Archive for Marsaglia

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!]

Truly random?!

Posted in Books, R, Statistics, University life with tags , , , , , , , on September 7, 2010 by xi'an

Having purchased the September edition of La Recherche because of its (disappointing!) coverage on black matter, I came by a short coverage on an Intel circuit producing “truly random” numbers… I already discussed this issue in an earlier post, namely that there is no reason physical generators are “more” random than congruential pseudo-random generators, but this short paper repeats the same misunderstanding on the role of “random” generators. The paper mentions dangers of pseudo-random generators for cryptography (but this is only when you know the deterministic function and the sequence of seeds used so far), while it misses the essential aspect of valid generators, namely that their distribution is exactly known (e.g., uniform) and, in the case of parallel generations, which seems to be the case for this circuit, that the generators are completely independent. La Recherche mentions that the entropy of the generator is really high, but this is more worrying than reassuring, as the Intel engineers do not have a more elaborate way to prove uniformity than a Monte Carlo experiment…

There is actually a deeper entry found on Technology Review. (Which may have been the source for the paper in the technology tribune of La Recherche.) The article mentions that the generator satisfied all benchmarks of “randomness” maintained by NIST. Those statistical tests sound much more reassuring than the entropy check mentioned by La Recherche, as they essentially reproduce Marsaglia’s DieHard benchmark… I remain rather skeptical about physical devices, as compared with mathematical functions, because of (a) non-reproducibility which is a negative feature despite what the paper says and of (b) instability of the device, which means that proven uniformity at time t does not induce uniformity at time t+1. Nonetheless, if the gains in execution are gigantic, it may be worth the approximation for most applications. But please stop using “true” in conjunction with randomness!!!