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

[wgould@stata.com (William Gould, StataCorp LP)]

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).

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