Not sure why this is noted now although as it happens I’m about to discuss it next week in my summer Seminar in Phil Stat.

]]>First, I did not intend to suggest that Prof. Mayo’s argument was meant as a “counterexample” to the LP.

Second, certainly an example in which the conditions of a “theorem” are satisfied while the theorem’s conclusions do not hold, can be described as a “counterexample” to the validity of the theorem.

Third, without getting into an analysis of what sort of arguments would be too “trivial” for particular commentators to write about, I can say that I am aware of Jim Berger (and other researchers for whom I have high regard) commenting on an article that appeared severely flawed to me. I would venture to speculate that they did so because the article in question was being published in a reputed journal.

I wonder why someone who is not even remotely talking about the relevant issue at hand, and hasn’t bothered to check out the exchange in Stat Sci, would wish to get involved in commenting on it. And I am a mathematician, logician, and philosopher. It may be that you “never saw it as a “proof” in a strict mathematical sense” but loads of stat texts present it as a theorem. If you are right, they should revise those sections to explain, “oh and by the way, we didn’t really mean theorem and proof here”. That is why, in my rejoinder to Evans, I call for the students who have been asked to complete the “proof” to be given their past credit due. No need to respond.

]]>I understood “counter example” in exactly the same sense. Having done research in mathematical logic with some of the best logicians alive, I understand the concept well. My comment still stands. I, like jaynes, never saw it as a “proof” in a strict mathematical sense for a variety of reasons (not all of which were given by you or Jaynes). I saw it as a rabbit hole of irrelevancies brought about by the desire of the mislead-n-misinformed to justify something already far clearer than the “assumptions” from which it was “derived”.

In the big picture I’ll stick to the sum/product rule and their implications like Bayes Theorem. The sum/product rule and mathematics derived from them trump philosopher’s intuitions every time. So much so that I refuse to listen to any philosophers of science who aren’t first rate mathematicians in their own right. They can neither see the implications of the mathematics themselves or understand the implications once explained.

If you wish to say the sum/product rule are only sometimes true –which is the central tenant of all rejections of Bayes—even though the mathematics indicates no such thing, then history will laugh at your foibles the way math students today laugh at the ancients who refused to accept negative solutions to equations because negative numbers weren’t philosophically sound or meaningful. It turns out, as it always does, that it was the philosophers that needed to be quietly forgotten, not the mathematician’s negative solutions.

P.s. Every well defined frequentist method that violates the likelihood principle in it’s range of applicability implied by the sum/product rules works on some narrow range of examples, but then gives horrendously absurd answers outside that range. The sum/product rule based Bayesian methods work just as well inside or outside that range.

That’s the real takeaway from debates about the likelihood principle.

Now you can scream your denials until the sun explodes, but that won’t change a single number in a single equation of the mathematics used to back that claim up.

]]>“Mayo’s argument amounts to “I like p-values, therefore they are a counter example to Birnbaum’s proof””

I say that he has clearly not bothered to understand the use of “counterexample” that is relevant here. I certainly do not assume p-values or any sampling theory, and my use of the term “counterexample” (as I clearly explain) is very different from finding an application that violates the principle (that was a GIVEN in line one). Instead it is the logician’s usage. A counterexample to an alleged valid argument from A to B is a model M such that A and not-B are true under M. But I also go much further and show any attempt to save the argument is unsound. See also my response to Evans. The mistake in the Birnbaum argument is actually a quantifier error. If theorems in stat were stated in terms of the relevant quantifiers, it would have been spotted earlier. It’s rather disappointing to see this kind of belittling of the logical issue on this blog.

]]>My argument does not fall under “counterexamples to the LP”, but a counterexample to a ‘proof’ that claims to be valid is one that shows the premises to be true and the conclusion false. But I also go much further than this (see my rejoinder to Evans). Texts should want to straighten out their claims to theorem hood, I should think. Were the matter so trivial, I doubt those commentators would have agreed to write.

]]>Thanks! I read Jim’s e-mail carefully before posting

to make sure I didn’t misrepresent what he wrote,

but I still hesitated. Anyway, I guess I’m glad

that the post was one of the proximate causes

of a Jefferys-Berger interaction!

Speaking of which, I attended a Physics seminar talk

by Penn State’s Eric Feigelson at GWU, earlier this month.

During the talk, he listed a number of topics and problems

in astronomy and statistical techniques that could be of use.

As a Bayesian, I was pleased to be able to mention

Bayesian model selection and the Cepheid distance scale

(Jefferys/Barnes/Berger/Mueller).

I asked Jim about this and what he wrote back to me is consistent with what Sudip says.

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