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Re: st: Re: Random start to random number sequence
email@example.com (William Gould, StataCorp LP)
Re: st: Re: Random start to random number sequence
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
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
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
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
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,
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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