Re: Small probabilities

From: Iain Strachan <igd.strachan@gmail.com>
Date: Tue Nov 08 2005 - 11:08:48 EST

While everyone has got interested in the point-picking-from-a-line example,
I don't believe that anyone has really addressed Bill's question about low
probability "eliminating chance". One can get lost in the philosophy of
picking a point from an infinite number of points, without seeing the real
point (which was to argue against Dembski's notion that low probability can
eliminate chance). I'd like to re-address this point. This is not to say
that low probability can detect "design", which is a separate issue.

Low probability by itself cannot "eliminate chance", because if every event
is low probability, then one of them has to happen. Bill states that the
probability of picking any point is zero yet a point is picked. To make it
less abstract and in the realm of the real world, consider 200 coin tosses.
You can say that the probability of any sequence occurring is 6.6x10^(-61) (
= 2^(-200)), which is exceptionally unlikely. Yet you toss a coin 200 times
and lo and behold you've just witnessed an event with probability 6.6e-61.
Clearly the low probability cannot eliminate chance by itself.

Something like this happens with a technique I work with, called "Hidden
Markov Models", which are used commonly in speech recognition (though I'm
using them in a medical application). When these models are used to
recognise speech, the speech signal is segmented into a number of frames,
say 10ms long, and each frame is signal processed to produce a vector of
numbers (usually some frequency domain analysis). Then in order to recognise
a word, one constructs a probabilistic model that evaluates a probability
for the entire sequence of these vectors. Now, the probability for the whole
lot is simply the product of the probabilities for each individual one, so
if there are many hundreds of samples, then you get incredibly small
probabilities. Now here lies a problem: you would like to have a number of
different models for different words that you might want to recognise, eg
"one" "two" "three" etc. But the length of time people take to say "one"
might vary a lot, and clearly it takes longer to say "seven" than it does to
say "one". So because there are many more samples in the sequence when you
say "seven", it will of necessity have a much lower probability, just as a
sequence of 200 coin tosses has a lower probability than a sequence of 100.
The raw probability isn't sufficient to discriminate between the two. But
what you can compute is an expected value of the probability churned out by
the model. If you say "one" into a model that is designed to recognise
"seven", the probability will be many orders of magnitude lower than if you
said "seven" (because the probability assigned to each of the vectors in the
10ms time frames will be much lower) so you can do the discrimination, and
the confidence you have in rejecting it could be given by the ratio of the
two probabilities.

Likewise, with a sequence of coins, Dembski uses the notion of
compressibility. Any arbitrary sequence of 200 coin tosses will on average
require 200 "bits" to describe it. But if you describe it as 50 reps of
HTHH, then clearly you have a much shorter description. Say this can be
fitted into 25 bits in some specification language. Now the number of 25 bit
strings is 2^25 and the number of 200 coin toss sequences is 2^200, so it
follows that the probability of getting a 200 sequence of coin tosses
describable in 25 bits is 2^(-175) = 2.08x10^(-53). This low probability can
be used to "eliminate" chance - you don't expect to get that kind of
repetition in a sequence of coin tosses.

All the above is not to say that this detects design as such. There may be a
naturalistic explanation of why you got 50 reps of HTHH. But it does clearly
detect non-randomness.

Hope this answers some of your question.
Iain

On 11/6/05, Bill Hamilton <williamehamiltonjr@yahoo.com> wrote:
>
> I read Dembski's response to Henry Morris
> (http://www.calvin.edu/archive/asa/200510/0514.html)
> and noted that it raised an old issue I've harped on before: that you can
> specify a probability below which chance is eliminated. There is a
> counterexample given (among other places) in Davenport and Root's book
> "Random
> Signals and Noise" (McGraw Hill, probably sometime in the early 60's) that
> goes
> like this:
> Draw a line 1 inch long. Randomly pick a single point on that line. The
> probability of picking any point on the line is identically zero. Yet a
> point
> is picked. Am I missing something?
>
> I will probably unsubscribe this evening, because I don't really have time
> during the week to read this list. However, I will watch the archive for
> responses and either resubscribe or resspond offline as appropriate.
>
> Bill Hamilton
> William E. Hamilton, Jr., Ph.D.
> 586.986.1474 (work) 248.652.4148 (home) 248.303.8651 (mobile)
> "...If God is for us, who is against us?" Rom 8:31
>
>
>
>
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Received on Tue Nov 8 11:11:26 2005

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