This paper by Andrew Gelman and Christian Hennig 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. Continue reading
Archive for Henri Poincaré
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.
Yesterday, 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:
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!
“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
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