**I**n the past weeks I have received and read several papers (and X validated entries)where the Bayes factor is used to compare priors. Which does not look right to me, not on the basis of my general dislike of Bayes factors!, but simply because this seems to clash with the (my?) concept of Bayesian model choice and also because data should not play a role in that situation, from being used to select a *prior*, hence at least twice to run the inference, to resort to a *single* parameter value (namely the one behind the data) to decide between two distributions, to having no asymptotic justification, to eventually favouring the prior concentrated on the maximum likelihood estimator. And more. But I fear that this reticence to test for prior adequacy also extends to the prior predictive, or Box’s p-value, namely the probability under this prior predictive to observe something “more extreme” than the current observation, to quote from David Spiegelhalter.

## Archive for the Statistics Category

## leave Bayes factors where they once belonged

Posted in Statistics with tags Bayes factors, Bayesian Analysis, Bayesian decision theory, cross validated, prior comparison, prior predictive, prior selection, The Bayesian Choice, The Beatles, using the data twice, xkcd on February 19, 2019 by xi'an## Bayesian intelligence in Warwick

Posted in pictures, Statistics, Travel, University life, Wines with tags ABC, AI, artificial intelligence, Bayesian Analysis, Bayesian intelligence, CRiSM, effective dimension, estimating constants, Monte Carlo integration, neural network, paradoxes, seminar, University of Warwick on February 18, 2019 by xi'an**T**his is an announcement for an exciting CRiSM Day in Warwick on 20 March 2019: with speakers

10:00-11:00 Xiao-Li Meng (Harvard): “Artificial Bayesian Monte Carlo Integration: A Practical Resolution to the Bayesian (Normalizing Constant) Paradox”

11:00-12:00 Julien Stoehr (Dauphine): “Gibbs sampling and ABC”

14:00-15:00 Arthur Ulysse Jacot-Guillarmod (École Polytechnique Fedérale de Lausanne): “Neural Tangent Kernel: Convergence and Generalization of Deep Neural Networks”

15:00-16:00 Antonietta Mira (Università della Svizzera italiana e Università degli studi dell’Insubria): “Bayesian identifications of the data intrinsic dimensions”

[whose abstracts are on the workshop webpage] and free attendance. The title for the workshop mentions Bayesian Intelligence: this obviously includes human intelligence and not just AI!

## Le Monde puzzle [#1085]

Posted in Books, Kids, R with tags dynamic programming, game theory, Le Monde, mathematical puzzle, minimax strategy, R on February 18, 2019 by xi'an**A **new Le Monde mathematical puzzle in the digit category:

Given 13 arbitrary relative integers chosen by Bo, Abigail can select any subset of them to be drifted by plus or minus one by Bo, repeatedly until Abigail reaches the largest possible number N of multiples of 5. What is the minimal possible value of N under the assumption that Bo tries to minimise it?

I got stuck on that one, as building a recursive functiion led me nowhere: the potential for infinite loop (add one, subtract one, add one, …) rather than memory issues forced me into a finite horizon for the R function, which then did not return anything substantial in a manageable time. Over the week and the swimming sessions, I thought of simplifying the steps, like (a) work modulo 5, (b) bias moves towards 1 or 4, away from 2 and 3, by keeping only one entry in 2 and 3, and all but one at 1 and 4, but could only produce five 0’s upon a sequence of attempts… With the intuition that only 3 entries should remain in the end, which was comforted by Le Monde solution the week after.

## 50/50 photography competition [another public image]

Posted in Statistics with tags competition, jatp, landscape photography, Old Man of Storr, Université Paris Dauphine, Vert le Futur on February 17, 2019 by xi'an## take a random integer

Posted in Books, Statistics with tags FiveThirtyEight, random variable, The Riddler on February 16, 2019 by xi'an**A** weird puzzle from FiveThirtyEight: what is the probability that the product of three random integers is a multiple of 100? Ehrrrr…, what is a random integer?! The solution provided by the Riddler is quite stunning

Reading the question charitably (since “random integer” has no specific meaning), there will be an answer if there is a limit for a uniform distribution of positive integers up to some number . But we can ignore that technicality, and make do with the idealization that since every second, fourth, fifth, and twenty-fifth integer are divisible by and , the chances of getting a random integer divisible by those numbers are , , , and .

as it acknowledges that the question is meaningless, then dismisses this as a “technicality” and still handles a Uniform random integer on {1,2,…,N} as N grows to infinity! Since all that matters is the remainder of the “random variable” modulo 100, this remainder will see its distribution vary as N moves to infinity, even though it indeed stabilises for $N$ large enough…

## say cheese [jatp]

Posted in Statistics with tags brains, british cheeses, cheesemonger, Instagram, jatp, Jericho Cheese, Little Clarendon Street, Oxford, pointfivegully, Stichelton, Stilton on February 16, 2019 by xi'an**A** picture taken at Jericho Cheese, Little Clarendon Street, Oxford, that I took this week when I visited this cheesemonger for the first time, after several years of passing by its tantalizing display of British cheeses! It happens to have become my most popular picture on Instagram, ranking above the fiery sunrise over the Calanques, and the alignment of brains at the Institute of Brain in Paris!

## undecidable learnability

Posted in Books, Statistics, Travel, University life with tags academic journals, incompleteness theorem, Kurt Gödel, learnability, Nature Machine Intelligence, Robins-Wasserman paradox, Vapnik-Chervonenkis on February 15, 2019 by xi'an

“There is an unknown probability distribution P over some finite subset of the interval [0,1]. We get to see m i.i.d. samples from P for m of our choice. We then need to find a finite subset of [0,1] whose P-measure is at least 2/3. The theorem says that the standard axioms of mathematics cannot be used to prove that we can solve this problem, nor can they be used to prove that we cannot solve this problem.”

In the first issue of the (controversial) nature machine intelligence journal, Ben-David et al. wrote a paper they present a s the machine learning equivalent to Gödel’s incompleteness theorem. The result is somewhat surprising from my layman perspective and it seems to only relate to a formal representation of statistical problems. Formal as in the Vapnik-Chervonenkis (PAC) theory. It sounds like, given a finite learning dataset, there are always features that cannot be learned if the size of the population grows to infinity, but this is hardly exciting…

The above quote actually makes me think of the Robbins-Wasserman counter-example for censored data and Bayesian tail prediction, but I am unsure the connection is anything more than sheer fantasy..!

~