## Fisher, Bayes, and predictive Bayesian inference [seminar]

Posted in Statistics with tags , , , , , , , , , on April 4, 2021 by xi'an

An interesting Foundations of Probability seminar at Rutgers University this Monday, at 4:30ET, 8:30GMT, by Sandy Zabell (the password is Angelina’s birthdate):

R. A. Fisher is usually perceived to have been a staunch critic of the Bayesian approach to statistics, yet his last book (Statistical Methods and Scientific Inference, 1956) is much closer in spirit to the Bayesian approach than the frequentist theories of Neyman and Pearson.  This mismatch between perception and reality is best understood as an evolution in Fisher’s views over the course of his life.  In my talk I will discuss Fisher’s initial and harsh criticism of “inverse probability”, his subsequent advocacy of fiducial inference starting in 1930, and his admiration for Bayes expressed in his 1956 book.  Several of the examples Fisher discusses there are best understood when viewed against the backdrop of earlier controversies and antagonisms.

## hard birthday problem

Posted in Books, Kids, R, Statistics with tags , , , , , , , , , on February 4, 2021 by xi'an

From an X validated question, found that WordPress now allows for direct link to pdf documents, like the above paper by my old friend Anirban Das Gupta! The question is about estimating a number M of individuals with N distinct birth dates over a year of T days. After looking around I could not find a simpler representation of the probability for N=r other than (1) in my answer,

$\frac{T!}{(\bar N-r)!}\frac{m!}{T^m} \sum_{(r_1,\ldots,r_m);\\\sum_1^m r_i=r\ \&\\\sum_1^m ir_i=m}1\Big/\prod_{j=1}^m r_j! (j!)^{r_j}$

borrowed from a paper by Fisher et al. (Another Fisher!) Checking Feller leads to the probability (p.102)

${T \choose r}\sum_{\nu=0}^r (-1)^{\nu}{r\choose\nu}\left(1-\frac{T-r+\nu}T \right)^m$

which fits rather nicely simulation frequencies, as shown using

apply(!apply(matrix(sample(1:Nb,T*M,rep=TRUE),T,M),1,duplicated),2,sum)


Further, Feller (1970, pp.103-104) justifies an asymptotic Poisson approximation with parameter$$\lambda(M)=\bar{N}\exp\{-M/\bar N\}$ from which an estimate of$M\$ can be derived. With the birthday problem as illustration (pp.105-106)!

It may be that a completion from N to (R¹,R²,…) where the components are the number of days with one birthdate, two birthdates, &tc. could help design an EM algorithm that would remove the summation in (1) but I did not spend more time on the problem (than finding a SAS approximation to the probability!).

## Don Fraser (1925-2020)

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , , , on December 24, 2020 by xi'an

I just received the very sad news that Don Fraser, emeritus professor of statistics at the University of Toronto, passed away this Monday, 21 December 2020. He was a giant of the field, with a unique ability for abstract modelling and he certainly pushed fiducial statistics much further than Fisher ever did. He also developed a theory of structural  inference that came close to objective Bayesian statistics, although he remained quite critical of the Bayesian approach (always in a most gentle manner, as he was a very nice man!). And most significantly contributed to high order asymptotics, to the critical analysis of ancilarity and sufficiency principles, and more beyond. (Statistical Science published a conversation with Don, in 2004, providing more personal views on his career till then.) I met with Don and Nancy rather regularly over the years, as they often attended and talked at (objective) Bayesian meetings, from the 1999 edition in Granada, to the last one in Warwick in 2019. I also remember a most enjoyable barbecue together, along with Ivar Ekeland and his family, during JSM 2018, on Jericho Park Beach, with a magnificent sunset over the Burrard Inlet. Farewell, Don!

## down with Galton (and Pearson and Fisher…)

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , , , on July 22, 2019 by xi'an

In the last issue of Significance, which I read in Warwick prior to the conference, there is a most interesting article on Galton’s eugenics, his heritage at University College London (UCL), and the overall trouble with honouring prominent figures of the past with memorials like named building or lectures… The starting point of this debate is a protest from some UCL students and faculty about UCL having a lecture room named after the late Francis Galton who was a professor there. Who further donated at his death most of his fortune to the university towards creating a professorship in eugenics. The protests are about Galton’s involvement in the eugenics movement of the late 18th and early 19th century. As well as professing racist opinions.

My first reaction after reading about these protests was why not?! Named places or lectures, as well as statues and other memorials, have a limited utility, especially when the named person is long dead and they certainly do not contribute in making a scientific theory [associated with the said individual] more appealing or more valid. And since “humans are [only] humans”, to quote Stephen Stigler speaking in this article, it is unrealistic to expect great scientists to be perfect, the more if one multiplies the codes for ethical or acceptable behaviours across ages and cultures. It is also more rational to use amphitheater MS.02 and lecture room AC.18 rather than associate them with one name chosen out of many alumni’s or former professors’.

Predictably, another reaction of mine was why bother?!, as removing Galton’s name from the items it is attached to is highly unlikely to change current views on eugenism or racism. On the opposite, it seems to detract from opposing the present versions of these ideologies. As some recent proposals linking genes and some form of academic success. Another of my (multiple) reactions was that as stated in the article these views of Galton’s reflected upon the views and prejudices of the time, when the notions of races and inequalities between races (as well as genders and social classes) were almost universally accepted, including in scientific publications like the proceedings of the Royal Society and Nature. When Karl Pearson launched the Annals of Eugenics in 1925 (after he started Biometrika) with the very purpose of establishing a scientific basis for eugenics. (An editorship that Ronald Fisher would later take over, along with his views on the differences between races, believing that “human groups differ profoundly in their innate capacity for intellectual and emotional development”.) Starting from these prejudiced views, Galton set up a scientific and statistical approach to support them, by accumulating data and possibly modifying some of these views. But without much empathy for the consequences, as shown in this terrible quote I found when looking for more material:

“I should feel but little compassion if I saw all the Damaras in the hand of a slave-owner, for they could hardly become more wretched than they are now…”

As it happens, my first exposure to Galton was in my first probability course at ENSAE when a terrific professor was peppering his lectures with historical anecdotes and used to mention Galton’s data-gathering trip to Namibia, literally measure local inhabitants towards his physiognomical views , also reflected in the above attempt of his to superpose photographs to achieve the “ideal” thief…

Posted in Books, Kids, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , on February 8, 2019 by xi'an

As a reading suggestion for my (last) OxWaSP Bayesian course at Oxford, I included the classic 1973 Marginalisation paradoxes by Phil Dawid, Mervyn Stone [whom I met when visiting UCL in 1992 since he was sharing an office with my friend Costas Goutis], and Jim Zidek. Paper that also appears in my (recent) slides as an exercise. And has been discussed many times on this  ‘Og.

Reading the paper in the train to Oxford was quite pleasant, with a few discoveries like an interesting pike at Fraser’s structural (crypto-fiducial?!) distributions that “do not need Bayesian improper priors to fall into the same paradoxes”. And a most fascinating if surprising inclusion of the Box-Müller random generator in an argument, something of a precursor to perfect sampling (?). And a clear declaration that (right-Haar) invariant priors are at the source of the resolution of the paradox. With a much less clear notion of “un-Bayesian priors” as those leading to a paradox. Especially when the authors exhibit a red herring where the paradox cannot disappear, no matter what the prior is. Rich discussion (with none of the current 400 word length constraint), including the suggestion of neutral points, namely those that do identify a posterior, whatever that means. Funny conclusion, as well:

“In Stone and Dawid’s Biometrika paper, B1 promised never to use improper priors again. That resolution was short-lived and let us hope that these two blinkered Bayesians will find a way out of their present confusion and make another comeback.” D.J. Bartholomew (LSE)

and another

“An eminent Oxford statistician with decidedly mathematical inclinations once remarked to me that he was in favour of Bayesian theory because it made statisticians learn about Haar measure.” A.D. McLaren (Glasgow)

and yet another

“The fundamentals of statistical inference lie beneath a sea of mathematics and scientific opinion that is polluted with red herrings, not all spawned by Bayesians of course.” G.N. Wilkinson (Rothamsted Station)

Lindley’s discussion is more serious if not unkind. Dennis Lindley essentially follows the lead of the authors to conclude that “improper priors must go”. To the point of retracting what was written in his book! Although concluding about the consequences for standard statistics, since they allow for admissible procedures that are associated with improper priors. If the later must go, the former must go as well!!! (A bit of sophistry involved in this argument…) Efron’s point is more constructive in this regard since he recalls the dangers of using proper priors with huge variance. And the little hope one can hold about having a prior that is uninformative in every dimension. (A point much more blatantly expressed by Dickey mocking “magic unique prior distributions”.) And Dempster points out even more clearly that the fundamental difficulty with these paradoxes is that the prior marginal does not exist. Don Fraser may be the most brutal discussant of all, stating that the paradoxes are not new and that “the conclusions are erroneous or unfounded”. Also complaining about Lindley’s review of his book [suggesting prior integration could save the day] in Biometrika, where he was not allowed a rejoinder. It reflects on the then intense opposition between Bayesians and fiducialist Fisherians. (Funny enough, given the place of these marginalisation paradoxes in his book, I was mistakenly convinced that Jaynes was one of the discussants of this historical paper. He is mentioned in the reply by the authors.)