Archive for Andrei Kolmogorov

from least squares to signal processing and particle filtering

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , on June 6, 2017 by xi'an

Nozer Singpurwalla, Nick. Polson, and Refik Soyer have just arXived a remarkable survey on the history of signal processing, from Gauß, Yule, Kolmogorov and Wiener, to Ragazzini, Shanon, Kálmán [who, I was surprised to learn, died in Gainesville last year!], Gibbs sampling, and the particle filters of the 1990’s.

Conditional love [guest post]

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , , , , , , , , , , , , , on August 4, 2015 by xi'an

[When Dan Simpson told me he was reading Terenin’s and Draper’s latest arXival in a nice Bath pub—and not a nice bath tub!—, I asked him for a blog entry and he agreed. Here is his piece, read at your own risk! If you remember to skip the part about Céline Dion, you should enjoy it very much!!!]

Probability has traditionally been described, as per Kolmogorov and his ardent follower Katy Perry, unconditionally. This is, of course, excellent for those of us who really like measure theory, as the maths is identical. Unfortunately mathematical convenience is not necessarily enough and a large part of the applied statistical community is working with Bayesian methods. These are unavoidably conditional and, as such, it is natural to ask if there is a fundamentally conditional basis for probability.

Bruno de Finetti—and later Richard Cox and Edwin Jaynes—considered conditional bases for Bayesian probability that are, unfortunately, incomplete. The critical problem is that they mainly consider finite state spaces and construct finitely additive systems of conditional probability. For a variety of reasons, neither of these restrictions hold much truck in the modern world of statistics.

In a recently arXiv’d paper, Alexander Terenin and David Draper devise a set of axioms that make the Cox-Jaynes system of conditional probability rigorous. Furthermore, they show that the complete set of Kolmogorov axioms (including countable additivity) can be derived as theorems from their axioms by conditioning on the entire sample space.

This is a deep and fundamental paper, which unfortunately means that I most probably do not grasp it’s complexities (especially as, for some reason, I keep reading it in pubs!). However I’m going to have a shot at having some thoughts on it, because I feel like it’s the sort of paper one should have thoughts on. Continue reading

about randomness (im Hamburg)

Posted in Statistics, Travel, University life with tags , , , , , , , , , , , , on February 20, 2013 by xi'an

exhibit in DESY campus, Hamburg, Germany, Feb. 19, 2013True randomness was the topic of the `Random numbers; fifty years later’ talk in DESY by Frederick James from CERN. I had discussed a while ago a puzzling book related to this topic. This talk went along a rather different route, focussing on random generators. James put this claim that there are computer based physical generators that are truly random. (He had this assertion that statisticians do not understand randomness because they do not know quantum mechanics.) He distinguished those from pseudo-random generators: “nobody understood why they were (almost) random”, “IBM did not know how to generate random numbers”… But then spent the whole talk discussing those pseudo-random generators. Among other pieces of trivia, James mentioned that George Marsaglia was the one exhibiting the hyperplane features of congruential generators. That Knuth achieved no successful definition of what randomness is in his otherwise wonderful books! James thus introduced Kolmogorov’s mixing (not Kolmogorov’s complexity, mind you!) as advocated by Soviet physicists to underlie randomness. Not producing anything useful for RNGs in the 60’s. He then moved to the famous paper by Ferrenberg, Landau and Wong (1992) that I remember reading more or less at the time. In connection with the phase transition critical slowing down phenomena in Ising model simulations. And connecting with the Wang-Landau algorithm of flipping many sites at once (which exhibited long-term dependences in the generators). Most interestingly, a central character in this story is Martin Lüscher, based in DESY, who expressed the standard generator of the time RCARRY into one studied by those Soviet mathematicians,

X’=AX

showing that it enjoyed Kolmogorov mixing, but with a very poor Lyapunov coefficient. I partly lost track there as RCARRY was not perfect. And on how this Kolmogorov mixing would relate to long-term dependencies. One explanation by James was that this property is only asymptotic. (I would even say statistical!) Also interestingly, the 1994 paper by Lüscher produces the number of steps necessary to attain complete mixing, namely 15 steps, which thus works as a cutoff point. (I wonder why a 15-step RCARRY is slower, since A15 can be computed at once… It may be due to the fact that A is sparse while A15 is not.) James mentioned that Marsaglia’s Die Hard battery of tests is now obsolete and superseded by Pierre Lecuyer’s TestU01.

In conclusion, I did very much like this presentation from an insider, but still do not feel it makes a contribution to the debate on randomness, as it stayed put on pseudorandom generators. To keep the connection with von Neumann, they all produce wrong answers from a randomness point of view, if not from a statistical one. (A final quote from the talk: “Among statisticians and number theorists who are supposed to be specialists, they do not know about Kolmogorov mixing.”) [Discussing with Fred James at the reception after the talk was obviously extremely pleasant, as he happened to know a lot of my Bayesian acquaintances!]

Decision systems and nonstochastic randomness

Posted in Books, Statistics, University life with tags , , , , , on October 26, 2011 by xi'an

Thus the informativity of stochastic experiment turned out to depend on the Bayesian system and to coincide to within the scale factor with the previous “value of information”.” V. Ivanenko, Decision systems and nonstochastic randomness, p.208

This book, Decision systems and nonstochastic randomness, written by the Ukrainian researcher Victor Ivanenko, is related to decision theory and information theory, albeit with a statistical component as well. It however works at a fairly formal level and the reading is certainly not light. The randomness it address is the type formalised by Andreï Kolmogorov (also covered in the book Randomness through Computation I [rather negatively] reviewed a few months ago, inducing angry comments and scathing criticisms in the process). The terminology is slightly different from the usual one, but the basics are those of decision theory as in De Groot (1970). However, the tone quickly gets much more mathematical and the book lost me early in Chapter 3 (Indifferent uncertainty) on a casual reading. The following chapter on non-stochastic randomness reminded me of von Mises for its use of infinite sequences, and of the above book for its purpose, but otherwise offered an uninterrupted array of definitions and theorems that sounded utterly remote from statistical problems. After failing to make sense of the chapter on the informativity of experiment in Bayesian decision problems, I simply gave up… I thus cannot judge from this cursory reading whether or not the book is “useful in describing real situations of decision-making” (p.208). It just sounds very remote from my centres of interest. (Anyone interested by writing a review?)

Randomness through computation

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , , on June 22, 2011 by xi'an

A few months ago, I received a puzzling advertising for this book, Randomness through Computation, and I eventually ordered it, despite getting a rather negative impression from reading the chapter written by Tomasso Toffoli… The book as a whole is definitely perplexing (even when correcting for this initial bias) and I would not recommend it to readers interested in simulation, in computational statistics or even in the philosophy of randomness. My overall feeling is indeed that, while there are genuinely informative and innovative chapters in this book, some chapters read more like newspeak than scientific material (mixing the Second Law of Thermodynamics, Gödel’s incompleteness theorem, quantum physics, and NP completeness within the same sentence) and do not provide a useful entry on the issue of randomness. Hence, the book is not contributing in a significant manner to my understanding of the notion. (This post also appeared on the Statistics Forum.) Continue reading