Here are the slides of my tutorial at O’ Bayes 2013 today, a pot-pourri of various, recent and less recent, criticisms (with, albeit less than usual, a certain proportion of recycled slides):
Archive for Jeffreys-Lindley paradox
“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 mentioned here a few days ago, I have been revising my paper on the Jeffreys-Lindley’s paradox paper for Philosophy of Science. It came as a bit of a (very pleasant) surprise that this journal was ready to consider a revised version of the paper given that I have no formal training in philosophy and that the (first version of the) paper was rather hurriedly made of a short text written for the 95th birthday of Dennis Lindley and of my blog post on Aris Spanos’ “Who should be afraid of the Jeffreys-Lindley paradox?“, recently published in Philosophy of Science. So I found both reviewers very supportive and I am grateful for their suggestions to improve both the scope and the presentation of the paper. It has been resubmitted and rearXived, and I am now waiting for the decision of the editorial team with the appropriate philosophical sense of detachment…
“In the asymptotic limit, the Bayesian cannot justify the strictly positive probability of H0 as an approximation to testing the hypothesis that the parameter value is close to θ0 — which is the hypothesis of real scientific interest”
While revising my Jeffreys-Lindley’s paradox paper for Philosophy of Science, it was suggested (to me) that I read the incoming paper by Jan Sprenger on this paradox. The paper is entitled Testing a Precise Null Hypothesis: The Case of Lindley’s Paradox and it defends the thesis that the regular Bayesian approach (hence the Bayes factor used in the Jeffreys-Lindley’s paradox) is forced to put a prior on the (point) null hypothesis when all that really matters is the vicinity of the null. (I think Andrew would agree there as he positively hates point null hypotheses. See also Rissanen’s perspective about maximal precision allowed by a give sample.) Sprenger then advocates the use of the log score for comparing the full model with the point-null sub-model, i.e. the posterior expectation of the Kullback-Leibler distance between both models:
rejoining José Bernardo and Phil Dawid on this ground.
While I agree about the notion that it is impossible to distinguish a small enough departure from the null from the null (no typo!), and I also support the argument that “all models are wrong”, hence point null should eventually—meaning with enough data—rejected, I find the Bayesian solution through the Bayes factor rather appealing because it uses the prior distribution to weight the alternative values of θ in order to oppose their averaged likelihood to the likelihood in θ0. (Note I did not mention Occam!) Further, while the notion of opposing a point null to the rest of the Universe may sound silly, what truly matters is the decisional setting, namely that we want to select a simpler model and use it for later purposes. It is therefore this issue that should be tested, rather than whether or not θ is exactly equal to θ0. I incidentally find it amusing that Sprenger picks the ESP experiment as his illustration in that this is a (the?) clearcut case where the point null: “there is no such thing as ESP” makes sense. Now, it can be argued that what the statistical experiment is assessing is the ESP experiment, for which many objective causes (beyond ESP!) may induce a departure from the null (and from the binomial model). But then this prevents any rational analysis of the test (as is indeed the case!).
The paper thus objects to the use of Bayes factors (and of p-values) to instead propose to compare scores in the Bernardo-Dawid spirit. As discussed earlier, it has several appealing features, from recovering the Kullback-Leibler divergence between models as a measure of fit to allowing for the incorporation of improper priors (a point Andrew may disagree with), to avoiding the double use of the data. It is however incomplete in that it creates a discrepancy or a disbalance between both models, thus making the comparison of more than two models difficult to fathom, and it does not readily incorporate the notion of nuisance parameters in the embedded model, seemingly forcing the inclusion of pseudo-priors as in the Bayesian analysis of Aitkin’s integrated likelihood.
After attending the first day of the RSS annual conference in Newcastle, I took the train to Coventry to join the Department of Statistics at the University of Warwick (this may sound confusing, but the University of Warwick is located in Coventry, not in Warwick, 8 miles south, and not to be confused with Coventry University, which is a former polytechnic; it is located in Warwickshire, though, which is why it took this name) where I now have a part-time professor position. I will thus be at the department a week at a time, every other five weeks or so, during the teaching terms, and I obviously look forward the huge opportunities to interact with faculty and students therein. The “first week at work” was quite smooth, not entirely surprising given my numerous previous visits to Warwick, with hardly any bureaucratic step in the instalment. This gave me the opportunity to start the revision of the Jeffreys-Lindley’s paradox paper for Philosophy of Science and to reconnoitre longer running routes…