Archive for A Treatise on Probability

someone who might benefit from increased contacts with the statistical community

Posted in Books, Statistics with tags , , , , , on July 23, 2012 by xi'an

A (kind of automated) email I got today:

Your name has come to our attention as someone who might benefit from increased contacts with the international statistical community. Given your professional interests and your statistical background (noting your publication ‘Reading Keynes’ Treatise on Probability’ in the journal International Statistical Review, volume 79, 2011), you should consider elected membership in the International Statistical Institute (ISI).

Hmmm, thanks but no thanks, I am not certain I need become a member of the ISI to increase my contacts with the international statistical community! (Disclaimer: This post makes fun of the anonymous emailing, not of the ISI!)

A seminar I will sadly miss

Posted in Books, Statistics, Travel, University life with tags , , , , , on January 4, 2011 by xi'an

Next Friday, January 7, Jean Claude Saut will give a seminar in Orsay (bâtiment 425, salle 113, 13:30) on “Autour de `A Treatise on Probability’ de John Maynard Keynes  (II)”. I would have liked very much to be there and hear about a mathematician’s views on this book, which are most likely orthogonal to mines… For those in Paris this week there also is a Big’MC seminar on Thursday (IHP, 3pm) with Jean-Louis Foulley on evidence computation (in connection with Ando’s book) and Gilles Celeux (on latent blocks).

key[ed/nes] in!

Posted in Books, Statistics, University life with tags , , , , on November 19, 2010 by xi'an

Great news in the mail today: my revision of Keynes’ A Treatise on Probability has been accepted by the International Statistical Review! With a very nice message from the editor:

It is an excellent revision and has addressed all the important points and more. I must also compliment you on your fluid and interesting writing style. It makes for very nice reading.

(In fact, this review of Keynes’ book is my first publication in this journal. This irrelevant point of information reminds me of an equally unimportant but enjoyable discussion Andrew Gelman and I had in the IHP cafeteria last year about the merits of publishing in new journals… )

Continue reading

“Reading Keynes” revised

Posted in Books, Statistics, University life with tags , on October 19, 2010 by xi'an

Over the past weeks, I have revised Reading Keynes’ Treatise on Probability and the new version is now arXived (and resubmitted). I have mostly focussed on the presentation of the Treatise within the (then) current scene and on the existing advances missed by Keynes, rather than studying the technical work on the principal averages any further. For the same reason, I also abstained from relating the book to the notion of uncertain probabilities, despite a nice suggestion by Michael Brady, as I felt this was not directly related to the statistical focus of the review.

reading Keynes [assessed]

Posted in Books, Statistics, University life with tags , , on October 1, 2010 by xi'an

A few days ago, I got the following review of my paper on Keynes’ A Treatise on Probability, review which, while being rather negative, is quite fair.

When I looked at the title, I was expecting a paper that would be of considerable general interest. Now that I have read the paper several times, I am not quite sure what I got out of the paper. While I admire the authors goal, the paper is unfortunately not very informative.. [It] comes across as a collection of disjoint points. Even within sections, the points being made are not cohesive, and the paper does not flow smoothly. Some of this is to be expected from the nature of the paper, as one must select key points to discuss. But the organization has to be better. Also, the material following the quotes do not really embellish on the point in the quote; in fact, sometimes there is no direct connection. Take for instance the first part of Section 2, before 2.1. The quote just sits there  the material doesn’t make any connection to it. The main point of the section is in the top para on page 3 but it doesn’t do much. There is a mention on lack of analyses of data but this seems to be a throw-away sentence. Continue reading


Posted in Books, Statistics with tags , , , , , , on March 31, 2010 by xi'an

Yesterday, I finally arXived my notes on Keynes’ book A Treatise On Probability, but, due to the new way the arXiv website operates, there is no indication of the page associated with the submitted paper before it gets accepted and I cannot thus prepare an Og’ entry until this acceptance, wasting perfect timing! Anyway, this is the first draft of the notes and it has not yet been submitted to a journal. As the new user interface on the arXiv webpage now displays all past papers, I added a category on our 2007 Annals paper with Randal Douc, Arnaud Guillin and Jean-Michel Marin, which means it appeared again in today’s list…

Today I completed my revision of the review of Burdzy’s The Search for Certainty over for Bayesian Analysis, so the new version will be on arXiv tomorrow morning. The changes are really minor as Bayesian Analysis mostly requested smoothing down my criticisms. I also added a few more quotes and some sentences in the conclusion. I wonder if this paper will appear with a discussion, since three are already written!

At last, let me point out three recent interesting postings on arXiv if I do not have time to discuss them more in depth, one by Peter Green on Colouring and breaking sticks: random distributions and heterogeneous clustering, one by Nicolas Chopin, Tony Lelièvre et Gabriel Stolz on Free energy methods for efficient exploration of mixture posterior densities, and one by Sophie Donnet and Jean-Michel Marin on An empirical Bayes procedure for the selection of Gaussian graphical models.

Keynes’ derivations

Posted in Books, Statistics with tags , , , , , on March 29, 2010 by xi'an

Chapter XVII of Keynes’ A Treatise On Probability contains Keynes’ most noteworthy contribution to Statistics, namely the classification of probability distributions such that the arithmetic/geometric/harmonic empirical mean/empirical median is also the maximum likelihood estimator. This problem was first stated by Laplace and Gauss (leading to Laplace distribution in connection with the median and to the Gaussian distribution for the arithmetic mean). The derivation of the densities f(x,\theta) of those probability distributions is based on the constraint the likelihood equation

\sum_{i=1}^n \dfrac{\partial}{\partial\theta}\log f(y_i,\theta) = 0

is satisfied for one of the four empirical estimate, using differential calculus (despite the fact that Keynes earlier derived Bayes’ theorem by assuming the parameter space to be discrete). Under regularity assumptions, in the case of the arithmetic mean, my colleague Eric Séré showed me this indeed leads to the family of distributions

f(x,\theta) = \exp\left\{ \phi^\prime(\theta) (x-\theta) - \phi(\theta) + \psi(x) \right\}\,,

where \phi and \psi are almost arbitrary functions under the constraints that \phi is twice differentiable and f(x,\theta) is a density in x. This means that \phi satisfies

\phi(\theta) = \log \int \exp \left\{ \phi^\prime(\theta) (x-\theta) + \psi(x)\right\}\, \text{d}x\,,

a constraint missed by Keynes.

While I cannot judge of the level of novelty in Keynes’ derivation with respect to earlier works, this derivation therefore produces a generic form of unidimensional exponential family, twenty-five years before their rederivation by Darmois (1935), Pitman (1936) and Koopman (1936) as characterising distributions with sufficient statistics of constant dimensions. The derivation of the distributions for which the geometric or the harmonic means are MLEs then follows by a change of variables, y=\log x,\,\lambda=\log \theta or y=1/x,\,\lambda=1/\theta, respectively. In those different derivations, the normalisation issue is treated quite off-handedly by Keynes, witness the function

f(x,\theta) = A \left( \dfrac{\theta}{x} \right)^{k\theta} e^{-k\theta}

at the bottom of page 198, which is not integrable in x unless its support is bounded away from 0 or \infty. Similarly, the derivation of the log-normal density on page 199 is missing the Jacobian factor 1/x (or 1/y_q in Keynes’ notations) and the same problem arises for the inverse-normal density, which should be

f(x,\theta) = A e^{-k^2(x-\theta)^2/\theta^2 x^2} \dfrac{1}{x^2}\,,

instead of A\exp k^2(\theta-x)^2/x (page 200). At last, I find the derivation of the distributions linked with the median rather dubious since Keynes’ general solution

f(x,\theta) = A \exp \left\{ \displaystyle{\int \dfrac{y-\theta}{|y-\theta|}\,\phi^{\prime\prime}(\theta)\,\text{d}\theta +\psi(x) }\right\}

(where the integral ought to be interpreted as a primitive) is such that the recovery of Laplace’s distribution, f(x,\theta)\propto \exp-k^2|x-\theta| involves setting (page 201)

\psi(x) = \dfrac{\theta-x}{|x-\theta|}\,k^2 x\,,

hence making \psi a function of \theta as well. The summary two pages later actually produces an alternative generic form, namely

f(x,\theta) = A \exp\left\{ \phi^\prime(\theta)\dfrac{x-\theta}{|x-\theta|}+\psi(x) \right\}\,,

with the difficulties that the distribution only vaguely depends on \theta, being then a step function times exp(\psi(x)) and that, unless \phi is properly calibrated, A also depends on \theta.

Given that this part is the most technical section of the book, this post shows why I am fairly disappointed at having picked this book for my reading seminar. There is no further section with innovative methodological substance in the remainder of the book, which now appears to me as no better than a graduate dissertation on the probabilistic and statistical literature of the (not that) late 19th century, modulo the (inappropriate) highly critical tone.