who’s afraid of the big B wolf?

Aris Spanos just published a paper entitled “Who should be afraid of the Jeffreys-Lindley paradox?” in the journal Philosophy of Science. This piece is a continuation of the debate about frequentist versus llikelihoodist versus Bayesian (should it be Bayesianist?! or Laplacist?!) testing approaches, exposed in Mayo and Spanos’ Error and Inference, and discussed in several posts of the ‘Og. I started reading the paper in conjunction with a paper I am currently writing for a special volume in  honour of Dennis Lindley, paper that I will discuss later on the ‘Og…

“…the postdata severity evaluation (…) addresses the key problem with Fisherian p-values in the sense that the severity evaluation provides the “magnitude” of the warranted discrepancy from the null by taking into account the generic capacity of the test (that includes n) in question as it relates to the observed data”(p.88)

First, the antagonistic style of the paper is reminding me of Spanos’ previous works in that it relies on repeated value judgements (such as “Bayesian charge”, “blatant misinterpretation”, “Bayesian allegations that have undermined the credibility of frequentist statistics”, “both approaches are far from immune to fallacious interpretations”, “only crude rules of thumbs”, &tc.) and rhetorical sleights of hand. (See, e.g., “In contrast, the severity account ensures learning from data by employing trustworthy evidence (…), the reliability of evidence being calibrated in terms of the relevant error probabilities” [my stress].) Connectedly, Spanos often resorts to an unusual [at least for statisticians] vocabulary that amounts to newspeak. Here are some illustrations: “summoning the generic capacity of the test”, ‘substantively significant”, “custom tailoring the generic capacity of the test”, “the fallacy of acceptance”, “the relevance of the generic capacity of the particular test”, yes the term “generic capacity” is occurring there with a truly high frequency.

reading classics (#7)

Last Monday, my student Li Chenlu presented the foundational 1962 JASA paper by Allan Birnbaum, On the Foundations of Statistical Inference. The very paper that derives the Likelihood Principle from the cumulated Conditional and Sufficiency principles and that had been discussed [maybe ad nauseam] on this ‘Og!!! Alas, thrice alas!, I was still stuck in the plane flying back from Atlanta as she was presenting her understanding of the paper, as the flight had been delayed four hours thanks to (or rather woe to!) the weather conditions in Paris the day before (chain reaction…):

I am sorry I could not attend this lecture and this for many reasons: first and  foremost, I wanted to attend every talk from my students both out of respect for them and to draw a comparison between their performances. My PhD student Sofia ran the seminar that day in my stead, for which I am quite grateful, but I do do wish I had been there… Second, this a.s. has been the most philosophical paper in the series.and I would have appreciated giving the proper light on the reasons for and the consequences of this paper as Li Chenlu stuck very much on the paper itself. (She provided additional references in the conclusion but they did not seem to impact the slides.)  Discussing for instance Berger’s and Wolpert’s (1988) new lights on the topic, as well as Deborah Mayo‘s (2010) attacks, and even Chang‘s (2012) misunderstandings, would have clearly helped the students.

the likelihood principle (sequel)

As mentioned in my review of Paradoxes in Scientific Inference I was a bit confused by this presentation of the likelihood principle and this led me to ponder for a week or so whether or not there was an issue with Birnbaum’s proof (or, much more likely, with my vision of it!). After reading again Birnbaum’s proof, while sitting down in a quiet room at ICERM for a little while, I do not see any reason to doubt it. (Keep reading at your own risk!)

My confusion was caused by mixing sufficiency in the sense of Birnbaum’s mixed experiment with sufficiency in the sense of our ABC model choice PNAS paper, namely that sufficient statistics are not always sufficient to select the right model. The sufficient statistics in the proof reduces the (2,x₂) observation from Model 2 to (1,x₁) from Model 1 when there is an observation x₁ that produces a likelihood proportional to the likelihood for x₂ and the statistic is indeed sufficient: the distribution of (2,x₂) given (1,x₁) does not depend on the parameter θ. Of course, the statistic is not sufficient (most of the time) for deciding between Model 1 and Model 2, but this model choice issue is foreign to Birnbaum’s construction.

That the likelihood principle does not hold…

Coming to Section III in Chapter Seven of Error and Inference, written by Deborah Mayo, I discovered that she considers that the likelihood principle does not hold (at least as a logical consequence of the combination of the sufficiency and of the conditionality principles), thus that  Allan Birnbaum was wrong…. As well as the dozens of people working on the likelihood principle after him! Including Jim Berger and Robert Wolpert [whose book sells for $214 on amazon!, I hope the authors get a hefty chunk of that ripper!!! Esp. when it is available for free on project Euclid...] I had not heard of  (nor seen) this argument previously, even though it has apparently created enough of a bit of a stir around the likelihood principle page on Wikipedia. It does not seem the result is published anywhere but in the book, and I doubt it would get past a review process in a statistics journal. [Judging from a serious conversation in Zürich this morning, I may however be wrong!]

The core of Birnbaum’s proof is relatively simple: given two experiments E¹ and E² about the same parameter θ with different sampling distributions f¹ and f², such that there exists a pair of outcomes (y¹,y²) from those experiments with proportional likelihoods, i.e. as a function of θ

one considers the mixture experiment E⁰ where E¹ and E² are each chosen with probability ½. Then it is possible to build a sufficient statistic T that is equal to the data (j,x), except when j=2 and x=y², in which case T(j,x)=(1,y¹). This statistic is sufficient since the distribution of (j,x) given T(j,x) is either a Dirac mass or a distribution on {(1,y¹),(2,y²)} that only depends on c. Thus it does not depend on the parameter θ. According to the weak conditionality principle, statistical evidence, meaning the whole range of inferences possible on θ and being denoted by Ev(E,z), should satisfy

Because the sufficiency principle states that

this leads to the likelihood principle

(See, e.g., The Bayesian Choice, pp. 18-29.) Now, Mayo argues this is wrong because

“The inference from the outcome (E^j,y^j) computed using the sampling distribution of [the mixed experiment] E⁰ is appropriately identified with an inference from outcome y^j based on the sampling distribution of E^j, which is clearly false.” (p.310)

This sounds to me like a direct rejection of the conditionality principle, so I do not understand the point. (A formal rendering in Section 5 using the logic formalism of A’s and Not-A’s reinforces my feeling that the conditionality principle is the one criticised and misunderstood.) If Mayo’s frequentist stance leads her to take the sampling distribution into account at all times, this is fine within her framework. But I do not see how this argument contributes to invalidate Birnbaum’s proof. The following and last sentence of the argument may bring some light on the reason why Mayo considers it does:

“The sampling distribution to arrive at Ev(E⁰,(j,y^j)) would be the convex combination averaged over the two ways that y^j could have occurred. This differs from the  sampling distributions of both Ev(E¹,y¹) and Ev(E²,y²).” (p.310)

Indeed, and rather obviously, the sampling distribution of the evidence Ev(E^*,z^*) will differ depending on the experiment. But this is not what is stated by the likelihood principle, which is that the inference itself should be the same for y¹ and y². Not the distribution of this inference. This confusion between inference and its assessment is reproduced in the “Explicit Counterexample” section, where p-values are computed and found to differ for various conditional versions of a mixed experiment. Again, not a reason for invalidating the likelihood principle. So, in the end, I remain fully unconvinced by this demonstration that Birnbaum was wrong. (If in a bystander’s agreement with the fact that frequentist inference can be built conditional on ancillary statistics.)

IMS Lecture Notes on line

When writing the review of Sober’s Evidence and Evolution: The Logic Behind the Science, I incidentaly found that all IMS Lecture Notes books are available on-line, free, through Project euclid. This is fantastic! This is for instance the case for Berger and Wolpert’s classic, The Likelihood Principle.