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Unformatted text preview: An introduction to
information theory and
entropy Tom Carter http://Cogs.csustan .edu/~ tom/SFI—CSSS Complex Systems Summer School June,2002 Our general topics: @ Measuring complexity @ Some probability background @ Basics of information theory @ Some entropy theory @ The Gibbs inequality @ A simple physical example (gases) @ Shannon’s communication theory @ Application to Biology (analyzing
genomes) @ Some other measures @ Some additional material @ Examples using Bayes’ Theorem @ Analog channels @ Application to Physics (lasers) @ References The quotes to @ Science, wisdom, and counting @ Being different — or random @ Surprise, information, and miracles @ Information (and hope) @ H (or S) for Entropy @ Thermodynamics @ Language, and putting things together
@ Tools Science, _wisdom, and
counting a “Science is organized knowledge. Wisdom is
organized life.”  Immanuel Kant “My own suspicion is that the universe is not
only stranger than we suppose, but stranger
than we can suppose.” — John Haldane “Not everything that can be counted counts,
and not everything that counts can be
counted!’ — Albert Einstein (1879—1955) “The laws of probability, so true in general,
so fallacious in particular  Edward Gibbon Measuring complexity 9 0 Workers in the field of complexity face a classic problem: how can we tell that the
system we are looking at is actually a
complex system? (i.e., should we even be
studying this system? :) Of course, in practice, we will study the
systems that interest us, for whatever
reasons, so the problem identified above
tends not to be a real problem. On the
other hand, having chosen a system to
study, we might well ask “How complex is
this system?” In this more general context, we probably
want at least to be able to compare two
systems, and be able to say that system
A is more complex than system B.
Eventually, we probably would like to have
some sort of numerical rating scale. 0 Various approaches to this task have been
proposed, among them: 1. Human observation and (subjective)
rating 2. Number of parts or distinct elements
(what counts as a distinct part?) 3. Dimension (measured how?) 4. Number of parameters controlling the
system 5. Minimal description (in which
language?) 6. Information content (how do we
define/measure information?) 7. Minimal generator/constructor (what
machines/methods can we use?) 8. Minimum energy/time to construct
(how would evolution count?) a Most (if not all) of these measures will
actually be measures associated with a
model of a phenomenon. Two observers
(of the same phenomenon?) may develop
or use very different models, and thus
disagree in their assessments of the
complexity. For example, in a very simple
case, counting the number of parts is
likely to depend on the scale at which the
phenomenon is viewed (counting atoms is
different from counting molecules, cells,
organs, etc.). We shouldn’t expect to be able to come
up with a single universal measure of
complexity. The best we are likely to have
is a measuring system useful by a
particular observer, in a particular
context, for a particular purpose. My first focus will be on measures related
to how surprising or unexpected an
observation or event is. This approach
has been described as information theory. 7 Being different — or
random to “The man who follows the crowd will usually
get no further than the crowd. The man who
walks alone is likely to find himself in places
no one has ever been before. Creativity in
living is not without its attendant difficulties,
for peculiarity breeds contempt. And the
unfortunate thing about being ahead of your
time is that when people finally realize you
were right, they’ll say it was obvious all along.
You have two choices in life: You can dissolve
into the mainstream, or you can be distinct.
To be distinct is to be different. To be
different, you must strive to be what no one
else but you can be. ” Alan AshleyPitt “Anyone who considers arithmetical methods
of producing random digits is, of course, in a
state of sin.”  John von Neumann (1903—1957) ...
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 Fall '09
 Carter
 Differential Equations, Equations

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