Last week, I received this new book of Deborah Mayo, which I was looking forward reading and annotating!, but thrice alas, the book had been sabotaged: except for the preface and acknowledgements, the entire book is printed upside down [a minor issue since the entire book is concerned] and with some part of the text cut on each side [a few letters each time but enough to make reading a chore!]. I am thus waiting for a tested copy of the book to start reading it in earnest!
“This is, in this revised version, an outstanding paper that covers the Jeffreys-Lindley paradox (JLP) in exceptional depth and that unravels the philosophical differences between different schools of inference with the help of the JLP. From the analysis of this paradox, the author convincingly elaborates the principles of Bayesian and severity-based inferences, and engages in a thorough review of the latter’s account of the JLP in Spanos (2013).” Anonymous
I have now received a second round of reviews of my paper, “On the Jeffreys-Lindleys paradox” (submitted to Philosophy of Science) and the reports are quite positive (or even extremely positive as in the above quote!). The requests for changes are directed to clarify points, improve the background coverage, and simplify my heavy style (e.g., cutting Proustian sentences). These requests were easily addressed (hopefully to the satisfaction of the reviewers) and, thanks to the week in Warwick, I have already sent the paper back to the journal, with high hopes for acceptance. The new version has also been arXived. I must add that some parts of the reviews sounded much better than my original prose and I was almost tempted to include them in the final version. Take for instance
“As a result, the reader obtains not only a better insight into what is at stake in the JLP, going beyond the results of Spanos (2013) and Sprenger (2013), but also a much better understanding of the epistemic function and mechanics of statistical tests. This is a major achievement given the philosophical controversies that have haunted the topic for decades. Recent insights from Bayesian statistics are integrated into the article and make sure that it is mathematically up to date, but the technical and foundational aspects of the paper are well-balanced.” Anonymous
As posted on her blog, Deborah Mayo is giving a lecture at JSM 2013 in Montréal about why Birnbaum’s derivation of the Strong Likelihood Principle (SLP) is wrong. Or, more accurately, why “WCP entails SLP”. It would have been a great opportunity to hear Deborah presenting her case and I am sorry I am missing this opportunity. (Although not sorry to be in the beautiful Dolomites at that time.) Here are the slides:
Deborah’s argument is the same as previously: there is no reason for the inference in the mixed (or Birnbaumized) experiment to be equal to the inference in the conditional experiment. As previously, I do not get it: the weak conditionality principle (WCP) implies that inference from the mixture output, once we know which component is used (hence rejecting the “and we don’t know which” on slide 8), should only be dependent on that component. I also fail to understand why either WCP or the Birnbaum experiment refers to a mixture (sl.13) in that the index of the experiment is assumed to be known, contrary to mixtures. Thus (still referring at slide 13), the presentation of Birnbaum’s experiment is erroneous. It is indeed impossible to force the outcome of y* if tail and of x* if head but it is possible to choose the experiment index at random, 1 versus 2, and then, if y* is observed, to report (E1,x*) as a sufficient statistic. (Incidentally, there is a typo on slide 15, it should be “likewise for x*”.)
Here is a comment on our rejoinder “the anti-Bayesian moment and its passing” with Andrew Gelman from Deborah Mayo, comment that could not make it through as a comment:
You assume that I am interested in long-term average properties of procedures, even though I have so often argued that they are at most necessary (as consequences of good procedures), but scarcely sufficient for a severity assessment. The error statistical account I have developed is a statistical philosop
hy. It is not one to be found in Neyman and Pearson, jointly or separately, except in occasional glimpses here and there (unfortunately). It is certainly not about well-defined accept-reject rules. If N-P had only been clearer, and Fisher better behaved, we would not have had decades of wrangling. However, I have argued, the error statistical philosophy explicates, and directs the interpretation of, frequentist sampling theory methods in scientific, as opposed to behavioural, contexts. It is not a complete philosophy…but I think Gelmanian Bayesians could find in it a source of “standard setting”.
You say “the prior is both a probabilistic object, standard from this perspective, and a subjective construct, translating qualitative personal assessments into a probability distribution. The extension of this dual nature to the so-called “conventional” priors (a very good semantic finding!) is to set a reference … against which to test the impact of one’s prior choices and the variability of the resulting inference. …they simply set a standard against which to gauge our answers.”
I think there are standards for even an approximate meaning of “standard-setting” in science, and I still do not see how an object whose meaning and rationale may fluctuate wildly, even in a given example, can serve as a standard or reference. For what?
Perhaps the idea is that one can gauge how different priors change the posteriors, because, after all, the likelihood is well-defined. That is why the prior and not the likelihood is the camel. But it isn’t obvious why I should want the camel. (camel/gnat references in the paper and response).
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!
First, there is a tendency to turn everything into a paradox: obviously, when writing a book about paradoxes, everything looks like a paradox! This means bringing into the picture every paradox known to man and then some, i.e., things that are either un-paradoxical (e.g., Gödel’s incompleteness result) or uninteresting in a scientific book (e.g., the birthday paradox, which may be surprising but is far from a paradox!). Fermat’s theorem is also quoted as a paradox, even though there is nothing in the text indicating in which sense it is a paradox. (Or is it because it is simple to express, hard to prove?!) Similarly, Brownian motion is considered a paradox, as “reconcil[ing] the paradox between two of the greatest theories of physics (…): thermodynamics and the kinetic theory of gases” (p.51) For instance, the author considers the MLE being biased to be a paradox (p.117), while omitting the much more substantial “paradox” of the non-existence of unbiased estimators of most parameters—which simply means unbiasedness is irrelevant. Or the other even more puzzling “paradox” that the secondary MLE derived from the likelihood associated with the distribution of a primary MLE may differ from the primary. (My favourite!)
“When the null hypothesis is rejected, the p-value is the probability of the type I error.” Paradoxes in Scientific Inference (p.105)
“The p-value is the conditional probability given H0.” Paradoxes in Scientific Inference (p.106)
Second, the depth of the statistical analysis in the book is often found missing. For instance, Simpson’s paradox is not analysed from a statistical perspective, only reported as a fact. Sticking to statistics, take for instance the discussion of Lindley’s paradox. The author seems to think that the problem is with the different conclusions produced by the frequentist, likelihood, and Bayesian analyses (p.122). This is completely wrong: Lindley’s (or Lindley-Jeffreys‘s) paradox is about the lack of significance of Bayes factors based on improper priors. Similarly, when the likelihood ratio test is introduced, the reference threshold is given as equal to 1 and no mention is later made of compensating for different degrees of freedom/against over-fitting. The discussion about p-values is equally garbled, witness the above quote which (a) conditions upon the rejection and (b) ignores the dependence of the p-value on a realized random variable. Continue reading →
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