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From |
wgould@stata.com (William Gould, StataCorp LP) |

To |
statalist@hsphsun2.harvard.edu |

Subject |
Re: st: Re: Random start to random number sequence |

Date |
Fri, 20 Aug 2010 12:35:07 -0500 |

Allan Reese made comments on my comments on psuedo-random-number generators (PRNGs), with all of which I agree except two. My first disagreement is merely about a possible misinterpretation of something Alan wrote and is not a real disagreement with Alan's point. Alan wrote, quoting himself, "... you can use the system clock which changes every second. This will not make the subsequent sequence any more (or less) random, but will make each session unique." In context, I agree with what Alan wrote. The statement quoted in isolation, however, might suggest to somebody that would okay to set the seed, draw a few random numbers, reset the seed, ... As I showed in my posting, and as Alan mentioned elsewhere in his, resetting the seed often with non-random numbers has the potential to greatly reduce the randomness of the pseudo-random numbers generated. I keep mentioning this because I fear this is a property not appreciated by many users of PRNGs. My second disagreement is purely philosphical. I wrote "Because you set the seed only once, we do not need to discuss randomness. Randomness is a property of sequences of numbers" and Alan wrote, "Disagree, but this is philosphy, not Stata." I apologize, but I just have to respond. I do not understand how, if we are talking about just one number, we could ever distinguish whether that number was the result of a random process, and because we cannot, I do not know why it would ever be meaningful to speculate about how the number was generated. Alan makes reference to Bell's Theorum, which produced Bell's Inequality which, when applied to the results of certain experiments, established something really deep about the universe. I don't think the reference bolstered Allan's argument. Bell's Inequality is applied to results across experiments. If we had data on only one example of the particular quantum weirdness, Bell's Inequality would be as useless for refuting its targeted hypothesis as any statistical test would be using the data on just one random number. -- Bill P.S. This is way off topic for this list, but for those curious, here's what Bell's Theorum -- the derivation of Bell's Inequality -- is all about. We have all at least heard about quantum theory know about the randomness that seems to be inherent in the theory. An interesting question is whether the randomness is real -- the universe really does have a random component -- or if instead there are unobserved variables -- physicists call them hidden variables -- that if we knew their values, would eliminate the randomness. In one particular experiment, two particles, "entangled" when near each other, continue later to exhibit certain correlations in their behavior We start off by knowing that the sum of the spin of two particles is zero. We do not, however, know the angle of the axis about which they are spinning. As an aside, If we could measure the angles, we would find that one is the opposite of the other (i.e., same axis, but spinning in opposite directions). In is not possible to measure the angle of spin directly. You can choose an angle, however, and ask whether it the particle is spinning about that axis, and obtain a yes or no answer, and asking the question will change the angle of spin. Actually, the answer you get will not be the answer to whether the particle was spinning about that axis, but whether it is now spinning about that axis. The two answers are, however, related. Say the particle is really spining around 0 degrees. If you ask the question the question about 0 degrees, the anwswer will be yes with p=1. If you ask the question about angle 180 degrees, the answer will be no with p=1. If you ask about 90 degrees, the answer will be yes or no with equal probability. Ask about other angles, and p changes linearly around the values just supplied. After you get your answer, regardless of the original angle, if the answer was yes, the particle will not be spinning around 0 degrees, and if the answer was no, the particle will now spinning around 180 degrees. In the two particle case I mentioned, we have two particles with known total spin 0. We also know both partlces are spinning and at the same rate. Thus, the first particle might have angle 0 degrees and the other, 180; or the first 1 degree and other, 181; etc. Here's the surpising experimental result: Ask the question about an angle for the first particle and if you subsequently ask the question about the same angle for the second particle, you will get the opposite answer with p=1. Thus the two answers for these "entangled" particles is correlated -1 across experiments: we will only see (yes,no) or (no, yes). Results are as as if the answer from the first particle is communicated to the second. The above experimental result conteinues to be observed even if the two measurements are made at a distance from each other and one shortly after the other. Shortly here can be less time than required by a signal to travel at the speed of light from one particle to the other. That bothered Einstein no end because his theory of relativity said nothing, not even information, could move between locations faster than the speed of light. Einstein and others speculated, just as you may now be speculating, that there are hidden (unobserved) variables that explain the correlation. In effect, the two particles "agreed" on how they would respond to certain stimuli in the future, and carried the information with them after they parted, thus leading to the observed correlation. Bell's Inequality specified the limits on how much hidden variables could affect the the correlation in this case. In repeated experiments, the universe exceeds those limits. The best explanation and derivation of this result intended for the more-than-merely mathematically inclined layperson -- and that would include me -- is in the book _The Emperor's New Mind_ by Roger Penrose, 1989, 602pp. A quarter of this book summarizes current physical theory, with math. If you enjoy that book, I highly recommend _The Road to Reality: A Complete Guide to the Laws of the Universe_, also by Roger Penrose, 2004, this one 1,136 pages! I sometimes he think he wrote this book for about six of us, but I see from Amazon that this book ranks 18th in matematical physics, so evidently I have an inflated opinion of myself. Anyway, the book is worth the cost just for his clear and concise derivations of the property of the exponential and related functions on the complex plane (people who use Stata for time series, take note). <end> * * For searches and help try: * http://www.stata.com/help.cgi?search * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/

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