Transcript Document

• Randomness
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Retirement Monte Carlo: Better allowance for randomness
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Finally, a newer method for determining the
adequacy of a retirement plan is Monte Carlo
Simulation. This method has been gaining
popularity and is now employed by many
financial planners. Monte Carlo retirement
calculators allow users to enter savings,
income and expense information and run
simulations of retirement scenarios. The
simulation results show the probability that
the retirement plan will be successful.
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Heuristics in judgment and decision making - Misperception of randomness
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These sequences have exactly the
same probability, but people tend to
see the more clearly patterned
sequences as less representative of
randomness, and so less likely to
result from a random process
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Statistical randomness
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A numeric sequence is said to be 'statistically
random' when it contains no recognizable
patterns or regularities; sequences such as
the results of an ideal dice|dice roll, or the
digits of pi|π exhibit statistical
randomness.[http://news.uns.purdue.edu/UN
S/html4ever/2005/050426.Fischbach.pi.html
Pi seems a good random number generator –
but not always the best], Chad Boutin,
Purdue University
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Statistical randomness
Statistical randomness does not
necessarily imply true randomness, i.e.,
objective unpredictability.
Pseudorandomness is sufficient for many
uses, such as statistics, hence the name
statistical randomness.
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Statistical randomness
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Long stretches of the same numbers,
even those generated by truly random
processes, would diminish the local
randomness of a sample (it might only
be locally random for sequences of
10,000 numbers; taking sequences of
less than 1,000 might not appear
random at all, for example).
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Statistical randomness
A sequence exhibiting a pattern is not
thereby proved not statistically random.
According to principles of Ramsey theory,
sufficiently large objects must necessarily
contain a given substructure (complete
disorder is impossible).
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Statistical randomness - Tests
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The first tests for random numbers were
published by M.G. Kendall and Bernard
Babington Smith in the Journal of the
Royal Statistical Society in 1938. They
were built on statistical tools such as
Pearson's chi-squared test that were
developed to distinguish whether
experimental phenomena matched their
theoretical probabilities. Pearson
developed his test originally by showing
that a number of dice experiments by
W.F.R. Weldon did not display random
behavior.
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Statistical randomness - Tests
Kendall and Smith's original four tests
were statistical hypothesis
testing|hypothesis tests, which took as
their null hypothesis the idea that each
number in a given random sequence had
an equal chance of occurring, and that
various other patterns in the data should
be also distributed equiprobably.
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Statistical randomness - Tests
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* The 'frequency test', was very basic:
checking to make sure that there were
roughly the same number of 0s, 1s, 2s,
3s, etc.
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Statistical randomness - Tests
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* The 'serial test', did the same thing but
for sequences of two digits at a time (00,
01, 02, etc.), comparing their observed
frequencies with their hypothetical
predictions were they equally distributed.
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Statistical randomness - Tests
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* The 'poker test', tested for certain
sequences of five numbers at a time
(aaaaa, aaaab, aaabb, etc.) based on
hands in the game poker.
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Statistical randomness - Tests
* The 'gap test', looked at the
distances between zeroes (00 would be
a distance of 0, 030 would be a
distance of 1, 02250 would be a
distance of 3, etc.).
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Statistical randomness - Tests
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Kendall and Smith differentiated local
randomness from true randomness in
that many sequences generated with
truly random methods might not
display local randomness to a given
degree mdash; very large sequences
might contain many rows of a single
digit
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Statistical randomness - Tests
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As random number sets became more
and more common, more tests, of
increasing sophistication were used.
Some modern tests plot random digits as
points on a three-dimensional plane,
which can then be rotated to look for
hidden patterns. In 1995, the statistician
George Marsaglia created a set of tests
known as the diehard tests, which he
distributes with a CD-ROM of 5 billion
pseudorandom numbers.
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Statistical randomness - Tests
Pseudorandom number generators
require tests as exclusive verifications
for their randomness, as they are
decidedly not produced by truly
random processes, but rather by
deterministic algorithms
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Statistical randomness - Tests
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* The Monobit test treats each output
bit of the random number generator as
a coin flip test, and determine if the
observed number of heads and tails
are close to the expected 50%
frequency. The number of heads in a
coin flip trail forms a binomial
distribution.
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Statistical randomness - Tests
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* The Wald–Wolfowitz runs test tests
for the number of bit transitions
between 0 bits, and 1 bits, comparing
the observed frequencies with
expected frequency of a random bit
sequence.
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Statistical randomness - Tests
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* Autocorrelation
test
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Probability - Relation to randomness
In a determinism|deterministic
universe, based on Newtonian
mechanics|Newtonian concepts,
there would be no probability if all
conditions were known (Laplace's
demon), (but there are situations in
which chaos theory|sensitivity to
initial conditions exceeds our ability
to measure them, i.e
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Probability - Relation to randomness
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Probability theory is required to describe
quantum phenomena.Burgi, Mark (2010)
Interpretations of Negative Probabilities, p
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Kolmogorov complexity - Kolmogorov randomness
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Kolmogorov randomness – also called
algorithmic randomness – defines a
string (usually of bits) as being
randomness|random if and only if it is
shorter than any computer program
that can produce that string
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Kolmogorov complexity - Kolmogorov randomness
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This definition, unlike the definition of
randomness for a finite string, is not
affected by which universal machine is
used to define prefix-free Kolmogorov
complexity.
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Randomness
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'Randomness' means lack of pattern or
predictability in events.The Oxford
English Dictionary defines random as
Having no definite aim or purpose; not
sent or guided in a particular direction;
made, done, occurring, etc., without
method or conscious choice; haphazard.
Randomness suggests a non:wikt:order|order or non:wikt:coherence|coherence in a
sequence of symbols or Procedure
(term)|steps, such that there is no
intelligible pattern or combination.
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Randomness
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In these situations, randomness implies a
measure of uncertainty, and notions of
haphazardness are irrelevant.
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Randomness
The fields of mathematics, probability, and
statistics use formal definitions of randomness
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Randomness
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Randomness is often used in statistics to
signify well-defined statistical properties.
Monte Carlo methods, which rely on
random input, are important techniques in
science, as, for instance, in Scientific
computing|computational
science.[http://www.people.fas.harvard.ed
u/~junliu/Workshops/workshop2007/ Third
Workshop on Monte Carlo Methods], Jun
Liu, Professor of Statistics, Harvard
University
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Randomness
Random selection is a method of
selecting items (often called units)
from a population where the
probability of choosing a specific item
is the proportion of those items in the
population
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Randomness - History
In ancient history, the concepts of
chance and randomness were
intertwined with that of fate. Many
ancient peoples threw dice to determine
fate, and this later evolved into games of
chance. Most ancient cultures used
various methods of divination to attempt
to circumvent randomness and
fate.Handbook to life in ancient Rome by
Lesley Adkins 1998 ISBN 0-19-512332-8
page 279Religions of the ancient world
by Sarah Iles Johnston 2004 ISBN 0-6741
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Randomness - History
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In the 1888 edition of his book The Logic
of Chance John Venn wrote a chapter on
The conception of randomness that
included his view of the randomness of the
digits of the number Pi by using them to
construct a random walk in two
dimensions.Annotated readings in the
history of statistics by Herbert Aron David,
2001 ISBN 0-387-98844-0 page 115
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Randomness - History
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The early part of the twentieth century saw
a rapid growth in the formal analysis of
randomness, as various approaches to the
mathematical foundations of probability
were introduced. In the mid- to latetwentieth century, ideas of algorithmic
information theory introduced new
dimensions to the field via the concept of
algorithmic randomness.
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Randomness - History
Although randomness had often been
viewed as an obstacle and a nuisance for
many centuries, in the twentieth century
computer scientists began to realize that
the deliberate introduction of randomness
into computations can be an effective tool
for designing better algorithms. In some
cases such randomized algorithms
outperform the best deterministic methods.
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Randomness - In the physical sciences
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In the 19th century, scientists used the
idea of random motions of molecules
in the development of statistical
mechanics to explain phenomena in
thermodynamics and gas laws|the
properties of gases.
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Randomness - In the physical sciences
Hidden variable theory|Hidden
variable theories reject the view that
nature contains irreducible
randomness: such theories posit that
in the processes that appear random,
properties with a certain statistical
distribution are at work behind the
scenes, determining the outcome in
each case.
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Randomness - In biology
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The modern evolutionary synthesis
ascribes the observed diversity of life
to natural selection, in which some
random genetic mutations are
retained in the gene pool due to the
systematically improved chance for
survival and reproduction that those
mutated genes confer on individuals
who possess them.
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Randomness - In biology
The characteristics of an organism
arise to some extent deterministically
(e.g., under the influence of genes and
the environment) and to some extent
randomly. For example, the density of
freckles that appear on a person's skin
is controlled by genes and exposure to
light; whereas the exact location of
individual freckles seems random.
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Randomness - In biology
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Randomness is important if an animal
is to behave in a way that is
unpredictable to others. For instance,
insects in flight tend to move about
with random changes in direction,
making it difficult for pursuing
predators to predict their trajectories.
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Randomness - In mathematics
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The mathematical theory of probability arose
from attempts to formulate mathematical
descriptions of chance events, originally in
the context of gambling, but later in
connection with physics. Statistics is used to
infer the underlying probability distribution of
a collection of empirical observations. For the
purposes of simulation, it is necessary to
have a large supply of Random
sequence|random numbers or means to
generate them on demand.
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Randomness - In mathematics
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Algorithmic information theory studies,
among other topics, what constitutes a
random sequence. The central idea is that
a string of bits is random if and only if it is
shorter than any computer program that
can produce that string (Kolmogorov
randomness)—this means that random
strings are those that cannot be data
compression|compressed. Pioneers of
this field include Andrey Kolmogorov and
his student Per Martin-Löf, Ray
Solomonoff, and Gregory Chaitin.
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Randomness - In mathematics
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In mathematics, there must be an infinite
expansion of information for randomness
to exist. This can best be seen with an
example. Given a random sequence of
three-bit numbers, each number can have
one of only eight possible values:
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Randomness - In mathematics
Therefore, as the random sequence
progresses, it must recycle previous
values. To increase the information
space, another bit may be added to
each possible number, giving 16
possible values from which to pick a
random number. It could be said that
the random four-bit number sequence
is more random than the three-bit one.
This suggests that true randomness
requires an infinite expansion of the
information space.
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Randomness - In mathematics
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Randomness occurs in numbers such as
binary logarithm|log (2) and pi. The
decimal digits of pi constitute an infinite
sequence and never repeat in a cyclical
fashion. Numbers like pi are also
considered likely to be normal
number|normal, which means their digits
are random in a certain statistical sense.
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Randomness - In mathematics
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Pi certainly seems to behave this way. In
the first six billion decimal places of pi,
each of the digits from 0 through 9 shows
up about six hundred million times. Yet
such results, conceivably accidental, do
not prove normality even in base 10, much
less normality in other number bases.
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Randomness - In statistics
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In statistics, randomness is commonly
used to create simple random samples.
This lets surveys of completely random
groups of people provide realistic data.
Common methods of doing this include
drawing names out of a hat or using a
random digit chart. A random digit
chart is simply a large table of random
digits.
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Randomness - In information science
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In information science, irrelevant or
meaningless data is considered noise.
Noise consists of a large number of
transient disturbances with a
statistically randomized time
distribution.
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Randomness - In information science
In communication theory,
randomness in a signal is called noise
and is opposed to that component of
its variation that is causally
attributable to the source, the signal.
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Randomness - In information science
In terms of the development of
random networks, for communication
randomness rests on the two simple
assumptions of Paul Erdős and Alfréd
Rényi who said that there were a fixed
number of nodes and this number
remained fixed for the life of the
network, and that all nodes were equal
and linked randomly to each
other.Laszso Barabasi, (2003), Linked,
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Randomness - In finance
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The random walk hypothesis considers
that asset prices in an organized Market
(economics)|market evolve at random, in
the sense that the expected value of their
change is zero but the actual value may
turn out to be positive or negative. More
generally, asset prices are influenced by a
variety of unpredictable events in the
general economic environment.
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Randomness - Randomness versus unpredictability
Randomness, as opposed to
unpredictability, is an objective property.
Determinists believe it is an objective fact
that randomness does not in fact exist.
Also, what appears random to one
observer may not appear random to
another. Consider two observers of a
sequence of bits, when only one of whom
has the cryptographic key needed to turn
the sequence of bits into a readable
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Randomness - Randomness versus unpredictability
One of the intriguing aspects of
random processes is that it is hard to
know whether a process is truly
random. An observer may suspect that
there is some key that unlocks the
message. This is one of the
foundations of superstition, but also a
motivation for discovery in science and
mathematics.
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Randomness - Randomness versus unpredictability
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Under the cosmological hypothesis of
determinism, there is no randomness
in the universe, only
predictability|unpredictability, since
there is only one possible outcome to
all events in the universe
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Randomness - Randomness versus unpredictability
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Some mathematically defined sequences,
such as the decimals of pi mentioned
above, exhibit some of the same
characteristics as random sequences, but
because they are generated by a
describable mechanism, they are called
pseudorandom. To an observer who does
not know the mechanism, a
pseudorandom sequence is unpredictable.
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Randomness - Randomness versus unpredictability
chaos theory|Chaotic systems are
unpredictable in practice due to their
extreme sensitivity to initial conditions.
Whether or not they are unpredictable in
terms of computability theory
(computation)|computability theory is a
subject of current research. At least in
some disciplines of computability theory,
the notion of randomness is identified with
computational unpredictability.
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Randomness - Randomness versus unpredictability
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Individual events that are random may still
be precisely described en masse, usually
in terms of probability or expected value
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Randomness - Randomness and politics
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Random selection can be an official
method to resolve Tie (draw)|tied
elections in some
jurisdictions.Municipal Elections Act
(Ontario, Canada) 1996, c
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Randomness - Randomness and religion
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Randomness can be seen as conflicting
with the deterministic ideas of some
religions, such as those where the
universe is created by an omniscient deity
who is aware of all past and future events.
If the universe is regarded to have a
purpose, then randomness can be seen as
impossible. This is one of the rationales for
religious opposition to evolution, which
states that non-random selection is
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Randomness - Randomness and religion
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Hindu and Buddhist philosophies state
that any event is the result of previous
events, as reflected in the concept of
karma, and as such there is no such
thing as a random event or a first
event.
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Randomness - Randomness and religion
In some religious contexts,
procedures that are commonly
perceived as randomizers are used for
divination. Cleromancy uses the
casting of bones or dice to reveal what
is seen as the will of the gods.
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Randomness - Randomness and religion
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Followers of Discordianism, who venerate
Eris (mythology)|Eris the Greco-Roman
goddess of chaos, have a strong belief in
randomness and unpredictability.
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Randomness - Applications and use of randomness
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In most of its mathematical, political,
social and religious use, randomness
is used for its innate fairness and lack
of bias.
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Randomness - Applications and use of randomness
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'Political': Athenian democracy was based
on the concept of isonomia (equality of
political rights) and used complex
allotment machines to ensure that the
positions on the ruling committees that ran
Athens were fairly allocated.
Sortition|Allotment is now restricted to
selecting jurors in Anglo-Saxon legal
systems and in situations where fairness is
approximated by randomization, such as
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Randomness - Applications and use of randomness
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Throughout history, randomness has
been used for games of chance and to
select out individuals for an unwanted
task in a fair way (see drawing
straws).
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Randomness - Applications and use of randomness
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'Sports': Some sports, including American
Football, use coin tosses to randomly
select starting conditions for games or
seed (sports)|seed tied teams for
playoffs|postseason play. The National
Basketball Association uses a weighted
NBA Draft Lottery|lottery to order teams in
its draft.
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Randomness - Applications and use of randomness
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'Mathematical': Random numbers are
also used where their use is
mathematically important, such as
sampling for opinion polls and for
statistical sampling in quality control
systems. Computational solutions for
some types of problems use random
numbers extensively, such as in the
Monte Carlo method and in genetic
algorithms.
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Randomness - Applications and use of randomness
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'Medicine': Random allocation of a clinical
intervention is used to reduce bias in
controlled trials (e.g., randomized
controlled trials).
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Randomness - Applications and use of randomness
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'Religious': Although not intended to
be random, various forms of
divination such as cleromancy see
what appears to be a random event as
a means for a divine being to
communicate their will. (See also Free
will and Determinism).
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Randomness - Generating randomness
# Randomness coming from the
environment (for example, Brownian
motion, but also hardware random number
generators)
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Randomness - Generating randomness
# Randomness coming from the initial
conditions. This aspect is studied by chaos
theory and is observed in systems whose
behavior is very sensitive to small
variations in initial conditions (such as
pachinko machines and dice).
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Randomness - Generating randomness
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# Randomness intrinsically generated
by the system. This is also called
pseudorandomness and is the kind
used in pseudo-random number
generators. There are many algorithms
(based on arithmetics or cellular
automaton) to generate pseudorandom
numbers. The behavior of the system
can be determined by knowing the
random seed|seed state and the
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Randomness - Generating randomness
The many applications of randomness
have led to many different methods for
generating random data. These methods
may vary as to how unpredictable or
statistical randomness|statistically random
they are, and how quickly they can
generate random numbers.
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Randomness - Generating randomness
Before the advent of computational
random number generators, generating
large amounts of sufficiently random
numbers (important in statistics) required a
lot of work. Results would sometimes be
collected and distributed as random
number tables.
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Randomness - Randomness measures and tests
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There are many Randomness tests|practical
measures of randomness for a binary
sequence. These include measures based on
frequency, discrete transforms, and
complexity, or a mixture of these. These
include tests by Kak, Phillips, Yuen, Hopkins,
Beth and Dai, Mund, and Marsaglia and
Zaman.Terry Ritter, Randomness tests: a
literature survey.
[http://www.ciphersbyritter.com/RES/RANDT
EST.HTM ciphersbyritter.com]
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Randomness - Misconceptions and logical fallacies
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Popular perceptions of randomness
are frequently mistaken, based on
fallacious reasoning or intuitions.
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Randomness - A number is due
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This argument is, In a random selection of
numbers, since all numbers eventually
appear, those that have not come up yet
are 'due', and thus more likely to come up
soon
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Randomness - A number is cursed or blessed
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In a random sequence of numbers, a
number may be said to be cursed
because it has come up less often in
the past, and so it is thought that it will
occur less often in the future
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Randomness - A number is cursed or blessed
In nature, events rarely occur with
perfectly equal frequency, so observing
outcomes to determine which events are
more probable makes sense. It is
fallacious to apply this logic to systems
designed to make all outcomes equally
likely, such as shuffled cards, dice, and
roulette wheels.
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Randomness - Odds are never dynamic
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In the beginning of a scenario, one might
calculate the odds of a certain event. The
fact is, as soon as one gains more
information about that situation, they may
need to re-calculate the odds.
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Randomness - Odds are never dynamic
If we are told that a woman has two
children, and one of them is a girl, what
are the odds that the other child is also a
girl? Considering this new child
independently, one might expect the odds
that the other child is female are 1/2 (50%)
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Randomness - Odds are never dynamic
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This technique provides insights in other
situations such as the Monty Hall problem,
a game show scenario in which a car is
hidden behind one of three doors, and two
goats are hidden as booby prizes behind
the others
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Applications of randomness
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Randomness has
many uses in art,
statistics,
cryptography,
gambling, etc.
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Applications of randomness
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These uses have different randomness
requirements, which leads to the use
of different randomization methods.
For example, applications in
cryptography have strict
requirements, whereas other uses
(such as generating a quote of the day)
can use a looser standard of
randomness.
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Applications of randomness - Divination
Many ancient cultures saw natural
events as signs from the Deity|gods;
many attempted to discover the
intentions of the gods through various
sorts of divination. The underlying
theory was that the condition of, say, a
chicken's liver, was connected with,
perhaps, the dangerous storms or
military or political fortune.
Divination is still practiced and on
1
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Applications of randomness - Games
1
Unpredictable (by the humans involved)
numbers (usually taken to be
randomness|random numbers) were first
investigated in the context of gambling
developing, sometimes, pathological forms
like Apophenia#Gambling|apophenia
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Applications of randomness - Games
It has been alleged that some gaming
machines' software is deliberately biased
to prevent true randomness, in the
interests of maximizing their owners'
revenue; the history of biased machines in
the gambling industry is the reason
government inspectors attempt to
supervise the machines—electronic
equipment has extended the range of
supervision
1
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Applications of randomness - Games
1
Random draws are often used to make a
decision where no rational or fair basis
exists for making a deterministic decision,
or to make unpredictable moves.
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Applications of randomness - Athenian democracy
1
Fifth century BC Athenian democracy
developed out of a notion of isonomia
(equality of political rights), and
random selection was a principal way
of achieving this fairness.Herodotus
3.80 Greek democracy (literally
meaning rule by the people) was
actually run by the people:
administration was in the hands of
committees Sortition|allotted from
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Applications of randomness - Modern use
1
Allotment is today restricted mainly to
the selection of jurors in Anglo-Saxon
legal systems like the UK and United
States. Proposals have been made for
its use in government such as a new
constitution for
Iraqhttp://www.sortition.org.uk and
various proposals for Upper Houses
chosen by Sortition|allotment. (See
Reform of the House of
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Applications of randomness - Science
1
Random numbers have uses in physics
such as Noise (electronics)|electronic
noise studies, engineering, and operations
research. Many methods of statistical
analysis, such as the Resampling
(statistics)#Bootstrap|bootstrap method,
require random numbers. Monte Carlo
methods in physics and computer science
require random numbers.
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Applications of randomness - Science
Random numbers
are often used in
parapsychology as a
test of precognition.
1
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Applications of randomness - Statistical sampling
1
Elements of statistical practice that
depend on randomness include:
choosing a representative statistical
sample|sample, disguising the
Clinical trial protocol|protocol of a
study from a participant (see
randomized controlled trial) and
Monte Carlo method|Monte Carlo
simulation.
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Applications of randomness - Statistical sampling
1
These applications are useful in auditing
(for determining samples - such as
invoices) and experimental design (for
example in the creation of double-blind
trials).
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Applications of randomness - Analysis
1
Many experiments in
physics rely on a
statistical analysis of
their output
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Applications of randomness - Simulation
1
In many scientific and engineering fields,
computer simulations of real phenomena
are commonly used. When the real
phenomena are affected by unpredictable
processes, such as radio noise or day-today weather, these processes can be
simulated using random or pseudorandom numbers.
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Applications of randomness - Simulation
1
Automatic random number generators
were first constructed to carry out
computer simulation of physical
phenomena, notably simulation of
neutron transport in nuclear fission.
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Applications of randomness - Simulation
Pseudo-random numbers are
frequently used in simulation of
statistical events, a very simple
example being the outcome of tossing
a coin. More complicated situations are
simulation of population genetics, or
the behaviour of sub-atomic particles.
Such simulation methods, often called
stochastic process|stochastic
methods, have many applications in
1
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Applications of randomness - Simulation
1
Some more speculative projects, such as
the Global Consciousness Project, monitor
fluctuations in the randomness of numbers
generated by many hardware random
number generators in an attempt to predict
the scope of an event in near future. The
intent is to prove that large scale events
that are about to happen build up a
pressure which affects the RNGs.
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Applications of randomness - Cryptography
1
A ubiquitous use of unpredictable random
numbers is in cryptography which
underlies most of the schemes which
attempt to provide security in modern
communications (e.g., confidentiality,
authentication, electronic commerce, etc.).
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Applications of randomness - Cryptography
1
For example, if a user wants to use an
encryption algorithm, it is best that they
select a random number as the key
(cryptography)|key
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Applications of randomness - Cryptography
1
x' = ax +b (mod m), given only five
consecutive values
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Applications of randomness - Cryptography
Truly random numbers are absolutely
required to be assured of the theoretical
security provided by the one-time pad
mdash; the only provably unbreakable
encryption algorithm. Furthermore, those
random sequences cannot be reused and
must never become available to any
attacker, which implies a continuously
operable generator. See Venona for an
example of what happens when these
1
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Applications of randomness - Cryptography
For cryptographic purposes, one
normally assumes some upper limit
on the work an adversary can do
(usually this limit is astronomically
sized)
1
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Applications of randomness - Cryptography
Since a requirement in cryptography is
high entropy (i.e., unpredictability to an
attacker), any published random sequence
is a poor choice, as are such sequences
as the digits in an irrational number such
as the Golden ratio|φ or even in
transcendental numbers such as Pi|π, or e
(mathematical constant)|e
1
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Applications of randomness - Cryptography
1
Since most cryptographic applications
require a few thousand bits at most,
slow random number generators
serve wellmdash;if they are actually
random. This use of random
generators is important; many
informed observers believe every
computer should have a way to
generate true random numbers.
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Applications of randomness - Literature, music and art
1
Some aesthetic theories claim to be based
on randomness in one way or another.
Little testing is done in these situations,
and so claims of reliance on and use of
randomness are generally abstract.
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Applications of randomness - Literature, music and art
1
An example of a need for randomness sometimes
occurs in arranging items in an art exhibit
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Applications of randomness - Literature, music and art
Dadaism, as well as many other
movements in art and letters, has
attempted to accommodate and
acknowledge randomness in various
ways. Often people mistake order for
randomness based on lack of
information; e.g., Jackson Pollock's
drip
1
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Applications of randomness - Literature, music and art
paintings, Helen Frankenthaler's
abstractions (e.g., For E.M.). Thus, in
some theories of art, all art is random in
that it's just paint and canvas (the
explanation of Frank Stella's work).
1
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Applications of randomness - Literature, music and art
1
Similarly, the unexpected ending is part of
the nature of interesting literature. An
example of this is Denis Diderot's novel
Jacques le fataliste (literally: James the
Fatalist; sometimes referred to as Jacques
the Fatalist or Jacques the Servant and his
Master). At one point in the novel, Diderot
speaks directly to the reader:
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Applications of randomness - Literature, music and art
Now I, as the author of this novel might
have them set upon by thieves, or I might
have them rest by a tree until the rain
stops, but in fact they kept on walking and
then near night-fall they could see the light
of an inn in the distance.
1
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Applications of randomness - Literature, music and art
(not an exact quote). Diderot was
making the point that the novel (then a
recent introduction to European
literature) seemed random (in the sense
of being invented out of thin air by the
author). See also Eugenio Montale,
Theatre of the Absurd.
1
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Applications of randomness - Literature, music and art
1
Randomness in music is generally thought
to be postmodern music|postmodern,
including John Cage's chance derived
Music of Changes, Iannis Xenakis'
stochastic music, aleatoric music,
indeterminate music, or generative music.
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Applications of randomness - Other uses
Random numbers are also used in
situations where fairness is approximated
by randomization, such as selecting jurors
and selective service|military draft
lotteries. In the Book of Numbers (33:54),
Moses commands the Israelites to
apportion the land by lot.
1
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Applications of randomness - Other uses
1
Other examples include selecting, or
generating, a Random Quote of the
Day for a website, or determining
which way a villain might move in a
computer game.
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Pseudorandomness
A 'pseudorandom' process is a
process that appears to be
randomness|random but is not.
Pseudorandom sequences typically
exhibit statistical randomness while
being generated by an entirely
Deterministic system|deterministic
causal process. Such a process is
easier to produce than a genuinely
random one, and has the benefit that it
1
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Pseudorandomness
To generate truly random numbers
requires precise, accurate, and repeatable
system measurements of absolutely nondeterministic processes
1
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Pseudorandomness - History
1
The generation of random numbers has
many uses (mostly in statistics, for
random sampling (statistics)|sampling,
and simulation). Before modern
computing, researchers requiring
random numbers would either generate
them through various means (dice,
playing cards|cards, roulette|roulette
wheels, etc.) or use existing random
number tables.
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Pseudorandomness - History
The first attempt to provide
researchers with a ready supply of
random digits was in 1927, when the
Cambridge University Press
published a table of 41,600 digits
developed by L.H.C. Tippett. In 1947,
the RAND Corporation generated
numbers by the electronic simulation
of a roulette wheel; the results were
eventually published in 1955 as A
1
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Pseudorandomness - History
John von Neumann was a pioneer in
computer-based random number generators.
In 1949, Derrick Henry Lehmer invented the
linear congruential generator, used in most
pseudorandom number generators today.
With the spread of the use of computers,
algorithmic pseudorandom number
generators replaced random number tables,
and true random number generators
(hardware random number generators) are
used in only a few cases.
1
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Pseudorandomness - Almost random
1
A pseudorandom variable is a variable
which is created by a deterministic
procedure (often a computer program
or subroutine) which (generally) takes
random bits as input. The
pseudorandom string will typically be
longer than the original random string,
but less random (less Information
entropy|entropic, in the information
theory sense). This can be useful for
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Pseudorandomness - Almost random
1
Pseudorandom number generators are
widely used in such applications as
computer modeling (e.g., Markov chains),
statistics, experimental design, etc.
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Pseudorandomness - Pseudorandomness in computational complexity
In theoretical computer science, a
probability distribution|distribution is
'pseudorandom' against a class of
adversaries if no adversary from the
class can distinguish it from the
uniform distribution with significant
advantage.Oded Goldreich.
Computational Complexity: A
Conceptual Perspective. Cambridge
University Press. 2008.
1
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Pseudorandomness - Pseudorandomness in computational complexity
This notion of pseudorandomness is
studied in computational complexity theory
and has applications to cryptography.
1
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Pseudorandomness - Pseudorandomness in computational complexity
1
Formally, let S and T be finite sets and
let 'F' = be a class of functions. A
probability distribution|distribution 'D'
over S is ε-'pseudorandom' against 'F'
if for every f in 'F', the total variation
distance|statistical distance between
the distributions f(X), where X is
sampled from 'D', and f(Y), where Y is
sampled from the uniform distribution
(discrete)|uniform distribution on S, is
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Pseudorandomness - Pseudorandomness in computational complexity
1
In typical applications, the class 'F' describes a
model of computation with bounded resources
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Pseudorandomness - Pseudorandomness in computational complexity
and one is interested in designing
distributions 'D' with certain properties that
are pseudorandom against 'F'. The
distribution 'D' is often specified as the
output of a pseudorandom generator.
1
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Pseudorandomness - Cryptography
Pseudorandom sequences are
deterministic and reproducible; all
that is required in order to discover
and reproduce a pseudorandom
sequence is the algorithm used to
generate it and the initial Seed
(randomness)|seed
1
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Pseudorandomness - Cryptography
There are many examples in
cryptographic history of cyphers,
otherwise excellent, in which random
choices were not random enough and
security was lost as a direct
consequence
1
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Pseudorandomness - Cryptography
1
Users and designers of cryptography
are strongly cautioned to treat their
randomness needs with the utmost
care. Absolutely nothing has changed
with the era of computerized
cryptography, except that patterns in
pseudorandom data are easier to
discover than ever before.
Randomness is, if anything, more
important than ever.
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Pseudorandomness - Monte Carlo method simulations
A Monte Carlo method simulation is
defined as any method that utilizes
sequences of random numbers to perform
the simulation. Monte Carlo simulations
are applied to many topics including
quantum chromodynamics, cancer
radiation therapy, traffic flow, stellar
evolution and VLSI design. All these
simulations require the use of random
numbers and therefore pseudorandom
1
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Pseudorandomness - Monte Carlo method simulations
1
A simple example of how a computer
would perform a Monte Carlo
simulation is the calculation of Pi|π
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Pseudorandomness - Monte Carlo method simulations
1
# generate N pseudorandom
independent x and y-values on
interval [0,1)
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Pseudorandomness - Monte Carlo method simulations
1
# Number of pts within
the quarter circle x^2 +
y^2
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Kolmogorov randomness - Definition
The Kolmogorov complexity can be
defined for any mathematical object, but
for simplicity the scope of this article is
restricted to strings
1
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Kolmogorov randomness - Definition
We could, alternatively, choose an
encoding for Turing machines, where
an encoding is a function which
associates to each Turing Machine 'M'
a bitstring . If 'M' is a Turing Machine
which, on input w, outputs string x,
then the concatenated string w is a
description of x. For theoretical
analysis, this approach is more suited
for constructing detailed formal
1
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Kolmogorov randomness - Definition
'function'
GenerateFixedString()
1
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Kolmogorov randomness - Definition
If a description of s, d(s), is of minimal
length (i.e. it uses the fewest bits), it is
called a 'minimal description' of s. Thus,
the length of d(s) (i.e. the number of bits in
the description) is the 'Kolmogorov
complexity' of s, written K(s). Symbolically,
1
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Kolmogorov randomness - Definition
1
The length of the shortest description will
depend on the choice of description
language; but the effect of changing
languages is bounded (a result called the
invariance theorem).
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Kolmogorov randomness - Informal treatment
1
There are some description languages
which are optimal, in the following
sense: given any description of an
object in a description language, I can
use that description in my optimal
description language with a constant
overhead. The constant depends only
on the languages involved, not on the
description of the object, or the object
being described.
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Kolmogorov randomness - Informal treatment
1
Here is an example of an optimal description
language. A description will have two parts:
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Kolmogorov randomness - Informal treatment
1
* The first part describes another
description language.
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Kolmogorov randomness - Informal treatment
* The second part is a
description of the
object in that language.
1
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Kolmogorov randomness - Informal treatment
1
In more technical terms, the first part of a
description is a computer program, with
the second part being the input to that
computer program which produces the
object as output.
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Kolmogorov randomness - Informal treatment
'The invariance theorem follows:' Given
any description language L, the optimal
description language is at least as efficient
as L, with some constant overhead.
1
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Kolmogorov randomness - Informal treatment
'Proof:' Any description D in L can be
converted into a description in the optimal
language by first describing L as a
computer program P (part 1), and then
using the original description D as input to
that program (part 2). The
1
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Kolmogorov randomness - Informal treatment
The length of P is a constant that
doesn't depend on D. So, there is at
most a constant overhead, regardless
of the object described. Therefore, the
optimal language is universal up to this
additive constant.
1
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Kolmogorov randomness - A more formal treatment
'Theorem': If K1 and K2 are the
complexity functions relative to
Turing complete description
languages L1 and L2, then there is a
constant c – which depends only on the
languages L1 and L2 chosen – such
that
1
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Kolmogorov randomness - A more formal treatment
1
'Proof': By symmetry, it suffices to prove
that there is some constant c such that for
all strings s
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Kolmogorov randomness - A more formal treatment
1
Now, suppose there is a program in
the language L1 which acts as an
interpreter (computing)|interpreter
for L2:
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Kolmogorov randomness - A more formal treatment
'function'
InterpretLanguage('strin
g' p)
1
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Kolmogorov randomness - A more formal treatment
: Running
InterpretLanguage on
input p returns the
result of running p.
1
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Kolmogorov randomness - A more formal treatment
1
Thus, if 'P' is a program in L2 which is a
minimal description of s, then
InterpretLanguage('P') returns the string s.
The length of this description of s is the
sum of
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Kolmogorov randomness - A more formal treatment
1
# The length of the program InterpretLanguage,
which we can take to be the constant c.
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Kolmogorov randomness - History and context
1
Algorithmic information theory is the area
of computer science that studies
Kolmogorov complexity and other
complexity measures on strings (or other
data structures).
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Kolmogorov randomness - History and context
The concept and theory of Kolmogorov
Complexity is based on a crucial theorem
first discovered by Ray Solomonoff, who
published it in 1960, describing it in A
Preliminary Report on a General Theory of
Inductive Inference
[http://world.std.com/~rjs/z138.pdf
revision], Nov., 1960. as part of his
invention of algorithmic probability. He
gave a more complete description in his
1
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Kolmogorov randomness - History and context
Andrey Kolmogorov later multiple
discovery|independently published this
theorem in Problems Inform.
Transmission in 1965. Gregory Chaitin
also presents this theorem in J. ACM –
Chaitin's paper was submitted October
1966 and revised in December 1968,
and cites both Solomonoff's and
Kolmogorov's papers.
1
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Kolmogorov randomness - History and context
1
Kolmogorov used this theorem to
define several functions of strings,
including complexity, randomness,
and information.
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Kolmogorov randomness - History and context
1
The general consensus in the scientific
community, however, was to associate
this type of complexity with
Kolmogorov, who was concerned with
randomness of a sequence, while
Algorithmic Probability became
associated with Solomonoff, who
focused on prediction using his
invention of the universal prior
probability distribution
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Kolmogorov randomness - History and context
There are several other variants of
Kolmogorov complexity or algorithmic
information. The most widely used one
is based on self-delimiting programs,
and is mainly due to Leonid Levin
(1974).
1
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Kolmogorov randomness - History and context
1
An axiomatic approach to Kolmogorov
complexity based on Blum axioms
(Blum 1967) was introduced by Mark
Burgin in the paper presented for
publication by Andrey Kolmogorov
(Burgin 1982).
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Kolmogorov randomness - Basic results
It is not hard to see that the minimal
description of a string cannot be too much
larger than the string itself - the program
GenerateFixedString above that outputs s
is a fixed amount larger than s.
1
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Kolmogorov randomness - Basic results
1
'Theorem': There is a constant c
such that
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Kolmogorov randomness - Uncomputability of Kolmogorov complexity
'Theorem': There exist strings of
arbitrarily large Kolmogorov
complexity. Formally: for each n ∈ ℕ,
there is a string s with K(s) ≥
n.However, an s with K(s) = n needn't
exist for every n. For example, if n isn't
a multiple of 7 bits, no ASCII program
can have a length of exactly n bits.
1
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Kolmogorov randomness - Uncomputability of Kolmogorov complexity
'Proof:' Otherwise all infinitely many
possible strings could be generated by the
finitely manyThere are 1 + 2 + 22 + 23 + ...
+ 2n = 2n+1 minus; 1 different program
texts of length up to n bits; cf. geometric
series. If program lengths are to be
multiples of 7 bits, even fewer program
texts exist. programs with a complexity
below n bits.
1
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Kolmogorov randomness - Uncomputability of Kolmogorov complexity
1
'Theorem': K is not a computable function.
In other words, there is no program which
takes a string s as input and produces the
integer K(s) as output.
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Kolmogorov randomness - Uncomputability of Kolmogorov complexity
1
The following indirect proof|indirect
'proof' uses a simple Pascal
(programming language)|Pascal-like
language to denote programs; for sake
of proof simplicity assume its
description (i.e. an interpreter
(computing)|interpreter) to have a
length of bits.
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Kolmogorov randomness - Uncomputability of Kolmogorov complexity
1
Assume for contradiction there
is a program
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Kolmogorov randomness - Uncomputability of Kolmogorov complexity
1
which takes as input a string s and returns
K(s); for sake of proof simplicity, assume
its length to be bits.
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Kolmogorov randomness - Uncomputability of Kolmogorov complexity
'function'
GenerateComplexString(
)
1
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Kolmogorov randomness - Uncomputability of Kolmogorov complexity
'if' KolmogorovComplexity(s)
mcolor|#8080ff|1color|#8080ff|2color|#808
0ff|3color|#8080ff|4color|#8080ff|5color|#8
080ff|6color|#8080ff|7color|#8080ff|8color|
#8080ff|9color|#8080ff|10color|#8080ff|11c
olor|#8080ff|12color|#8080ff|13color|#8080
ff|14Main| Chain rule for Kolmogorov
complexitycitation needed|date=July
2014nobr|1=NL = 2Lnowrap|1=L = 0, 1, ...,
n minus; 1nobr|N0 + N1 + ..
1
https://store.theartofservice.com/the-randomness-toolkit.html
Kolmogorov randomness - Uncomputability of Kolmogorov complexity
'function'
GenerateProvablyParadox
icalString()
1
https://store.theartofservice.com/the-randomness-toolkit.html
Kolmogorov randomness - Uncomputability of Kolmogorov complexity
'return'
GenerateProvablyComplexString(n
0)
1
https://store.theartofservice.com/the-randomness-toolkit.html
Kolmogorov randomness - Uncomputability of Kolmogorov complexity
(note that n0 is hard-coded into the
above function, and the summand
log2(n0) already allows for its
encoding). The program
GenerateProvablyParadoxicalString
outputs a string s for which there exists
an L such that K(s)≥L can be formally
proved in 'S' with L≥n0. In particular,
K(s)≥n0 is true. However, s is also
described by a program of length
1
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Kolmogorov randomness - Uncomputability of Kolmogorov complexity
1
Similar ideas are used to
prove the properties of
Chaitin's constant.
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Kolmogorov randomness - Minimum message length
1
The minimum message length principle of
statistical and inductive inference and
machine learning was developed by Chris
Wallace (computer scientist)|C.S
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Kolmogorov randomness - Conditional versions
1
The conditional [Kolmogorov] complexity
of two strings K(x|y) is, roughly speaking,
defined as the Kolmogorov complexity of x
given y as an auxiliary input to the
procedure.
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Kolmogorov randomness - Conditional versions
There is also a length-conditional
complexity K(x|l(x)), which is the
complexity of x given the length of x as
known/input.
1
https://store.theartofservice.com/the-randomness-toolkit.html
Representativeness heuristic - Randomness
Irregularity and
local
representativeness
affect judgments of
randomness.
1
https://store.theartofservice.com/the-randomness-toolkit.html
Representativeness heuristic - Randomness
1
Things that do not appear to have any logical
sequence are regarded as representative of
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Representativeness heuristic - Randomness
1
randomness and thus more likely to occur. E.g.
THTHTH as a series of coin tosses would
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Representativeness heuristic - Randomness
Local representativeness is an
assumption wherein people rely on the
law of small numbers, whereby small
1
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Representativeness heuristic - Randomness
samples are perceived to represent their
population to the same extent as large samples
1
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Representativeness heuristic - Randomness
1
(Tversky and Kahneman, 1971). A small
sample which appears randomly
distributed would reinforce the belief,
under the assumption of local
representativeness, that the population
is randomly distributed. Conversely, a
small sample with a skewed distribution
would weaken this belief. If a coin toss is
repeated several times and the majority
of the results consists of 'heads', the
assumption of local representativeness
will cause the observer to believe the
coin is biased toward 'heads'.
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History of randomness
1
In ancient 'history', the concepts of chance
and 'randomness' were intertwined with
that of fate. Many ancient peoples threw
dice to determine fate, and this later
evolved into games of chance. At the
same time, most ancient cultures used
various methods of divination to attempt to
circumvent randomness and
fate.Handbook to Life in Ancient Rome,
Lesley Adkins, 1998 ISBN 0-19-512332-8
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History of randomness
1
The Chinese were perhaps the earliest
people to formalize odds and chance
3,000 years ago. The Ancient Greek
philosophy|Greek philosophers discussed
randomness at length, but only in nonquantitative forms. It was only in the
sixteenth century that Italian
mathematicians began to formalize the
odds associated with various games of
chance. The invention of modern calculus
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History of randomness
1
The early part of the twentieth century
saw a rapid growth in the formal
analysis of randomness, and
mathematical foundations for
probability were introduced, leading to
its axiomatization in 1933. At the same
time, the advent of quantum mechanics
changed the scientific perspective on
determinacy. In the mid to late 20thcentury, ideas of algorithmic
information theory introduced new
dimensions to the field via the concept of
algorithmic randomness.
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History of randomness - Antiquity to the Middle Ages
1
In ancient history, the concepts of chance
and randomness were intertwined with
that of fate. Pre-Christian people along the
Mediterranean threw dice to determine
fate, and this later evolved into games of
chance.What is Random?: Chance and
Order in Mathematics and Life, Edward J.
Beltrami, 1999, Springer ISBN 0-38798737-1 pp. 2-4 There is also evidence of
games of chance played by ancient
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History of randomness - Antiquity to the Middle Ages
Chinese, dating back to 2100
BC.Encyclopedia of Leisure and
Outdoor Recreation, John Michael
Jenkins, 2004 ISBN 0-415-25226-1 p.
194 The Chinese used dice before the
Europeans, and have a long history of
playing games of chance.Audacious
Angles of China, Elise Mccormick,
2007 ISBN 1-4067-5332-7 p. 158
1
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History of randomness - Antiquity to the Middle Ages
1
313 However, Western philosophy focused
on the non-mathematical aspects of
chance and randomness until the 16th
century.
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History of randomness - Antiquity to the Middle Ages
She cites studies by Daniel
Kahneman|Kahneman and Amos
Tversky|Tversky; these concluded that
statistical principles are not learned from
everyday experience because people do
not attend to the detail necessary to gain
such knowledge.Randomness, Deborah J
1
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History of randomness - Antiquity to the Middle Ages
1
Around 400 BC, Democritus presented
a view of the world as governed by the
unambiguous laws of order and
considered randomness as a
subjective concept that only
originated from the inability of
humans to understand the nature of
events
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History of randomness - Antiquity to the Middle Ages
1
He viewed randomness as a genuine
and widespread part of the world, but
as subordinate to necessity and
order.Aristotle's Physics: a Guided
Study, Joe Sachs, 1995 ISBN 0-81352192-0 p
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History of randomness - Antiquity to the Middle Ages
Around 300 BC Epicurus proposed the
concept that randomness exists by itself,
independent of human knowledge. He
believed that in the atomic world, atoms
would swerve at random along their paths,
bringing about randomness at higher
levels.Epicurus: an Introduction, John M.
Rist, 1972 ISBN 0-521-08426-1 p. 52
1
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History of randomness - Antiquity to the Middle Ages
1
For several centuries thereafter, the idea of chance
continued to be intertwined with fate
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History of randomness - Antiquity to the Middle Ages
1
Aristotle's classification of events into the
three classes: certain, probable and
unknowable was adopted by Roman
philosophers, but they had to reconcile it
with deterministic Christian teachings in
which even events unknowable to man
were considered to be predetermined by
God
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History of randomness - Antiquity to the Middle Ages
1
While he believed in the existence of
randomness, he rejected it as an
explanation of the end-directedness of
nature, for he saw too many patterns
in nature to have been obtained by
chance.The treatise on the divine
nature: Summa theologiae I, 1-13, by
Saint Thomas Aquinas, Brian J
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History of randomness - Antiquity to the Middle Ages
1
893Randomness,
Deborah J
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History of randomness - 17th–19th centuries
1
The work of Pascal and Fermat influenced
Gottfried Wilhelm Leibniz|Leibniz's work
on the infinitesimal calculus, which in turn
provided further momentum for the formal
analysis of probability and randomness.
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History of randomness - 17th–19th centuries
1
Three centuries later, the same concept was
formalized as algorithmic randomness by A
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History of randomness - 17th–19th centuries
The Doctrine of Chances, the first
textbook on probability theory was
published in 1718 and the field
continued to grow
thereafter.Schneider, Ivo (2005),
Abraham De Moivre, The Doctrine of
Chances (1718, 1738, 1756), in
Grattan-Guinness, I., Landmark
Writings in Western Mathematics
1640-1940, Amsterdam: Elsevier, p
1
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History of randomness - 17th–19th centuries
1
While the mathematical elite was making
progress in understanding randomness
from the 17th to the 19th century, the
public at large continued to rely on
practices such as fortune telling in the
hope of taming chance
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History of randomness - 17th–19th centuries
The term entropy, which is now a key
element in the study of randomness, was
coined by Rudolf Clausius in 1865 as he
studied heat engines in the context of the
second law of thermodynamics. Clausius
was the first to state entropy always
increases.Great physicists by William H.
Cropper 2004 ISBN page 93
1
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History of randomness - 17th–19th centuries
1
From the time of Isaac Newton|Newton
until about 1890, it was generally believed
that if one knows the initial state of a
system with great accuracy, and if all the
forces acting on the system can be
formulated with equal accuracy, it would
be possible, in principle, to make
predictions of the state of the universe for
an infinitely long time
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History of randomness - 17th–19th centuries
1
During the 19th century, as probability
theory was formalized and better
understood, the attitude towards
randomness as nuisance began to be
questioned. Goethe wrote:
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History of randomness - 17th–19th centuries
1
is built from necessities
and randomness;
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History of randomness - 17th–19th centuries
The words of Goethe proved
prophetic, when in the 20th century
randomized algorithms were
discovered as powerful tools.Design
and Analysis of Randomized
Algorithms, Juraj Hromkovič, 2005
ISBN 3-540-23949-9 p
1
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History of randomness - 20th century
1
During the 20th century, the five main
Probability
interpretations|interpretations of
probability theory (e.g., classical,
logical, frequency, propensity and
subjective) became better understood,
were discussed, compared and
contrasted.[http://plato.stanford.edu/en
tries/probability-interpret/ Stanford
Encyclopedia of Philosophy] A
significant number of application areas
were developed in this century, from
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History of randomness - 20th century
This approach led him to suggest a
definition of randomness that was later
refined and made mathematically rigorous
by Alonzo Church by using computable
functions in 1940.Companion
Encyclopedia of the History and
Philosophy Volume 2, Ivor GrattanGuinness 0801873975 p
1
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History of randomness - 20th century
Alberto Coffa, Randomness and
knowledge, in PSA 1972: proceedings of
the 1972 Biennial Meeting Philosophy of
Science Association, Volume 20, Springer,
1974 ISBN 90-277-0408-2 p
1
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History of randomness - 20th century
1
The advent of quantum mechanics in
the early 20th century and the
formulation of the Heisenberg
uncertainty principle in 1927 saw the
end to the Newtonian mindset among
physicists regarding the determinacy of
nature. In quantum mechanics, there is
not even a way to consider all
observable elements in a system as
random variables at once, since many
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History of randomness - 20th century
Karl Popper echoed the same
sentiment as Aristotle in viewing
randomness as subordinate to order
when he wrote that the concept of
chance is not opposed to the concept
of law in nature, provided one
considers the laws of chance.Karl
Popper, The Logic of Scientific
Discovery p
1
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History of randomness - 20th century
1
Hence if a stochastic system has entropy
zero it has no randomness and any
increase in entropy increases randomness
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History of randomness - 20th century
Martingale (probability
theory)|Martingales for the study of
chance and betting strategies were
introduced by Paul Lévy
(mathematician)|Paul Lévy in the 1930s
and were formalized by Joseph L
1
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History of randomness - 20th century
1
Chaitin's Omega number later related
randomness and the halting
probability for programs.Thinking
about Gödel and Turing, Gregory J
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History of randomness - 20th century
1
McCamy 2001 ISBN 0-387-98993-5 page
20The effortless economy of science?
by Philip Mirowski 2004 ISBN 0-82233322-8 page 255 In his 1997 he defined
seven states of randomness ranging
from mild to wild, with traditional
randomness being at the mild end of
the scale.Fractals and scaling in
finance by Benoît Mandelbrot 1997
ISBN 0-387-98363-5 pages 136-142
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History of randomness - 20th century
Despite mathematical advances,
reliance on other methods of dealing
with chance, such as fortune telling
and astrology continued in the 20th
century
1
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History of randomness - 20th century
During the 20th century, Limit
(mathematics)|limits in dealing with
randomness were better understood
1
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History of randomness - 20th century
In the late 1970s and early 1980s,
computer science|computer
scientists began to realize that the
deliberate introduction of randomness
into computations can be an effective
tool for designing better algorithms.
In some cases, such randomized
algorithms outperform the best
deterministic methods.Design and
Analysis of Randomized Algorithms,
Juraj Hromkovič 2005 ISBN 3-54023949-9 p. 4
1
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Random ballot - Randomness in other electoral systems
1
#It is often observed that candidates
who are placed in a high position on
the ballot-paper will receive extra
votes as a result, from voters who are
apathetic (especially in elections with
compulsory voting) or who have a
strong preference for a party but are
indifferent among individual
candidates representing that party
(when there are two or more)
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Random ballot - Randomness in other electoral systems
#In some Single Transferable Vote
(STV) systems of proportional
representation, an elected candidate's
surplus of votes over and above the
Droop quota|quota is transferred by
selecting the required number of ballot
papers at random
1
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Algorithmic randomness
1
Intuitively, an 'algorithmically random
sequence' (or 'random sequence') is an
infinite Sequence#Infinite sequences in
theoretical computer science|sequence
of binary digits that appears random to
any algorithm. The notion can be
applied analogously to sequences on
any finite alphabet (e.g. decimal digits).
Random sequences are key objects of
study in algorithmic information theory.
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Algorithmic randomness
As different types of algorithms are
sometimes considered, ranging from
algorithms with specific bounds on
their running time to algorithms which
may ask questions of an oracle, there
are different notions of randomness.
The most common of these is known as
'Martin-Löf randomness' (or '1randomness'), but stronger and weaker
forms of randomness also exist. The
1
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Algorithmic randomness
1
Because infinite sequences of binary
digits can be identified with real
numbers in the unit interval, random
binary sequences are often called
'random real numbers'. Additionally,
infinite binary sequences correspond
to characteristic functions of sets of
natural numbers; therefore those
sequences might be seen as sets of
natural numbers.
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Algorithmic randomness
1
The class of all Martin-Löf random (binary)
sequences is denoted by RAND or MLR.
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Algorithmic randomness - History
This contrasts with the idea of
randomness in probability; in that
theory, no particular element of a
sample space can be said to be
random.
1
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Algorithmic randomness - History
1
The thesis that the definition of Martin-Löf
randomness correctly captures the
intuitive notion of randomness has been
called the 'Martin-Löf–Chaitin Thesis'; it is
somewhat similar to the Church–Turing
thesis.Jean-Paul Delahaye,
[http://books.google.com/books?id=EDoXd
ozqYQCpg=PA145source=gbs_toc_rcad=0_
0 Randomness, Unpredictability and
Absence of Order], in Philosophy of
Probability, p
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Algorithmic randomness - Three equivalent definitions
1
Martin-Löf's original definition of a
random sequence was in terms of
constructive null covers; he defined a
sequence to be random if it is not
contained in any such cover
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Algorithmic randomness - Three equivalent definitions
1
* 'Kolmogorov complexity' (Schnorr
1973, Levin 1973): Kolmogorov
complexity can be thought of as a
lower bound on the algorithmic
compressibility of a finite sequence
(of characters or binary digits)
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Algorithmic randomness - Three equivalent definitions
:An infinite sequence S is Martin-Löf
random if and only if there is a constant c
such that all of S's finite Prefix (computer
science)|prefixes are c-incompressible.
1
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Algorithmic randomness - Three equivalent definitions
1
* 'Constructive null covers' (Martin-Löf 1966): This is
Martin-Löf's original definition
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Algorithmic randomness - Three equivalent definitions
1
:A sequence is defined to be Martin-Löf
random if it is not contained in any
G_\delta set determined by a
constructive null cover.
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Algorithmic randomness - Three equivalent definitions
1
* 'Constructive martingales' (Schnorr
1971): A Martingale (probability
theory)|martingale is a function
d:\^*\to[0,\infty) such that, for all finite
strings w, d(w) = (d(w^\smallfrown 0) +
d(w^\smallfrown 1))/2, where
a^\smallfrown b is the concatenation of
the strings a and b
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Algorithmic randomness - Three equivalent definitions
1
:A sequence is Martin-Löf random if and only if no
constructive martingale succeeds on it.
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Algorithmic randomness - Three equivalent definitions
1
:(Note that the definition of martingale
used here differs slightly from the one
used in probability theory
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Algorithmic randomness - Interpretations of the definitions
The Kolmogorov complexity
characterization conveys the intuition that
a random sequence is incompressible: no
prefix can be produced by a program
much shorter than the prefix.
1
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Algorithmic randomness - Interpretations of the definitions
The null cover characterization
conveys the intuition that a random
real number should not have any
property that is “uncommon”
1
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Algorithmic randomness - Interpretations of the definitions
The martingale characterization
conveys the intuition that no effective
procedure should be able to make
money betting against a random
sequence
1
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Algorithmic randomness - Properties and examples of Martin-Löf random sequences
1
* Chaitin's constant|Chaitin's halting probability Ω is
an example of a random sequence.
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Algorithmic randomness - Properties and examples of Martin-Löf random sequences
* RANDc (the Complement (set
theory)|complement of RAND) is a
Measure (mathematics)|measure 0
subset of the set of all infinite
sequences. This is implied by the fact
that each constructive null cover
covers a measure 0 set, there are only
countable|countably many constructive
null covers, and a countable union of
measure 0 sets has measure 0. This
1
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Algorithmic randomness - Properties and examples of Martin-Löf random sequences
* Every random
sequence is Normal
number|normal.
1
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Algorithmic randomness - Properties and examples of Martin-Löf random sequences
* There is a constructive null cover of
RANDc. This means that all effective tests
for randomness (that is, constructive null
covers) are, in a sense, subsumed by this
universal test for randomness, since any
sequence that passes this single test for
randomness will pass all tests for
randomness. (Martin-Löf 1966)
1
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Algorithmic randomness - Properties and examples of Martin-Löf random sequences
* There is a universal constructive
martingale 'd'. This martingale is universal
in the sense that, given any constructive
martingale d, if d succeeds on a
sequence, then 'd' succeeds on that
sequence as well. Thus, 'd' succeeds on
every sequence in RANDc (but, since 'd' is
constructive, it succeeds on no sequence
in RAND). (Schnorr 1971)
1
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Algorithmic randomness - Properties and examples of Martin-Löf random sequences
1
* The class RAND is a \Sigma^0_2 subset
of Cantor space, where \Sigma^0_2 refers
to the second level of the arithmetical
hierarchy. This is because a sequence S is
in RAND if and only if there is some open
set in the universal effective null cover that
does not contain S; this property can be
seen to be definable by a \Sigma^0_2
formula.
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Algorithmic randomness - Properties and examples of Martin-Löf random sequences
1
* There is a random sequence which is
\Delta^0_2, that is, computable
relative to an oracle for the Halting
problem. (Schnorr 1971) Chaitin's
Chaitin's constant|Ω is an example of
such a sequence.
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Algorithmic randomness - Properties and examples of Martin-Löf random sequences
1
* No random sequence is Decidability
(logic)|decidable, computably
enumerable, or computably
enumerable|co-computablyenumerable. Since these correspond
to the \Delta_1^0, \Sigma_1^0, and
\Pi_1^0 levels of the arithmetical
hierarchy, this means that \Delta_2^0
is the lowest level in the arithmetical
hierarchy where random sequences
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Algorithmic randomness - Properties and examples of Martin-Löf random sequences
1
* Every sequence is Turing reducible
to some random sequence. (Kučera
1985/1989, Péter Gács|Gács 1986).
Thus there are random sequences of
arbitrarily high Turing degree.
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Algorithmic randomness - Relative randomness
1
As each of the equivalent definitions of a
Martin-Löf random sequence is based on
what is computable by some Turing
machine, one can naturally ask what is
computable by a Turing oracle machine
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Algorithmic randomness - Relative randomness
An important result relating to relative
randomness is Michiel van Lambalgen|van
Lambalgen's theorem, which states that if
C is the sequence composed from A and B
by interleaving the first bit of A, the first bit
of B, the second bit of A, the second bit of
B, and so on, then C is algorithmically
random if and only if A is algorithmically
random, and B is algorithmically random
relative to A
1
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Algorithmic randomness - Stronger than Martin-Löf randomness
However, the halting probability
Chaitin's constant|Ω is \Delta^0_2 and
1-random; it is only after 2-randomness
is reached that it is impossible for a
random set to be \Delta^0_2.
1
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Algorithmic randomness - Weaker than Martin-Löf randomness
1
Additionally, there are several notions of
randomness which are weaker than
Martin-Löf randomness. Some of these
are weak 1-randomness, Schnorr
randomness, computable randomness,
partial computable randomness.
Additionally, Kolmogorov-Loveland
randomness is known to be no stronger
than Martin-Löf randomness, but it is not
known whether it is actually weaker.
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Algorithmic randomness - Weaker than Martin-Löf randomness
1
At the opposite end of the randomness
spectrum there is the notion of a Ktrivial set. These sets are antirandom
in that all initial segment have the
least K-complexity up to a constant b.
That is, K(w) \leq K(|w|) + b for each
initial segment w.
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Per Martin-Löf - Randomness and Kolmogorov complexity
1
In 1964 and 1965, Martin-Löf studied in
Moscow under the supervision of Andrei
N. Kolmogorov. He wrote a 1966 article
On the definition of random sequences
that gave the first suitable definition of a
random sequence.
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Per Martin-Löf - Randomness and Kolmogorov complexity
1
The thesis that the definition of Martin-Löf
randomness correctly captures the
intuitive notion of randomness has been
called the Martin-Löf-Gregory
Chaitin|Chaitin Thesis; it is somewhat
similar to the Church–Turing thesis.JeanPaul Delahaye,
[http://books.google.com/books?id=EDoXd
ozqYQCpg=PA145source=gbs_toc_rcad=0_
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Per Martin-Löf - Randomness and Kolmogorov complexity
1
flipping a coin to produce each bit will
randomly produce a string), algorithmic
randomness refers to the string itself
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Per Martin-Löf - Randomness and Kolmogorov complexity
1
An algorithmically random sequence
is an infinite sequence of characters,
all of whose prefixes (except possibly
a finite number of exceptions) are
strings that are close to
algorithmically random (their length
is within a constant of their
Kolmogorov complexity).
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Randomization function - Randomness
In theory, randomization functions are
assumed to be truly random, and yield an
unpredictably different function every time
the algorithm is executed
1
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Randomization function - Randomness
1
In practice, however, the main concern is
that some bad cases for the deterministic
algorithm may occur in practice much
more often than it would be predicted by
chance
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Fooled by Randomness
1
'Fooled by Randomness: The Hidden Role
of Chance in Life and in the Markets' is a
book by Nassim Nicholas Taleb that deals
with the fallibility of human knowledge.
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Fooled by Randomness - Reaction
The book was selected by Fortune
(magazine)|Fortune as one of the 75 Smartest
Books of All Time
1
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Fooled by Randomness - Thesis
1
Taleb sets forth the idea that modern
humans are often unaware of the
existence of randomness. They tend to
explain random outcomes as nonrandom.
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Fooled by Randomness - Thesis
1
# overestimate causality, e.g., they see
elephants in the clouds instead of
understanding that they are in fact
randomly shaped clouds that appear to
our eyes as elephants (or something
else);
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Fooled by Randomness - Thesis
Other misperceptions of
randomness that are discussed
include:
1
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Fooled by Randomness - Thesis
1
* Survivorship bias. We see the winners
and try to learn from them, while forgetting
the huge number of losers.
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Fooled by Randomness - Thesis
1
* Skewed distributions
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Fooled by Randomness - Editions
1
*In 2004, TEXERE published a
revamped second edition.
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Fooled by Randomness - Editions
1
*In 2005, Random House published a
softback edition with more changes.
(ISBN ISBN 1-58799-190-X, New York :
Random house, 2005)
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Fooled by Randomness - Editions
*In 2005, a French version
appeared, with many unique
changes.
1
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Fooled by Randomness - Editions
*The book has been translated into 20
languages,[http://www.fooledbyrandomnes
s.com fooled by randomness] and is
reported to have sold over half a million
copies.
1
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Fooled by Randomness - Editions
1
*Further editions have been published
by Penguin (softback, May 2007) and
Random House (hardback, October
2008.)
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Randomness tests
1
If a selected set of data fails the tests,
then parameters can be changed or
other randomized data can be used
which does pass the tests for
randomness.
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Randomness tests
There are many practical measures of
randomness for a binary sequence. These
include measures based on statistical
tests, Hadamard transform|transforms,
and complexity or a mixture of these. The
use of Hadamard transform to measure
randomness was proposed by Subhash
Kak|S. Kak and developed further by
Phillips, Yuen, Hopkins, Beth and Dai,
Mund, and George Marsaglia|Marsaglia
1
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Randomness tests
1
Terry Ritter, Randomness
tests: a literature survey,
webpage:
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Randomness tests - Background
1
The issue of randomness is an important
philosophical and theoretical question.
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Randomness tests - Background
1
Many random number generators in use
today generate what are called random
sequences but they are actually the result
of prescribed algorithms and so they are
called pseudo-random number generators.
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Randomness tests - Background
1
These generators do not always generate
sequences which are sufficiently random
and generate very repetitive patterns such
as the infamous RANDU which fails many
randomness tests dramatically including
the Spectral Test.
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Randomness tests - Background
The use of an ill-conceived random
number generator will result in
invalid experiments, due to the lack of
randomness.
1
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Randomness tests - Background
1
Tests for randomness are not restricted
to analysing the output of pseudorandom number generators, they can
also be used to determine whether a
set of data has a recognisable pattern
to it.
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Randomness tests - Background
For example Wolfram used
randomness tests on the output of Rule
30 to examine its potential for
generating random numbers, though it
was shown to have an effective key
size far smaller than its actual size and
to perform poorly on a chi-squared
test..
1
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Randomness tests - Specific tests for randomness
1
Several of these tests, which are of linear
complexity, provide spectral measures of
randomness. T. Beth and Z-D. DaiBeth, T.
and Z-D. Dai. 1989. On the Complexity of
Pseudorandomness|Pseudo-Random
Sequences -- or: If You Can Describe a
Sequence It Can't be Random. Advances
in Cryptology -- EUROCRYPT '89. 533543. Springer-Verlag showed that
Kolmogorov complexity and linear
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Randomness tests - Specific tests for randomness
Y. Wang.Y.ongge Wang 1999. Linear
complexity versus pseudorandomness: on
Beth and Dai's result. In: Proc. Asiacrypt
99, pages 288--298. LNCS 1716, Springer
Verlag On the other hand, Y. Wang
showed that for Martin-Lof random
sequences, the Kolmogorov complexity is
essentially the same as linear complexity.
1
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Randomness tests - Specific tests for randomness
1
These practical tests make it possible
to compare and contrast the
randomness of string (computer
science)|strings. On probabilistic
grounds, all strings, say of length 64,
have the same randomness. However,
consider the two following strings:
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Randomness tests - Specific tests for randomness
1
Using linear Hadamard spectral tests
(see Hadamard transform), the first of
these sequences will be found to be of
much less randomness than the
second one, which agrees with
intuition.
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Online poker - Randomness of the shuffle
Many critics question whether the
operators of such games - especially
those located in jurisdictions separate
from most of their players - might be
engaging in fraud themselves.
1
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Online poker - Randomness of the shuffle
Internet discussion forums are rife with
allegations of non-random card dealing,
possibly to favour house-employed players
or computer game bot|bots (poker-playing
software disguised as a human opponent),
or to give multiple players good hands
thus increasing the bets and the rake, or
simply to prevent new players from losing
so quickly that they become discouraged
1
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Online poker - Randomness of the shuffle
Many players claim to see lots of bad
beats with large hands pitted against
others all too often at a rate that seems to
be a lot more common than in live games.
However, this might actually be caused by
the higher hands per hour at on-line
cardrooms. Since online players get to
see more hands, their likelihood of seeing
more improbable bad beats or randomly
large pots is similarly increased.
1
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Online poker - Randomness of the shuffle
1
Many new players fail to understand
that there is a great deal of variance in
poker (like most card games) whether
the game is played live or online. On
the other hand, newcomers who
experience a run of bad luck are more
likely to suspect foul play when simple
variance is the most likely cause.
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Online poker - Randomness of the shuffle
1
Many online poker sites are certified
by bodies such as the Kahnawake
Gaming Commission and major
auditing firms like
PricewaterhouseCoopers to review the
fairness of the random number
generator,[http://www.pokerstars.co
m/poker/rng/ Random number
generator analysis], shuffle, and
payouts for some sites.
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