## empirical Bayes, reference priors, entropy & EM

Posted in Mountains, Statistics, Travel, University life with tags , , , , , , , , , , , on January 9, 2017 by xi'an

Klebanov and co-authors from Berlin arXived this paper a few weeks ago and it took me a quiet evening in Darjeeling to read it. It starts with the premises that led Robbins to introduce empirical Bayes in 1956 (although the paper does not appear in the references), where repeated experiments with different parameters are run. Except that it turns non-parametric in estimating the prior. And to avoid resorting to the non-parametric MLE, which is the empirical distribution, it adds a smoothness penalty function to the picture. (Warning: I am not a big fan of non-parametric MLE!) The idea seems to have been Good’s, who acknowledged using the entropy as penalty is missing in terms of reparameterisation invariance. Hence the authors suggest instead to use as penalty function on the prior a joint relative entropy on both the parameter and the prior, which amounts to the average of the Kullback-Leibler divergence between the sampling distribution and the predictive based on the prior. Which is then independent of the parameterisation. And of the dominating measure. This is the only tangible connection with reference priors found in the paper.

The authors then introduce a non-parametric EM algorithm, where the unknown prior becomes the “parameter” and the M step means optimising an entropy in terms of this prior. With an infinite amount of data, the true prior (meaning the overall distribution of the genuine parameters in this repeated experiment framework) is a fixed point of the algorithm. However, it seems that the only way it can be implemented is via discretisation of the parameter space, which opens a whole Pandora box of issues, from discretisation size to dimensionality problems. And to motivating the approach by regularisation arguments, since the final product remains an atomic distribution.

While the alternative of estimating the marginal density of the data by kernels and then aiming at the closest entropy prior is discussed, I find it surprising that the paper does not consider the rather natural of setting a prior on the prior, e.g. via Dirichlet processes.

## Turing’s Bayesian contributions

Posted in Books, Kids, pictures, Running, Statistics, University life with tags , , , , , , , , , , , , on March 17, 2015 by xi'an

Following The Imitation Game, this recent movie about Alan Turing played by Benedict “Sherlock” Cumberbatch, been aired in French theatres, one of my colleagues in Dauphine asked me about the Bayesian contributions of Turing. I first tried to check in Sharon McGrayne‘s book, but realised it had vanished from my bookshelves, presumably lent to someone a while ago. (Please return it at your earliest convenience!) So I told him about the Bayesian principle of updating priors with data and prior probabilities with likelihood evidence in code detecting algorithms and ultimately machines at Bletchley Park… I could not got much farther than that and hence went checking on Internet for more fodder.

“Turing was one of the independent inventors of sequential analysis for which he naturally made use of the logarithm of the Bayes factor.” (p.393)

I came upon a few interesting entries but the most amazìng one was a 1979 note by I.J. Good (assistant of Turing during the War) published in Biometrika retracing the contributions of Alan Mathison Turing during the War. From those few pages, it emerges that Turing’s statistical ideas revolved around the Bayes factor that Turing used “without the qualification `Bayes’.” (p.393) He also introduced the notion of ban as a unit for the weight of evidence, in connection with the town of Banbury (UK) where specially formatted sheets of papers were printed “for carrying out an important classified process called Banburismus” (p.394). Which shows that even in 1979, Good did not dare to get into the details of Turing’s work during the War… And explains why he was testing simple statistical hypothesis against simple statistical hypothesis. Good also credits Turing for the expected weight of evidence, which is another name for the Kullback-Leibler divergence and for Shannon’s information, whom Turing would visit in the U.S. after the War. In the final sections of the note, Turing is also associated with Gini’s index, the estimation of the number of species (processed by Good from Turing’s suggestion in a 1953 Biometrika paper, that is, prior to Turing’s suicide. In fact, Good states in this paper that “a very large part of the credit for the present paper should be given to [Turing]”, p.237), and empirical Bayes.

## JSM 2010 [day 2]

Posted in Books, pictures, R, Running, Statistics, University life with tags , , , , , , , on August 3, 2010 by xi'an

After a very good early run in Stanley Park, I went to a morning session on new statistical challenges in genetics, but unfortunately could not keep focussed enough (due to a very short night, still being not tuned to Pacific time!) so I ended up chatting with Sid Chib at the Springer booth about the future of R and the drawback of it running too slowly… The second session of the morning I attended was the I.J. Good memorial session (although there were many alternative choices I could have made at the same time!) where Steve Fienberg, Jim Berger, Adrian Raftery and David Banks gave different perspectives on the life and influence of this leading figure. After his work in Bletchley Park along Alan Turing during the war, already using Bayes factors introduced a few years earlier by Harold Jeffreys, I.J. Good contributed very much to the Bayesian revival of the 50’s. (A fact not mentioned this morning is that he was a consultant for 2001: A Space Odyssey!) The afternoon session on Bayesian processing of massive data systems was somehow compulsory since I was talking in this session! While the talks were interestingly diverse, there were however again very people in the room, making me feel the attendance was much lower than last year. As the day ended earlier to let free space to the presidential address, this eventually came as a less exciting day (but left me time for an early evening swim plus two mixers!)…