Re: Chex Wat: Pi is "random" and "not predictable"?
- From: Seanpit <seanpitnospam@xxxxxxxxxxxxxxxxxxxxxxxxxxx>
- Date: Fri, 03 Aug 2007 08:29:48 -0700
On Aug 2, 10:26 pm, "R. Baldwin" <res0k...@xxxxxxxxxxxxxxxxxxxx>
wrote:
"Seanpit" <seanpitnos...@xxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote in message
We are talking about betting strategies, or "martingales", in general
here. And, it is most certainly possible to use an algorithm as a
basis for a betting strategy.
A finite state martingale advances one state at a time, one symbol at a
time, and each state has one bet output for each symbol in the symbol set.
This is not true of algorithms. They can have many states that do not change
their output sequence. They can move forward or backward through the
sequence. They can alter the sequence throughout. They can output a sequence
in unpredictable order. So, no, in the general case an agorithm cannot be
used as the basis for a betting strategy on the succeeding output of a
sequence.
I'm not talking about "the general case". I'm talking about a
specific case. The fact is that an algorithm can also be used to
predict what number will come next in a series.
The odds are far worse than even. The odds are approximately zero. If you
pick a starting point in a normal number at random, you will almost
certainly not synchronize any algorithm with the number.
That's true. I never said otherwise.
On re-reading what you said, I see I misread your sentence about odds. I
apologize for that. But the rest of your paragraph is wrong, and my proof of
why it is wrong is still applicable.
Interesting . . . We'll take a look.
What I said, and pay close
attention this time, is that if the algorithm does happen to work in
that it does actually make correct predictions over time, the
hypothesis that it is in fact the correct algorithm and that it has
started at the correct position gains predictive value with each
successful prediction.
And, this is wrong. Remember, the context here is normal numbers. As I
showed in the proof that you snipped below, even if your algorithm appears
to find a match to a segment of the normal number, the conditional
probability that it really is a match is effectively zero. No matter how
long a finite test you allow for your algorithm, there are infinitely many
other substrings within the normal number that match that finite sequence.
That's true, but you forget that there are different kinds of
infinities. Some infinities are actually "bigger" than other
infinities. As it turns out, the infinity of "correct" sequence
predictions decreases exponentially relative to the increase in the
infinity of non-correct possibilities with each increase in size.
Therefore, the odds of actually landing on one of the correct
infinities drop exponentially with each successful prediction.
Your argument that picking the correct position, at random, is
essentially impossible is correct when it comes to a normal number.
That's not the point. The point is that if the algorithm does happen
to work over time, however unlikely this might be, the odds that the
virtually impossible correct position was actually chosen gains
predictive value over time.
No, sir. There were two distinct consequences of the proof I gave you. The
first was about the probability of finding an apparent match. The second
was, given that you have found an apparent match, the conditional
probability that it really is a match. It is this second probability we are
discussing here, and it gains zero predictive value over time on a normal
number.
Let's take pi for instance. we'll use decimal notation, and ignore the radix
point. Assuming the strong conjecture is correct and pi is normal:
The probability of finding "3" in pi at a randomly selected index is 1/10.
Given that "3" is found, the probability that it is at index 0 is 0.
The probability of finding "31" in pi at a randomly selected index is 1/100.
Given that "31" is found, the probability that it is at index 0 is 0.
The probability of finding "314" in pi at a randomly selected index is
1/1000.
Given that "314" is found, the probability that it is at index 0 is 0.
And so on.
No matter how many matching digits you have, the probability that you have
found a particular starting point in a normal number is zero. There is NO
predictive value AT ALL for getting more matching digits as you increase
your test string length, because of how normal numbers work
You are figuring your probabilities incorrectly. The probability that
the correct position has been found is a function of the odds of
hitting upon any sequence that would produce the correct prediction
divided by those that would not.
Beyond this, the odds of picking a correct starting point for an
algorithm in a normal number may not be as "impossible" as you think.
You yourself admit that if space explorers happened to come across an
alien radio signal repeating the first 100 digits of pi over and over
again, it would be pretty good evidence of non-random bias and even of
ETI.
I did not admit that. I said it would probably indicate terrestrial origin
for the digits, and provided my reasoning.
You can't even really say that given your above reasoning. Why?
Because, according to you, no matter how many times the first 100
digits of pi were repeated, it would provide no predictive value of
bias at all - regardless of terrestrial or extraterrestrial origin. I
mean really, it could just be completely random chance or the product
of some downstream portion of pi. It would be meaningless basically -
according to you anyway.
< snip another irrelevant proof >
If this were not possible, there would be no way for SETI scientists
or anyone else to use pattern matching as evidence of bias production
of anything. For example, you evolutionists couldn't say that genetic
sequence homology is clear evidence of common origin - since such a
pattern could have been the result of purely random happenstance.
Hardly. This entire discussion is about normal numbers. You can't put a
normal number in DNA. DNA is finite. All finite strings correspond to
rational numbers. No rational number is normal. Take normal numbers (as pi
is considered to be) out of the picture, and we must move into an entirely
different area of math.
You forget that DNA could simply be a partial expression of a normal
number. The discussion here is about predicting what will come next
in a normal number based on a finite sequence. The fact that any
sequence matches, to include DNA, could actually be the result of
purely random happenstance - a subsection of a normal random sequence
generator. There is no way you can absolutely prove otherwise based
on analysis of DNA sequences or radiosignals themselves.
If my formula for pi happened to successfully predict a few million
digits in an incoming sequence, without error, would you actually call
this result "trivial" as a "prediction algorithm"? You might think
this result impossible for all practical purposes, but if such a
result were actually realized, it would be far from "trivial". I dare
say you would be all by yourself with this conclusion of yours if SETI
scientists actually did find such a match in a radiosignal proved to
be coming from outer space.
Yes, I would call it trivial.
Then you hold a very different position compared to various statements
of SETI scientists. I mean really, what would you say to the
following comment by Seth Shostak, a senior astronomer at the SETI
Institute:
"Perhaps the extraterrestrials will preface their message with a
string of prime numbers, or maybe the first fifty terms of the ever-
popular Fibonacci series.
Well, there's no doubt that such tags would convey intelligence."
- - Seth Shostak, How to Sort Signs of Artificial Life from the Real
Thing, SETI Thursday, Space.com, January 30, 2003
In short, it seems very much like you would be forced to call any such
signal, to include a million repeats of the first 100 digits of pi,
"trivial" and "worthless". Even if such a signal came from a
terrestrial source you wouldn't be able to detect it as a likely
biased signal because of your argument that it could have been the
product of any random origin in pi or any portion of any other normal
number - to include all random normal numbers.
Assuming the conjecture about irrational
numbers holds, those several million digits may be matched within pi at an
infinite number of places. They may be matched within e at an infinite
number of places. They may be matched within any other irrational number at
an infinite number of places. They may be matched within any real number at
an infinite number of places. Having no idea of the alien broadcast's
starting point, and no idea of their numeral system, and no idea how they
map signal modulation to their numeral system, there isn't much point in
getting excited about several million digits that I can match to just about
anything (and which I can't see arriving on a short snapshot of a narrow
band extrasolar signal, anyway).
You can't match several million digits of a randomly produced binary
sequence to just about anything. That's the whole point. Even though
there is a theoretical infinity to the possible number of different
matches to an infinite number of normal numbers, the infinity of a non-
match is much much more infinite. Therein lies the strength of
scientific induction.
On the other hands, if I found a short number of decimal digits matching the
starting sequence of pi lying around in a human computer network, I would
conclude it is an artifact simply because we have so many copies of short
starting sequences of pi in our data processing environment, and very few
copies of intermediate substrings of normal numbers, and most of them are
either in our conventional decimal or binary numeral systems, which we know
humans use, and all that makes for good odds.
So, basically, you can detect humans because you know something about
humans, but it would be impossible for you to detect an alien signal
because you don't know anything else about aliens? Try explaining that
to SETI scientists. I'm afraid you are quite alone on this one . . .
Sean Pitman
www.DetectingDesign.com
.
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