At 02:39 PM 9/6/00 +0100, Richard wrote:
>From: Brian D Harper <bharper@postbox.acs.ohio-state.edu>
>
>
> >At 01:57 PM 8/29/00 +0100, Richard wrote:
>
>Brian, I'm going to take the liberty of addressing just part of your post
>for now. I think it will be difficult to make further progress until we
>settle on what we mean by "random".
OK
> >Yes, pattern can be very subjective of course, so let me
> >back up. The key issue with algorithmic randomness
> >is compressibility. A string that cannot be compressed
> >is said to be random.
>
>OK. This is the crucial point. I'm using the word random in the statistical
>(i.e. probabilistic) sense. Your algorithmic sense of the word is quite a
>different one. It is devoid of probabilistic content. Let's be clear in
>future which one we're using at any time.
OK, I'll try :).
> >Generally speaking, a pattern is
> >what one uses to compress the string. Generically, this
> >kind of agrees with the everyday notion of pattern, but
> >there may be exceptions. Humans seem particularly
> >good at "seeing" patterns which aren't really there. Likewise,
> >patterns in the sense we're speaking of here may not be
> >obvious at all.
> >
> >You mention above there are three possibilities. Actually
> >from this point of view there is a continuum of possibilities,
> >however, purposeful is not included. Purpose is subjective,
> >it can't be measured. The reason for the continuum is that
> >the notion of randomness admits of degrees. Something
> >may be very nearly incompressible but not quite. One
> >can make this notion precise by introducing per cent
> >compressible. Something only 5% compressible is
> >nearly random. Something 95% compressible is very
> >ordered.
>
>Yes, algorithmic randomness (as you've defined it) admits of different
>degrees. Statistical randomness does not, strictly speaking (although
>different degrees of variability are sometimes referred to as degrees of
>randomness).
Here is a point of contention, I think. I cannot understand referring
to an entire process as random when that process actually contains
many elements, only a few of which are actually random. Here I
mean random in the statistical sense.
Glenn Morton (who is no longer with us) was always talking about
Sierpinski's Gasket (SG). Here there is a stochastic element yet there
is also a pattern which always forms. In view of the latter, I
would be more inclined to call this deterministic. It is a matter
of degree, of course, but the overall effect seems deterministic to me.
From the algorithmic point of view, we would call this (SG) simple since
it is the result of a very short algorithm.
Anyway, using your notions it seems we would be forced to
refer to the engineering design process that I referred to earlier
as a random process since it contains a stochastic element.
But the main point for me in all of this is an attempt to
associate meaning, purpose etc. with a technical word
such as random, no matter how its defined. For example,
even if I were to concede for sake of argument that evolution
is a random process, I still would insist that this says nothing
about purpose.
> >In my view, evolution fits on this scale towards the upper
> >end. It is not random but highly complex.
>
>Having just established that algorithmic randomness is a matter of degree,
>what do you mean by "it's not random"? Relative to what?
Well, as usual I'm thinking of the results of the process. If one looks
at the fossil record, say, one will find all kinds of patterns. How
can we infer common descent without these? So, to clarify,
I would say that the results of evolution are clearly not (algorithmically)
random.
> >I don't know if this helps. I think our key difference is the
> >process versus result distinction.
> >
> >Oh, I almost forgot to mention the "dilemma" which Chaitin
> >solved. This is usually described something like this. Suppose
> >you have the following two strings which supposedly represent
> >tosses of a fair coin. 1=heads, 2=tails:
> >
> >(A) 0101010101010101010101010101010101010101010101010101010101010101
> >
> >(B) 1110101010010010101001010011000111100111100110011111000011010011
> >
> >Is either of the results more surprising than the other? From the point of
>view
> >of probability theory we cannot really say since the probability of any
> >specific
> >sequence is the same.
> >
> >What we learn from the above considerations is that if a number is
> >defined to be random in view of the process by which the number is
> >produced, then an obviously ordered result such as the first sequence
> >above must be considered random since the random process of coin
> >flipping produces that result with a probability equal to that of any
> >other specific sequence.
>
>To clarify, you are talking about statistical randomness here, right?
Yes.
> >In the words of Gregory Chaitin "The conclusion
> >is singularly unhelpful in distinguishing the random from the orderly.
>
>But what does Chaitin mean by random here? Since he hasn't yet introduced
>his algorithmic randomness, he must mean statistical randomness. So his
>objective is to detect statistical randomness.
No, that is not the objective. The objective is an objective and intrinsic
measure of order and complexity. Random lies at the upper end
of the complexity scale. How does the theory of probability distinguish
these if a stochastic process can produce both ordered and random
results? Its not like we're throwing out the theory of probability here.
What we're doing is carefully separating two issues. In this way,
the combination of the two theories will be made much stronger.
Let's go back to my coin tossing example. By combining algorithmic
information theory and probability theory we can overcome the
difficulty alluded to. What we can say is that a stochastic process
involving tossing a fair coin is extremely unlikely to produce any
patterned result. Not just the specific result (a) but *any* patterned
result. The unlikeliness of this can be made as arbitrarily close
to zero probability as you like by increasing the length of the sequence.
This gives justification for our being surprised by (a).
> >Clearly a more sensible definition of randomness is required, one that
> >does not contradict the intuitive concept of a 'patternless' number."
> >(Chaitin, G.J. (1975). "Randomness and Mathematical Proof," <Scientific
> >American>, 232 (May 1975):47-52.)
> >
> >As might be expected from the above, the "more sensible definition
> >of randomness" is Chaitin's :).
>
>But it is *not* sensible to redefine a word (totally changing its meaning)
>just because you are having difficulty detecting the presence of the
>phenomenon.
>
>Effectively, Chaitin is saying that we can't definitively detect statistical
>randomness, so he defines a measure which we *can* detect, and, confusingly,
>calls this randomness too. But what use is this measure, unless it's used to
>detect statistical randomness? Why do we care whether a set of data is
>patterned or not? And, if algorithmic randomness *is* used to detect
>statistical randomness, then it is really statistical randomness that we're
>interested in, and algorithmic randomness is just a marker for it.
>Also, what relevance does algorithmic randomness have to our original
>subject? We were talking about whether random variation and natural
>selection (or evolution) are random processes. But you can't talk about the
>algorithmic randomness of a process. You can talk about whether contemporary
>organisms exhibit a greater or lesser degree of algorithmic randomness (i.e.
>they are more or less patterned), but what's the relevance of that?
OK, I think things have become confused on the issue of process
and result. Speaking as an experimentalist :) I have to say that
the results are the most important aspect of science. Trying to detect
patterns in data and thus compress data into theories is, perhaps,
the most essential aspect of what scientists do. This particular
application is, BTW, the motivation behind the development of
algorithmic information theory for one of its co-founders, Solomonoff.
As an example, consider the mountains of data that Kepler got
from Tycho Brahe. The tremendous compression of that data into
first Kepler's and then Newton's laws tells us something.
The compression of available data into the theory of evolution is
not so great and not so elegant. Nor is it complete. This also
says something. There is no proof of course. Above I gave an
example of strong statements one could make if one combines
algorithmic and probability theory. But before you can make such
statements you have to know the probability distribution. To say
that there is one begs the question, of course.
Before I start rambling on too much, let me try to summarize
what I think are our two most important disagreements.
1) Calling the entire process of evolution random because it
contains some random elements. In particular referring to
selection as random. True, individual organisms and even
species are guaranteed nothing due to accidents. Nevertheless,
the overall effect of selection is one of ordering.
2) Associating any technical definition of randomness with
lack of meaning or purpose. This really is the most important
issue for me and is what prompted my original response to
Bertvan.
>[...]
> >phew, it's hard to find time for such demanding replies (i.e. those
> >requiring that I think :)
>
>Me too. Most of what passes for discussion here is just rote repetition of
>much-rehearsed arguments (and I include my own stuff). It's good to have a
>tough debate for a change. ;-)
>
>Richard Wein (Tich)
Brian Harper
Associate Professor
Mechanical Engineering
The Ohio State University
"One never knows, do one?"
-- Fats Waller
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