Archive for Henri Poincaré

beyond subjective and objective in Statistics

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , on August 28, 2015 by xi'an

“At the level of discourse, we would like to move beyond a subjective vs. objective shouting match.” (p.30)

This paper by Andrew Gelman and Christian Henning calls for the abandonment of the terms objective and subjective in (not solely Bayesian) statistics. And argue that there is more than mere prior information and data to the construction of a statistical analysis. The paper is articulated as the authors’ proposal, followed by four application examples, then a survey of the philosophy of science perspectives on objectivity and subjectivity in statistics and other sciences, next to a study of the subjective and objective aspects of the mainstream statistical streams, concluding with a discussion on the implementation of the proposed move.

“…scientists and the general public celebrate the brilliance and inspiration of greats such as Einstein, Darwin, and the like, recognizing the roles of their personalities and individual experiences in shaping their theories and discoveries” (p.2)

I do not see the relevance of this argument, in that the myriad of factors leading, say, Marie Curie or Rosalind Franklin to their discoveries are more than subjective, as eminently personal and the result of unique circumstance, but the corresponding theories remain within a common and therefore objective corpus of scientific theories. Hence I would not equate the derivation of statistical estimators or even less the computation of statistical estimates to the extension or negation of existing scientific theories by scientists.

“We acknowledge that the “real world” is only accessible to human beings through observation, and that scientific observation and measurement cannot be independent of human preconceptions and theories.” (p.4)

The above quote reminds me very much of Poincaré‘s

“It is often said that experiments should be made without preconceived ideas. That is impossible. Not only would it make every experiment fruitless, but even if we wished to do so, it could not be done. Every man has his own conception of the world, and this he cannot so easily lay aside.” Henri Poincaré, La Science et l’Hypothèse

The central proposal of the paper is to replace `objective’ and `subjective’ with less value-loaded and more descriptive terms. Given that very few categories of statisticians take pride in their subjectivity, apart from a majority of Bayesians, but rather use the term as derogatory for other categories, I fear the proposal stands little chance to see this situation resolved. Even though I agree we should move beyond this distinction that does not reflect the complexity and richness of statistical practice. As the discussion in Section 2 makes it clear, all procedures involve subjective choices and calibration (or tuning), either plainly acknowledged or hidden under the carpet. Which is why I would add (at least) two points to the virtues of subjectivity:

  1. Spelling out unverifiable assumptions about the data production;
  2. Awareness of calibration of tuning parameters.

while I do not see consensus as necessarily a virtue. The following examples in Section 3 are all worth considering as they bring more details, albeit in specific contexts, to the authors’ arguments. Most of them give the impression that the major issue stands with the statistical model itself, which may be both the most acute subjectivity entry in statistical analyses and the least discussed one. Including the current paper, where e.g. Section 3.4 wants us to believe that running a classical significance test is objective and apt to detect an unfit model. And the hasty dismissal of machine learning in Section 6 is disappointing, because one thing machine learning does well is to avoid leaning too much on the model, using predictive performances instead. Furthermore, apart from Section 5.3, I actually see little in the paper about the trial-and-error way of building a statistical model and/or analysis, while subjective inputs from the operator are found at all stages of this construction and should be spelled out rather than ignored (and rejected).

“Yes, Bayesian analysis can be expressed in terms of subjective beliefs, but it can also be applied to other settings that have nothing to do with beliefs.” (p.31)

The survey in Section 4 about what philosophy of sciences says about objectivity and subjectivity is quite thorough, as far as I can judge, but does not expand enough the issue of “default” or all-inclusive statistical solutions, used through “point-and-shoot” software by innumerate practitioners in mostly inappropriate settings, with the impression of conducting “the” statistical analysis. This false feeling of “the” proper statistical analysis and its relevance for this debate also transpire through the treatment of statistical expertises by media and courts. I also think we could avoid the Heisenberg principle to be mentioned in this debate, as it does not really contribute anything useful. More globally, the exposition of a large range of notions of objectivity is as often the case in philosophy not conclusive and I feel nothing substantial comes out of it… And that it is somehow antagonistic with the notion of a discussion paper, since every possible path has already been explored. Even forking ones. As a non-expert in philosophy, I would not feel confident in embarking upon a discussion on what realism is and is not.

“the subjectivist Bayesian point of view (…) can be defended for honestly acknowledging that prior information often does not come in ways that allow a unique formalization” (p.25)

When going through the examination of the objectivity of the major streams of statistical analysis, I get the feeling of exploring small worlds (in Lindley‘s words) rather than the entire spectrum of statistical methodologies. For instance, frequentism seems to be reduced to asymptotics, while completely missing the entire (lost?) continent of non-parametrics. (Which should not be considered to be “more” objective, but has the advantage of loosening the model specification.) While the error-statistical (frequentist) proposal of Mayo (1996) seems to consume a significant portion [longer than the one associated with the objectivist Bayesianism section] of the discussion with respect to its quite limited diffusion within statistical circles. From a Bayesian perspective, the discussions of subjective, objective, and falsificationist Bayes do not really bring a fresh perspective to the debate between those three branches, apart from suggesting we should give up such value loaded categorisations. As an O-Bayes card-carrying member, I find the characterisation of the objectivist branch somehow restrictive, by focussing solely on Jaynesmaxent solution. Hence missing the corpus of work on creating priors with guaranteed frequentist or asymptotic properties. Like matching priors. I also find the defence of the falsificationist perspective, i.e. of Gelman and Shalizi (2013) both much less critical and quite extensive, in that, again, this is not what one could call a standard approach to statistics. Resulting in an implicit (?) message that this may the best way to proceed.

In conclusion, on the positive side [for there is a positive side!], the paper exposes the need to spell out the various inputs (from the operator) leading to a statistical analysis, both for replicability or reproducibility, and for “objectivity” purposes, although solely conscious choices and biases can be uncovered this way. It also reinforces the call for model awareness, by which I mean a critical stance on all modelling inputs, including priors!, a disbelief that any model is true, applying to statistical procedures Popper’s critical rationalism. This has major consequences on Bayesian modelling in that, as advocated in Gelman and Shalizi (2013) , as well as Evans (2015), sampling and prior models should be given the opportunity to be updated when they are inappropriate for the data at hand. On the negative side, I fear the proposal is far too idealistic in that most users (and some makers) of statistics cannot spell out their assumptions and choices, being unaware of those. This is in a way [admitedly, with gross exaggeration!] the central difficulty with statistics that almost anyone anywhere can produce an estimate or a p-value without ever being proven wrong. It is therefore difficult to perceive how the epistemological argument therein [that objective versus subjective is a meaningless opposition] is going to profit statistical methodology, even assuming the list of Section 2.3 was to be made compulsory. The eight deadly sins listed in the final section would require expert reviewers to vanish from publication (and by expert, I mean expert in statistical methodology), while it is almost never the case that journals outside our field make a call to statistics experts when refereeing a paper. Apart from banning all statistics arguments from a journal, I am afraid there is no hope for a major improvement in that corner…

All in all, the authors deserve a big thank for making me reflect upon those issues and (esp.) back their recommendation for reproducibility, meaning not only the production of all conscious choices made in the construction process, but also through the posting of (true or pseudo-) data and of relevant code for all publications involving a statistical analysis.

failures and uses of Jaynes’ principle of transformation groups

Posted in Books, Kids, R, Statistics, University life with tags , , , , on April 14, 2015 by xi'an

This paper by Alon Drory was arXived last week when I was at Columbia. It reassesses Jaynes’ resolution of Bertrand’s paradox, which finds three different probabilities for a given geometric event depending on the underlying σ-algebra (or definition of randomness!). Both Poincaré and Jaynes argued against Bertrand that there was only one acceptable solution under symmetry properties. The author of this paper, Alon Drory, argues this is not the case!

“…contrary to Jaynes’ assertion, each of the classical three solutions of Bertrand’s problem (and additional ones as well!) can be derived by the principle of transformation groups, using the exact same symmetries, namely rotational, scaling and translational invariance.”

Drory rephrases as follows:  “In a circle, select at random a chord that is not a diameter. What is the probability that its length is greater than the side of the equilateral triangle inscribed in the circle?”.  Jaynes’ solution is indifferent to the orientation of one observer wrt the circle, to the radius of the circle, and to the location of the centre. The later is the one most discussed by Drory, as he argued that it does not involve an observer but the random experiment itself and relies on a specific version of straw throws in Jaynes’ argument. Meaning other versions are also available. This reminded me of an earlier post on Buffon’s needle and on the different versions of the needle being thrown over the floor. Therein reflecting on the connection with Bertrand’s paradox. And running some further R experiments. Drory’s alternative to Jaynes’ manner of throwing straws is to impale them on darts and throw the darts first! (Which is the same as one of my needle solutions.)

“…the principle of transformation groups does not make the problem well-posed, and well-posing strategies that rely on such symmetry considerations ought therefore to be rejected.”

In short, the conclusion of the paper is that there is an indeterminacy in Bertrand’s problem that allows several resolutions under the principle of indifference that end up with a large range of probabilities, thus siding with Bertrand rather than Jaynes.

Ulam’s grave [STAN post]

Posted in Books, Kids, pictures, Travel, University life with tags , , , , , , , on July 27, 2014 by xi'an

ulamSince Stan Ulam is buried in Cimetière du Montparnasse, next to CREST, Andrew and I paid his grave a visit on a sunny July afternoon. Among elaborate funeral constructions, the Aron family tomb is sober and hidden behind funeral houses. It came as a surprise to me to discover that Ulam had links with France to the point of him and his wife being buried in Ulam’s wife family vault. Since we were there, we took a short stroll to see Henri Poincaré’s tomb in the Poincaré-Boutroux vault (missing Henri’s brother, the French president Raymond Poincaré). It came as a surprise that someone had left a folder with the cover of 17 equations that changed the World on top of the tomb). Even though the book covers Poincaré’s work on the three body problem as part of Newton’s formula. There were other mathematicians in this cemetery, but this was enough necrophiliac tourism for one day.

poincare

Luke and Pierre at big’MC

Posted in Linux, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , on May 19, 2014 by xi'an

crossing Rue Soufflot on my way to IHP from Vieux Campeur, March 28, 2013Yesterday, Luke Bornn and Pierre Jacob gave a talk at our big’MC ‘minar. While I had seen most of the slides earlier, either at MCMski IV,  Banff, Leuven or yet again in Oxford, I really enjoyed those talks as they provided further intuition about the techniques of Wang-Landau and non-negative unbiased estimators, leading to a few seeds of potential ideas for even more potential research. For instance, I understood way better the option to calibrate the Wang-Landau algorithm on levels of the target density rather than in the original space. Which means (a) a one-dimensional partition target (just as in nested sampling); (b) taking advantage of the existing computations of the likelihood function; and (b) a somewhat automatic implementation of the Wang-Landau algorithm. I do wonder why this technique is not more popular as a default option. (Like, would it be compatible with Stan?) The impossibility theorem of Pierre about the existence of non-negative unbiased estimators never ceases to amaze me. I started wondering during the seminar whether a positive (!) version of the result could be found. Namely, whether perturbations of the exact (unbiased) Metropolis-Hastings acceptance ratio could be substituted in order to guarantee positivity. Possibly creating drifted versions of the target…

One request in connection with this post: please connect the Institut Henri Poincaré to the eduroam wireless network! The place is dedicated to visiting mathematicians and theoretical physicists, it should have been the first one [in Paris] to get connected to eduroam. The cost cannot be that horrendous so I wonder what the reason is. Preventing guests from connecting to the Internet towards better concentration? avoiding “parasites” taking advantage of the network? ensuring seminar attendees are following the talks? (The irony is that Institut Henri Poincaré has a local wireless available for free, except that it most often does not work with my current machine. And hence wastes much more of my time as I attempt to connect over and over again while there.) Just in connection with IHP, a video of Persi giving a talk there about Poincaré, two years ago:

Bayes on the radio

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , on November 10, 2012 by xi'an

In relation with the special issue of Science & Vie on Bayes’ formula, the French national radio (France Culture) organised a round table with Pierre Bessière, senior researcher in physiology at Collège de France, Dirk Zerwas, senior researcher in particle physics in Orsay, and Hervé Poirier, editor of Science & Vie. And myself (as I was quoted in the original paper). While I am not particularly fluent in oral debates, I was interested by participating in this radio experiment, if only to bring some moderation to the hyperbolic tone found in the special issue. (As the theme was “Is there a universal mathematical formula? “, I was for a while confused about the debate, thinking that maybe the previous blogs on Stewart’s 17 Equations and Mackenzie’s Universe in Zero Words had prompted this invitation…)

As it happened [podcast link], the debate was quite moderate and reasonable, we discussed about the genesis, the dark ages, and the resurgimento of Bayesian statistics within statistics, the lack of Bayesian perspectives in the Higgs boson analysis (bemoaned by Tony O’Hagan and Dennis Lindley), and the Bayesian nature of learning in psychology. Although I managed to mention Poincaré’s Bayesian defence of Dreyfus (thanks to the Theory that would not die!), Nate Silver‘s Bayesian combination of survey results, and the role of the MRC in the MCMC revolution, I found that the information content of a one-hour show was in the end quite limited, as I would have liked to mention as well the role of Bayesian techniques in population genetic advances, like the Asian beetle invasion mentioned two weeks ago… Overall, an interesting experience, maybe not with a huge impact on the population of listeners, and a confirmation I’d better stick to the written world!

17 equations that changed the World (#2)

Posted in Books, Statistics with tags , , , , , , , , , , , , , , , , , on October 16, 2012 by xi'an

(continuation of the book review)

If you placed your finger at that point, the two halves of the string would still be able to vibrate in the sin 2x pattern, but not in the sin x one. This explains the Pythagorean discovery that a string half as long produced a note one octave higher.” (p.143)

The following chapters are all about Physics: the wave equation, Fourier’s transform and the heat equation, Navier-Stokes’ equation(s), Maxwell’s equation(s)—as in  The universe in zero word—, the second law of thermodynamics, E=mc² (of course!), and Schrödinger’s equation. I won’t go so much into details for those chapters, even though they are remarkably written. For instance, the chapter on waves made me understand the notion of harmonics in a much more intuitive and lasting way than previous readings. (This chapter 8 also mentions the “English mathematician Harold Jeffreys“, while Jeffreys was primarily a geophysicist. And a Bayesian statistician with major impact on the field, his Theory of Probability arguably being the first modern Bayesian book. Interestingly, Jeffreys also was the first one to find approximations to the Schrödinger’s equation, however he is not mentioned in this later chapter.) Chapter 9 mentions the heat equation but is truly about Fourier’s transform which he uses as a tool and later became a universal technique. It also covers Lebesgue’s integration theory, wavelets, and JPEG compression. Chapter 10 on Navier-Stokes’ equation also mentions climate sciences, where it takes a (reasonable) stand. Chapter 11 on Maxwell’s equations is a short introduction to electromagnetism, with radio the obvious illustration. (Maybe not the best chapter in the book.) Continue reading

17 equations that changed the World (#1)

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , , on October 15, 2012 by xi'an

I do not know if it is a coincidence or if publishers were competing for the same audience: after reviewing The universe in zero word: The story of mathematics as told through equations, in this post (and in CHANCE, to appear in 25(3)!), I noticed Ian Stewart’s 17 equations That Changed the World, published in 2011, and I bought a copy to check the differences between both books.

I am quite glad I did so, as I tremendously enjoyed this book, both for its style and its contents, both entertaining and highly informative. This does not come as a big surprise, given Stewart’s earlier books and their record, however this new selection and discussion of equations is clearly superior to The universe in zero word! Maybe because it goes much further in its mathematical complexity, hence is more likely to appeal to the mathematically inclined (to borrow from my earlier review). For one thing, it does not shy away from inserting mathematical formulae and small proofs into the text, disregarding the risk of cutting many halves of the audience (I know, I know, high powers of (1/2)…!) For another, 17 equations That Changed the World uses the equation under display to extend the presentation much much further than The universe in zero word. It is also much more partisan (in an overall good way) in its interpretations and reflections about the World.

In opposition with The universe in zero word, formulas are well-presented, each character in the formula being explained in layman terms. (Once again, the printer could have used better fonts and the LaTeX word processor.) The (U.K. edition, see tomorrow!) cover is rather ugly, though, when compared with the beautiful cover of The universe in zero word. But this is a minor quibble! Overall, it makes for an enjoyable, serious and thought-provoking read that I once again undertook mostly in transports (planes and métros). Continue reading

Follow

Get every new post delivered to your Inbox.

Join 903 other followers