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the complexity of the context in which even simple webs are embedded results in a
degradation of a scientist's ability to make predictions. The exact deterministic future
predicted by simple dynamical equations is replaced with a distribution of alternative
futures. In our examination of the DDM model we obtained analytic forms for the prob-
able decisions an individual can make, given uncertain and incomplete information.
In this way we do not predict a person's decision, but rather we provide a functional
form for the likelihood that the person will choose one future over another. This is a
rather pompous way of saying that we do not know what a person's decision will be,
but some decisions are more likely than others. We determine the odds and anticipate a
given choice, giving one possibility more weight than another, but we do predict not the
choice, but only its probability.
4.1.1
Simple random walks
Random walks date from two 1905 articles in the magazine
Nature
. In the first article
the biostatistician Karl Pearson posed a question that probably would not be published
today. The question frames a mathematics problem in a form whose solution would
provide insight into a variety of scientific phenomena. The question he asked was the
following:
A man starts from a point 0 and walks
l
yards in a straight line: he then turns through any angle
whatever and walks another
l
yards in a second straight line. He repeats this process
n
times. I
require the probability that after these
n
stretches he is at a distance between
r
and
r
+
dr
from
his starting point 0. The problem is one of considerable interest, but I have only succeeded in
obtaining an integrated solution for two stretches. I think, however, that a solution ought to be
found, if only in the form of a series of powers of 1
/
n
where
n
is large.
Lord Rayleigh immediately saw the generic form of Pearson's question and
responded in the same issue of
Nature
with the following answer:
This problem, proposed by Prof. Karl Pearson in the current number of
Nature
, is the same as
that of the composition of
n
iso-periodic vibrations of unit amplitude and of phase distributed at
random, considered in
Philosophical Magazine
, X. p. 73, 1880, XLVII. p. 246, 1899 (
Scientific
Papers
, I. p. 491; IV. p. 370). If
n
be very great, the probability sought is
2
n
−
1
e
−
r
2
/
n
rdr
Probably methods similar to those employed in the papers referred to would avail for the
development of an approximate expression applicable when
n
is only moderately great.
The ability to strip away irrelevant detail and focus on the important aspects of a prob-
lem is the hallmark of a good modeler and being right is the sign of a great modeler. In
this regard Lord Rayleigh's explanation of the random walk is reminiscent of Newton's
thinking about the speed of sound in air, at least in terms of the clarity of the approach.
The Bernoulli sequence
Of course, history has endorsed Lord Rayleigh's intuition concerning the equivalence
of a random walk to the superposition of acoustic modes with random phases. A simple
