uniformly most powerful Bayesian tests???

“The difficulty in constructing a Bayesian hypothesis test arises from the requirement to specify an alternative hypothesis.”

Vale Johnson published (and arXived) a paper in the Annals of Statistics on uniformly most powerful Bayesian tests. This is in line with earlier writings of Vale on the topic and good quality mathematical statistics, but I cannot really buy the arguments contained in the paper as being compatible with (my view of) Bayesian tests. A “uniformly most powerful Bayesian test” (acronymed as UMBT)  is defined as

“UMPBTs provide a new form of default, nonsubjective Bayesian tests in which the alternative hypothesis is determined so as to maximize the probability that a Bayes factor exceeds a specified threshold”

which means selecting the prior under the alternative so that the frequentist probability of the Bayes factor exceeding the threshold is maximal for all values of the parameter. This does not sound very Bayesian to me indeed, due to this averaging over all possible values of the observations x and comparing the probabilities for all values of the parameter θ rather than integrating against a prior or posterior and selecting the prior under the alternative with the sole purpose of favouring the alternative, meaning its further use when the null is rejected is not considered at all and catering to non-Bayesian theories, i.e. trying to sell Bayesian tools as supplementing p-values and arguing the method is objective because the solution satisfies a frequentist coverage (at best, this maximisation of the rejection probability reminds me of minimaxity, except there is no clear and generic notion of minimaxity in hypothesis testing).

Olli à/in/im Paris

Warning: Here is an old post from last October I can at last post since Olli just arXived the paper on which this talk was based (more to come, before or after Olli’s talk in Roma!).

Oliver Ratman came to give a seminar today at our Big’MC seminar series. It was an extension of the talk I attended last month in Bristol:

10:45 Oliver Ratmann (Duke University and Imperial College) – “Approximate Bayesian Computation based on summaries with frequency properties”

Approximate Bayesian Computation (ABC) has quickly become a valuable tool in many applied fields, but the statistical properties obtained by choosing a particular summary, distance function and error threshold are poorly understood. In an effort to better understand the effect of these ABC tuning parameters, we consider summaries that are associated with empirical distribution functions. These frequency properties of summaries suggest what kind of distance function are appropriate, and the validity of the choice of summaries can be assessed on the fly during Monte Carlo simulations. Among valid choices, uniformly most powerful distances can be shown to optimize the ABC acceptance probability. Considering the binding function between the ABC model and the frequency model of the summaries, we can characterize the asymptotic consistency of the ABC maximum-likelhood estimate in general situations. We provide examples from phylogenetics and dynamical systems to demonstrate that empirical distribution functions of summaries can often be obtained without expensive re-simulations, so that the above theoretical results are applicable in a broad set of applications. In part, this work will be illustrated on fitting phylodynamic models that capture the evolution and ecology of interpandemic influenza A (H3N2) to incidence time series and the phylogeny of H3N2’s immunodominant haemagglutinin gene.

I however benefited enormously from hearing the talk again and also from discussing the fundamentals of his approach before and after the talk (in the nearest Aussie pub!). Olli’s approach is (once again!) rather iconoclastic in that he presents ABC as a testing procedure, using frequentist tests and concepts to build an optimal acceptance condition. Since he manipulates several error terms simultaneously (as before), he needs to address the issue of multiple testing but, thanks to a switch between acceptance and rejection, null and alternative, the individual α-level tests get turned into a global α-level test.

reading classics (#11)

Today was my last Reading Seminar class and the concluding paper chosen by the student was Tukey’s “The future of data analysis“, a 1962 Annals of Math. Stat. paper. Unfortunately, reading this paper required much more maturity and background than the student could afford, which is the reason why this last presentation is not posted on this page… Given the global and a-theoretical perspective of the paper, it was quite difficult to interpret without further delving into Tukey’s work and without a proper knowledge of what was Data Analysis in the 1960’s. (The love affair of French statisticians with data analysis was then at its apex, but it has very much receded since then!) Being myself unfamiliar with this paper, and judging mostly from the sentences pasted by the student in his slides, I cannot tell how much of the paper is truly visionary and how much is cheap talk: focussing on trimmed and winsorized means does not sound like offering a very wide scope for data analysis… I liked the quote “It’s easier to carry a slide rule than a desk computer, to say nothing of a large computer”! (As well as the quote from Azimov “The sound of panting“…. (Still, I am unsure I will keep the paper within the list next year!)

Overall, despite a rather disappointing lower tail of the distribution of the talks, I am very happy with the way the seminar proceeded this year and the efforts produced by the students to assimilate the papers, the necessary presentation skills including building a background in LaTeX and Beamer for most students. I thus think almost all students will pass this course and do hope those skills will be profitable for their future studies…

reading classics (#9)

In today’s classics seminar, my student Bassoum Abou presented the 1981 paper written by Charles Stein for the Annals of Statistics, Estimating the mean of a normal distribution, recapitulating the advances he made on Stein estimators, minimaxity and his unbiased estimator of risk. Unfortunately; this student missed a lot about paper and did not introduce the necessary background…So I am unsure at how much the class got from this great paper… Here are his slides (watch out for typos!)

 Historically, this paper is important as this is one of the very few papers published by Charles Stein in a major statistics journal, the other publications being made in conference proceedings. It contains the derivation of the unbiased estimator of the loss, along with comparisons with posterior expected loss.

reading classics (#8)

In today’s classics seminar, my student Dong Wei presented the historical paper by Neyman and Pearson on efficient  tests: “On the problem of the most efficient tests of statistical hypotheses”, published in the Philosophical Transactions of the Royal Society, Series A. She had a very hard time with the paper… It is not an easy paper, to be sure, and it gets into convoluted and murky waters when it comes to the case of composite hypotheses testing. Once again, it would have been nice to broaden the view on testing by including some of the references given in Dong Wei’s slides:

Listening to this talk, while having neglected to read the original paper for many years (!), I was reflecting on the way tests, Type I & II, and critical regions were introduced, without leaving any space for a critical (!!) analysis of the pertinence of those concepts. This is an interesting paper also because it shows the limitations of such a notion of efficiency. Apart from the simplest cases, it is indeed close to impossible to achieve this efficiency because there is no most powerful procedure (without restricting the range of those procedures). I also noticed from the slides that Neyman and Pearson did not seem to use a Lagrange multiplier to achieve the optimal critical region. (Dong Wei also inverted the comparison of the sufficient and insufficient statistics for the test on the variance, as the one based on the sufficient statistic is more powerful.) In any case, I think I will not keep the paper in my list for next year, maybe replacing it with the Karlin-Rubin (1956) UMP paper…