Everything about Information Theory totally explained
Information theory is a branch of
applied mathematics and
engineering involving the quantification of
information. Historically, information theory was developed to find fundamental limits on compressing and reliably
communicating data. Since its inception it has broadened to find applications in many other areas, including
statistical inference,
networks other than communication networks as in
neurobiology, the evolution and function of molecular codes, model selection in ecology, thermal physics,
quantum computing, plagiarism detection and other forms of
data analysis.
A key measure of information that comes up in the theory is known as
information entropy, which is usually expressed by the average number of bits needed for storage or communication. Intuitively, entropy quantifies the uncertainty involved in a
random variable. For example, a fair coin flip will have less entropy than a roll of a die.
Applications of fundamental topics of information theory include
lossless data compression (for example
ZIP files),
lossy data compression (for example
MP3s), and
channel coding (for example for
DSL lines). The field is at the crossroads of
mathematics,
statistics,
computer science,
physics,
neurobiology, and
electrical engineering. Its impact has been crucial to success of the
Voyager missions to deep space, the invention of the CD, the feasibility of mobile phones, the development of the
Internet, the study of
linguistics and of human perception, the understanding of
black holes, and numerous other fields. Important sub-fields of information theory are source coding, channel coding, algorithmic complexity theory, algorithmic information theory, and measures of information.
Overview
The main concepts of information theory can be grasped by considering the most widespread means of human communication: language. Two important aspects of a good language are as follows: First, the most common words (for example, "a", "the", "I") should be shorter than less common words (for example, "benefit", "generation", "mediocre"), so that sentences won't be too long. Such a tradeoff in word length is analogous to
data compression and is the essential aspect of
source coding. Second, if part of a sentence is unheard or misheard due to noise—for example, a passing car—the listener should still be able to glean the meaning of the underlying message. Such robustness is as essential for an electronic communication system as it's for a language; properly building such robustness into communications is done by
channel coding. Source coding and channel coding are the fundamental concerns of information theory.
Note that these concerns have nothing to do with the
importance of messages. For example, a platitude such as "Thank you; come again" takes about as long to say or write as the urgent plea, "Call an ambulance!" while clearly the latter is more important and more meaningful. Information theory, however, doesn't involve message importance or meaning, as these are matters of the quality of data rather than the quantity of data, the latter of which is determined solely by probabilities.
Information theory is generally considered to have been founded in 1948 by
Claude Shannon in his seminal work, "
A Mathematical Theory of Communication." The central paradigm of classical information theory is the engineering problem of the transmission of information over a noisy channel. The most fundamental results of this theory are Shannon's
source coding theorem, which establishes that, on average, the number of
bits needed to represent the result of an uncertain event is given by its
entropy; and Shannon's
noisy-channel coding theorem, which states that
reliable communication is possible over
noisy channels provided that the rate of communication is below a certain threshold called the channel capacity. The channel capacity can be approached by using appropriate encoding and decoding systems.
Information theory is closely associated with a collection of pure and applied disciplines that have been investigated and reduced to engineering practice under a variety of rubrics throughout the world over the past half century or more:
adaptive systems,
anticipatory systems,
artificial intelligence,
complex systems,
complexity science,
cybernetics,
informatics,
machine learning, along with
systems sciences of many descriptions. Information theory is a broad and deep mathematical theory, with equally broad and deep applications, amongst which is the vital field of
coding theory.
Coding theory is concerned with finding explicit methods, called
codes, of increasing the efficiency and reducing the net error rate of data communication over a noisy channel to near the limit that Shannon proved is the maximum possible for that channel. These codes can be roughly subdivided into
data compression (source coding) and
error-correction (channel coding) techniques. In the latter case, it took many years to find the methods Shannon's work proved were possible. A third class of information theory codes are cryptographic algorithms (both
codes and
ciphers). Concepts, methods and results from coding theory and information theory are widely used in
cryptography and
cryptanalysis.
See the article ban (information) for a historical application.
Information theory is also used in
information retrieval,
intelligence gathering,
gambling,
statistics, and even in
musical composition.
Historical background
The landmark event that established the discipline of information theory, and brought it to immediate worldwide attention, was the publication of
Claude E. Shannon's classic paper "
A Mathematical Theory of Communication" in the
Bell System Technical Journal in July and October of 1948.
Prior to this paper, limited information theoretic ideas had been developed at Bell Labs, all implicitly assuming events of equal probability.
Harry Nyquist's 1924 paper,
Certain Factors Affecting Telegraph Speed, contains a theoretical section quantifying "intelligence" and the "line speed" at which it can be transmitted by a communication system, giving the relation
, where
W is the speed of transmission of intelligence,
m is the number of different voltage levels to choose from at each time step, and
K is a constant.
Ralph Hartley's 1928 paper,
Transmission of Information, uses the word
information as a measurable quantity, reflecting the receiver's ability to distinguish that one sequence of symbols from any other, thus quantifying information as
, where
S was the number of possible symbols, and
n the number of symbols in a transmission. The natural unit of information was therefore the decimal digit, much later renamed the
hartley in his honour as a unit or scale or measure of information.
Alan Turing in 1940 used similar ideas as part of the statistical analysis of the breaking of the German second world war
Enigma ciphers.
Much of the mathematics behind information theory with events of different probabilities was developed for the field of
thermodynamics by
Ludwig Boltzmann and
J. Willard Gibbs. Connections between information-theoretic entropy and thermodynamic entropy, including the important contributions by
Rolf Landauer in the 1960s, are explored in
Entropy in thermodynamics and information theory.
In Shannon's revolutionary and groundbreaking paper, the work for which had been substantially completed at Bell Labs by the end of 1944, Shannon for the first time introduced the qualitative and quantitative model of communication as a statistical process underlying information theory, opening with the assertion that
» "The fundamental problem of communication is that of reproducing at one point, either exactly or approximately, a message selected at another point."
With it came the ideas of
Ways of measuring information
Information theory is based on
probability theory and
statistics. The most important quantities of information are
entropy, the information in a
random variable, and
mutual information, the amount of information in common between two random variables. The former quantity indicates how easily message data can be
compressed while the latter can be used to find the communication rate across a
channel.
The choice of logarithmic base in the following formulae determines the
unit of
information entropy that's used. The most common unit of information is the
bit, based on the
binary logarithm. Other units include the
nat, which is based on the
natural logarithm, and the
hartley, which is based on the
common logarithm.
In what follows, an expression of the form
is considered by convention to be equal to zero whenever
p is. This is justified because
is the
binary entropy function:
» :
A binary erasure channel (BEC) with erasure probability p is a binary input, ternary output channel. The possible channel outputs are 0, 1, and a third symbol 'e' called an erasure. The erasure represents complete loss of information about an input bit. The capacity of the BEC is 1 - p bits per channel use.
» :
Applications to other fields
Intelligence uses and secrecy applications
Information theoretic concepts apply to cryptography and cryptanalysis. Turing's information unit, the ban, was used in the Ultra project, breaking the German Enigma machine code and hastening the end of WWII in Europe. Shannon himself defined an important concept now called the unicity distance. Based on the redundancy of the plaintext, it attempts to give a minimum amount of ciphertext necessary to ensure unique decipherability.
Information theory leads us to believe it's much more difficult to keep secrets than it might first appear. A brute force attack can break systems based on public-key cryptography or on most commonly used methods of private-key cryptography, such as block ciphers. The security of such methods comes from the assumption that no known attack can break them in a practical amount of time.
Information theoretic security refers to methods such as the one-time pad that are not vulnerable to such brute force attacks. In such cases, the positive conditional mutual information between the plaintext and ciphertext (conditioned on the key) can ensure proper transmission, while the unconditional mutual information between the plaintext and ciphertext remains zero, resulting in absolutely secure communications. In other words, an eavesdropper wouldn't be able to improve his or her guess of the plaintext by gaining knowledge of the ciphertext but not of the key. However, as in any other cryptographic system, care must be used to correctly apply even information-theoretically secure methods; the Venona project was able to crack the one-time pads of the Soviet Union due to their improper reuse.
Pseudorandom number generation
Cryptographically secure pseudorandom number generators need effectively random seeds, which can be obtained via extractors. The measure of sufficient randomness for extractors is min-entropy, a value related to Shannon entropy through Rényi entropy; Rényi entropy is also used in evaluating randomness in cryptographic systems. Although related, the distinctions among these measures mean that a random variable with high Shannon entropy isn't necessarily satisfactory for use in an extractor.
Miscellaneous applications
Information theory also has applications in gambling and investing, black holes, bioinformatics, and music.
Further Information
Get more info on 'Information Theory'.
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