## a paradox in decision-theoretic interval estimation (solved)

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , , , on October 4, 2012 by xi'an

In 1993, we wrote a paper [with George Casella and Gene/Juinn Hwang] on the paradoxical consequences of using the loss function

$\text{length}(C) - k \mathbb{I}_C(\theta)$

(published in Statistica Sinica, 3, 141-155) since it led to the following property: for the standard normal mean estimation problem, the regular confidence interval is dominated by the modified confidence interval equal to the empty set when is too large… This was first pointed out by Jim Berger and the most natural culprit is the artificial loss function where the first part is unbounded while the second part is bounded by k. Recently, Paul Kabaila—whom I met in both Adelaide, where he quite appropriately commented about the abnormal talk at the conference!,  and Melbourne, where we met with his students after my seminar at the University of Melbourne—published a paper (first on arXiv then in Statistics and Probability Letters) where he demonstrates that the mere modification of the above loss into

$\dfrac{\text{length}(C)}{\sigma} - k \mathbb{I}_C(\theta)$

solves the paradox:! For Jeffreys’ non-informative prior, the Bayes (optimal) estimate is the regular confidence interval. besides doing the trick, this nice resolution explains the earlier paradox as being linked to a lack of invariance in the (earlier) loss function. This is somehow satisfactory since Jeffreys’ prior also is the invariant prior in this case.

## Kant, Platon, Bayes, & Le Monde…

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , on July 2, 2012 by xi'an

In the weekend edition of Le Monde I bought when getting out of my plane back from Osaka, and ISBA 2012!, the science leaflet has a (weekly) tribune by a physicist called Marco Zito that discussed this time of the differences between frequentist and Bayesian confidence intervals. While it is nice to see this opposition debated in a general audience daily like Le Monde, I am not sure the tribune will bring enough light to help to the newcomer to reach an opinion about the difference! (The previous tribune considering Bayesian statistics was certainly more to my taste!)

Since I cannot find a link to the paper, let me sum up: the core of the tribune is to wonder what does 90% in 90% confidence interval mean? The Bayesian version sounds ridiculous since “there is a single true value of [the parameter] M and it is either in the interval or not” [my translation]. The physicist then goes into stating that the probability is in fact “subjective. It measures the degree of conviction of the scientists, given the data, for M to be in the interval. If those scientists were aware of another measure, they would use another interval” [my translation]. Darn… so many misrepresentations in so few words! First, as a Bayesian, I most often consider there is a true value for the parameter associated with a dataset but I still use a prior and a posterior that are not point masses, without being incoherent, simply because the posterior only summarizes what I know about the  parameter, but is obviously not a property of the true parameter. Second, the fact that the interval changes with the measure has nothing to do with being Bayesians. A frequentist would also change her/his interval with other measures…Third, the Bayesian “confidence” interval is but a tiny (and reductive) part of the inference one can draw from the posterior distribution.

From this delicate start, things do not improve in the tribune: the frequentist approach is objective and not contested by Marco Zito, as it sounds eminently logical. Kant is associated with Bayes and Platon with the frequentist approach, “religious wars” are mentioned about both perspectives debating endlessly about the validity of their interpretation (is this truly the case? In the few cosmology papers I modestly contributed to, referees’ reports never objected to the Bayesian approach…) The conclusion makes one wonders what is the overall point of this tribune: superficial philosophy (“the debate keeps going on and this makes sense since it deals with the very nature of research: can we know and speak of the world per se or is it forever hidden to us? (…) This is why doubt and even distrust apply about every scientific result and also in other settings.”) or criticism of statistics (“science (or art) of interpreting results from an experiment”)? (And to preamp a foreseeable question: no, I am not writing to the journal this time!)