perhaps out in left field) who, though they still use the Bayesian umbrella, were flatly denying the very idea that Bayesian updating fits anything they actually do in statistics.3 Obeisance to Bayesian reasoning remained, but on some kind of a priori philosophical grounds. Doesn’t the methodology used in practice really need a philosophy of its own? I say it does, and I want to provide this. Continue reading
U-Phil: I would like to open up this post, together with Gandenberger’s (Oct. 30, 2012), to reader U-Phils, from December 6- 19 (< 1000 words) for posting on this blog (please see # at bottom of post). Where Gandenberger claims, “Birnbaum’s proof is valid and his premises are intuitively compelling,” I have shown that if Birnbaum’s premises are interpreted so as to be true, the argument is invalid. If construed as formally valid, I argue, the premises contradict each other. Who is right? Gandenberger doesn’t wrestle with my critique of Birnbaum, but I invite you (and Greg!) to do so. I’m pasting a new summary of my argument below.
The main premises may be found on pp. 11-14. While these points are fairly straightforward (and do not require technical statistics), they offer an intriguing logical, statistical and linguistic puzzle. The following is an overview of my latest take on the Birnbaum argument. See also “Breaking Through the Breakthrough” posts: Dec. 6 and Dec 7, 2011.
Gandenberger also introduces something called the methodological likelihood principle. A related idea for a U-Phil is to ask: can one mount a sound, non-circular argument for that variant? And while one is at it, do his methodological variants of sufficiency and conditionality yield plausible principles?
Graduate students and others invited!
New Summary of Mayo Critique of Birnbaum’s Argument for the SLP
See also a (draft) of the full PAPER corresponding to this summary, a later and more satisfactory draft is here. Yet other links to the Strong Likelihood Principle SLP: Mayo 2010; Cox & Mayo 2011 (appendix).
Evidential Meaning and Methods of Inference
PhD student, History and Philosophy of Science
Master’s student, Statistics
University of Pittsburgh
Bayesian methods conform to the Likelihood Principle, while frequentist methods do not. Thus, proofs of the Likelihood Principle* such as Birnbaum’s (1962) appear to be threats to frequentist positions. Deborah Mayo has recently argued that Birnbaum’s proof is no threat to frequentist positions because it is invalid (Ch. 7(III) in Mayo and Spanos 2010). In my view, Birnbaum’s proof is valid and his premises are intuitively compelling. Nevertheless, I agree with Professor Mayo that the proof, properly understood, does not imply that frequentist methods should not be used.
There are actually at least two different Likelihood Principles: one, which I call the Evidential Likelihood Principle, says that the evidential meaning of an experimental outcome with respect to a set of hypotheses depends only on its likelihood function for those hypothesis (i.e., the function that maps each of those hypotheses to the probability it assigns to that outcome, defined up to a constant of proportionality); the other, which I call the Methodological Likelihood Principle, says that a statistical method should not be used if it can generate different conclusions from outcomes that have the same likelihood function, without a relevant difference in utilities or prior probabilities. Continue reading
In writing a new chapter on the Strong Likelihood Principle [i] the past few weeks, I noticed a passage in G. Casella and R. Berger (2002) that in turn recalled a puzzling remark noted in my Jan. 3, 2012 post. The post began:
A question arose from a Bayesian acquaintance:
“Although the Birnbaum result is of primary importance for sampling theorists, I’m still interested in it because many Bayesian statisticians think that model checking violates the (strong) likelihood principle (SLP), as if this principle is a fundamental axiom of Bayesian statistics”.
But this is puzzling for two reasons. First, if the LP does not preclude testing for assumptions (and he is right that it does not[ii]), then why not simply explain that rather than appeal to a disproof of something that actually never precluded model testing? To take the disproof of the LP as grounds to announce: “So there! Now even Bayesians are free to test their models” would seem only to ingrain the original fallacy.
You can read the rest of the original post here.
The remark in G. Casella and R. Berger seems to me equivocal on this point: Continue reading
17 February 1890--29 July 1962
Note: I find this to be an intriguing, if perhaps little-known, discussion, long before the conflicts reflected in the three articles (the “triad”) below, Here Fisher links his tests to the Neyman and Pearson lemma in terms of power. I invite your deconstructions/comments.
by R.A. Fisher, F.R.S.
Proceedings of the Royal Society, Series A, 144: 285-307 (1934)
To Thomas Bayes must be given the credit of broaching the problem of using the concepts of mathematical probability in discussing problems of inductive inference, in which we argue from the particular to the general; or, in statistical phraselogy, argue from the sample to the population, from which, ex hypothesi, the sample was drawn. Bayes put forward, with considerable caution, a method by which such problems could be reduced to the form of problems of probability. His method of doing this depended essentially on postulating a priori knowledge, not of the particular population of which our observations form a sample, but of an imaginary population of populations from which this population was regarded as having been drawn at random. Clearly, if we have possession of such a priori knowledge, our problem is not properly an inductive one at all, for the population under discussion is then regarded merely as a particular case of a general type, of which we already possess exact knowledge, and are therefore in a position to draw exact deductive inferences.
This is a first draft of part II of the presentation begun in the December 6 blog post. This completes the proposed presentation. I expect errors, and I will be grateful for feedback! (NOTE: I did not need to actually rip a cover of EGEK to obtain this effect!)
SEVEN:NOW FOR THE BREAKTHROUGH
You have observed y”, the .05 significant result from E”,the optional stopping rule, ending at n = 100.
Birnbaum claims he can show that you, as a frequentist error statistician, must grant that it is equivalent to having fixed n= 100 at the start (i.e., experiment E’)
The (strong) LikelihoodPrinciple (LP) is a universal conditional claim:
If two data sets y’and y” from experiments E’ and E” respectively, have likelihood functions which are functions of the same parameter(s) µ
and are proportional to each other, then y’ and y”should lead to identical inferential conclusions about µ Continue reading
brakes on the ‘breakthrough’
I am going to post a FIRST draft (for a brief presentation next week in Madrid). [I thank David Cox for the idea!] I expect errors, and I will be very grateful for feedback! This is part I; part II will be posted tomorrow. These posts may disappear once I’ve replaced them with a corrected draft. I’ll then post the draft someplace.
If you wish to share queries/corrections please post as a comment or e-mail: firstname.lastname@example.org. (ignore Greek symbols that are not showing correctly, I await fixes by Elbians.) Thanks much!
ONE: A Conversation between Sir David Cox and D. Mayo (June, 2011)
Toward the end of this exchange, the issue of the Likelihood Principle (LP) arose:
COX: It is sometimes claimed that there are logical inconsistencies in frequentist theory, in particular surrounding the strong Likelihood Principle (LP). I know you have written about this, what is your view at the moment.
MAYO: What contradiction?
COX: Well, that frequentist theory does not obey the strong LP. Continue reading
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
I am guilty of not having provided the detailed responses that are owed to the several entries in Christian Robert’s blog on Mayo and Spanos (eds.), ERROR AND INFERENCE: Recent Exchanges on Experimental Reasoning Reliability, and the Objectivity and Rationality of Science (E.R.R.O.R.S.) (2010, CUP). Today, I couldn’t resist writing a (third) follow-up comment having to do with my argument on the (strong) Likelihood Principle, even though I wasn’t planning to jump into that issue on this blog just yet. Having been lured to react, and even sketch the argument, I direct interested readers to his blog:
As you can guess, hard copies of our book play a useful role in propping open doors to breeze through marble floors in a wheelchair! Since I’m nearly free of it (thanks to the ministrations of the recovery team here at Chatfield Chateau), a picture seemed in order!
For an interesting, longish review of the book that I just encountered by Adam La Caze (Note Dame Philosophical Reviews) see: http://ndpr.nd.edu/news/24435-error-and-inference-recent-exchanges-on-experimental-reasoning-reliability-and-the-objectivity-and-rationality-of-science/