[Paleopsych] Science Week: Mathematics: On Random Numbers
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Mathematics: On Random Numbers
http://scienceweek.com/2005/sw050527-6.htm
The following points are made by Gianpietro Malescio (Nature 2005
434:1073):
1) Making predictions is one of the main goals of science.
Traditionally this implies writing down, and solving, the equations
governing the system under investigation. When this method proves
impossible we often turn to a stochastic approach. The term
"stochastic" encompasses a variety of techniques that are based on a
common feature: using unpredictable entities --random numbers -- to
make predictions possible.
2) The origins of stochastic simulation can be traced to an experiment
performed in the 18th century by Georges Louis Leclerc, Comte de
Buffon ((1701-1788). Leclerc repeatedly tossed a needle at random on
to a board ruled with parallel straight lines. From his observations,
he derived the probability that the needle would intersect a line.
Subsequently, Pierre Simon de Laplace (1749-1827) saw in this
experiment a way to obtain a statistical estimate of pi.
3) Later, the advent of mechanical calculating machines allowed
numerical "experiments" such as that performed in 1901 by William
Thomson (Lord Kelvin) (1824-1907) to demonstrate the equipartition
theorem of the internal energy of a gas. Enrico Fermi (1901-1954) was
probably the first to apply statistical sampling to research problems,
while studying neutron diffusion in the early 1930s. During their work
on the Manhattan Project, Stanislaw Ulam (1909-1986), John von Neumann
(1903-1957) and Nicholas Metropolis (1915-1999) rediscovered Fermi's
method. They established the use of random numbers as a formal
methodology, generating the "Monte Carlo method" -- named after the
city famous for its gambling facilities. Today, stochastic simulation
is used to study a wide variety of problems (many of which are not at
all probabilistic), ranging from the economy to medicine, and from
traffic flow to biochemistry or the physics of matter.
4) When the temporal evolution of a system cannot be studied by
traditional means, random numbers can be used to generate an
"alternative" evolution. Starting with a possible configuration,
small, random changes are introduced to generate a new arrangement:
whenever this is more stable than the previous one, it replaces it,
usually until the most stable configuration is reached. Randomness
cannot tell us where the system likes to go, but allows the next best
thing: exploration of the space of the configurations while avoiding
any bias that might exclude the region of the possible solution. If we
are able to guess the probability distribution of the configurations,
then instead of conducting a uniform random search we can perform an
"importance" sampling, focusing our search on where the solution is
more likely to be found.
5) Optimization problems are often solved using stochastic algorithms
that mimic biological evolution. Although it may sound vaguely
unpleasant, we come from a random search. In nature, new genetic
variants are introduced through random changes (mutations) in the
genetic pool while additional variability is provided by the random
mixing of parent genes (by recombination). Randomness allows organisms
to explore new "designs" which the environment checks for fitness,
selecting those most suited to survival. But the optimal solution is
not found once and for ever. A continually changing environment means
evolution is an on-going process; it does not produce the "perfect"
organism, but rather a dynamic balance of myriad organisms within an
ecosystem.
References:
1. Metropolis, N. Los Alamos Sci. 15, 125-130 (1987)
2. Chaitin, G. J. Sci. Am. 232, 47-52 (1975)
3. Calude, C. S. & Chaitin, G. J. Nature 400, 319-320 (1999)
Nature http://www.nature.com/nature
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