|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{Use dmy dates|date=July 2013}}
| | I'm Lionel and I live with my husband and our 2 children in Kobenhavn K, in the REGION SJALLAND south part. My hobbies are Card collecting, Kiteboarding and Chainmail making.<br><br>My weblog [https://plus.google.com/u/0/108459281883728562902/posts hot melt adhesive] |
| [[File:Entropy flip 2 coins.jpg|thumb|120px|2 bits of entropy.]]
| |
| In [[information theory]], '''entropy''' is a measure of the uncertainty in a [[random variable]].<ref>{{cite book|author=Ihara, Shunsuke|title=Information theory for continuous systems|publisher=World Scientific|year=1993|isbn=978-981-02-0985-8|page=2|url=http://books.google.com/books?id=rwg5ndTuTC4C&pg=PA2}}</ref> In this context, the term usually refers to the '''Shannon entropy''', which quantifies the [[expected value]] of the [[self-information|information]] contained in a message.<ref>In this context, a 'message' means a specific realization of the random variable.</ref> Entropy is typically measured in [[bit]]s, [[nat (information)|nat]]s, or [[ban (information)|ban]]s.<ref>{{cite book|author=Brillouin, Léon|title=Science & Information Theory |publisher=Dover Publications|year=2004|isbn=978-0-486-43918-1|page=293 |url=http://books.google.com/books?id=DWo7lVRVnhcC&pg=PA293 }}</ref> Shannon entropy is the average unpredictability in a random variable, which is equivalent to its [[information content]]. Shannon entropy provides an absolute limit on the best possible [[lossless]] encoding or [[data compression|compression]] of any communication, assuming that<ref>{{cite book|authors=Goise, Francois & Olla, Stefano|title=Entropy methods for the Boltzmann equation: lectures from a special semester at the Centre Émile Borel, Institut H. Poincaré, Paris, 2001|publisher=Springer|year=2008|isbn=978-3-540-73704-9|page=14|url=http://books.google.com/books?id=K3meGQ9ntjgC&pg=PA14}}</ref> the communication may be represented as a sequence of [[independent and identically distributed random variables]].
| |
| | |
| A single toss of a [[fair coin]] has an entropy of one bit. A series of two fair coin tosses has an entropy of two bits. The number of fair coin tosses is its entropy in bits. This random selection between two outcomes in a sequence over time, whether the outcomes are equally probable or not, is often referred to as a [[Bernoulli process]]. The entropy of such a process is given by the [[binary entropy function]]. The [[entropy rate]] for a fair coin toss is one bit per toss. However, if the coin is not fair, then the uncertainty, and hence the entropy rate, is lower. This is because, if asked to predict the next outcome, we could choose the most frequent result and be right more often than wrong. The difference between what we know, or predict, and the information that the unfair coin toss reveals to us is less than one heads-or-tails "message", or bit, per toss.<ref name="Schneier, B page 234">Schneier, B: ''Applied Cryptography'', Second edition, page 234. John Wiley and Sons.</ref>
| |
| | |
| This definition of "entropy" was introduced by [[Claude E. Shannon]] in his 1948 paper "[[A Mathematical Theory of Communication]]".<ref name=shannonPaper>{{cite journal |author=Shannon, Claude E. |authorlink=Claude Shannon |title=[[A Mathematical Theory of Communication]] |journal=Bell System Technical Journal |volume=27 |issue=3 |pages=379–423 |date=July/October 1948 }} ([http://www.alcatel-lucent.com/bstj/vol27-1948/articles/bstj27-3-379.pdf PDF])</ref>
| |
| | |
| ==Introduction==
| |
| Entropy is a measure of ''unpredictability'' of ''information content''. To get an informal, intuitive understanding of the connection between these three English terms, consider the example of a poll on some political issue. Usually, such polls happen because the outcome of the poll isn't already known. In other words, the outcome of the poll is relatively ''unpredictable'', and actually performing the poll and learning the results gives some new ''information''; these are just different ways of saying that the ''entropy'' of the poll results is large. Now, consider the case that the same poll is performed a second time shortly after the first poll. Since the result of the first poll is already known, the outcome of the second poll can be predicted well and the results should not contain much new information; in this case the entropy of the second poll results is small.
| |
| | |
| Now consider the example of a coin toss. When the coin is fair, that is, when the probability of heads is the same as the probability of tails, then the entropy of the coin toss is as high as it could be. This is because there is no way to predict the outcome of the coin toss ahead of time—the best we can do is predict that the coin will come up heads, and our prediction will be correct with probability 1/2. Such a coin toss has one bit of entropy since there are two possible outcomes that occur with equal probability, and learning the actual outcome contains one bit of information. Contrarily, a coin toss with a coin that has two heads and no tails has zero entropy since the coin will always come up heads, and the outcome can be predicted perfectly.
| |
| | |
| English text has fairly low entropy. In other words, it is fairly predictable. Even if we don't know exactly what is going to come next, we can be fairly certain that, for example, there will be many more e's than z's, or that the combination 'qu' will be much more common than any other combination with a 'q' in it and the combination 'th' will be more common than 'z', 'q', or 'qu'. Uncompressed, English text has between 0.6 and 1.3 [[bit]]s of entropy for each character of message.<ref name="Schneier, B page 234"/><ref name="Shannon, Claude E. 1951">Shannon, Claude E.: Prediction and entropy of printed English, ''[[The Bell System Technical Journal]]'', 30:50–64, January 1951.</ref>
| |
| | |
| If a [[Data compression|compression]] scheme is lossless—that is, you can always recover the entire original message by decompressing—then a compressed message has the same quantity of information as the original, but communicated in fewer characters. That is, it has more information per character, or a higher entropy. This means a compressed message is more unpredictable, because there is less redundancy. Roughly speaking, [[Shannon's source coding theorem]] says that a lossless compression scheme cannot compress messages, on average, to have more than one bit of information per bit of message. The entropy of a message multiplied by the length of that message is a measure of how much information the message contains.
| |
| | |
| Shannon's theorem also implies that no lossless compression scheme can compress ''all'' messages. If some messages come out smaller, at least one must come out larger due to the [[pigeonhole principle]]. In practical use, this is generally not a problem, because we are usually only interested in compressing certain types of messages, for example English documents as opposed to gibberish text, or digital photographs rather than noise, and it is unimportant if a compression algorithm makes some unlikely or uninteresting sequences larger. However, the problem can still arise even in everyday use when applying a compression algorithm to already compressed data; for example, making a ZIP file of music in the [[FLAC]] audio format is unlikely to achieve much extra savings in space.
| |
| | |
| ==Definition==
| |
| Named after [[H-theorem|Boltzmann's H-theorem]], Shannon denoted the entropy ''H'' (Greek letter [[Eta]]) of a [[discrete random variable]] ''X'' with possible values {''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>} and [[probability mass function]] ''P(X)'' as,
| |
| | |
| :<math>H(X) = E[I(X)] = E[-\ln(P(X))].</math>
| |
| | |
| Here E is the [[expected value|expected value operator]], and ''I'' is the [[self-information|information content]] of ''X''.<ref>{{cite book|author=Borda, Monica|title=Fundamentals in Information Theory and Coding|publisher=Springer|year=2011|isbn=978-3-642-20346-6|page=11|url=http://books.google.com/books?id=Lyte2yl1SPAC&pg=PA11}}</ref><ref>{{cite book|authors=Han, Te Sun & Kobayashi, Kingo|title=Mathematics of Information and Coding|publisher=American Mathematical Society|year=2002|isbn=978-0-8218-4256-0|pages=19–20|url=http://books.google.com/books?id=VpRESN24Zj0C&pg=PA19}}</ref>
| |
| ''I''(''X'') is itself a random variable.
| |
| | |
| When taken from a finite sample, the entropy can explicitly be written as
| |
| :<math>H(X) = \sum_{i} {P(x_i)\,I(x_i)} = -\sum_{i} {P(x_i) \log_b P(x_i)}</math>
| |
| | |
| where ''b'' is the [[base (exponentiation)|base]] of the [[logarithm]] used. Common values of ''b'' are 2, [[e (mathematical constant)|Euler's number {{math|''e''}}]], and 10, and the unit of entropy is [[bit]] for ''b'' = 2, [[Nat (information)|nat]] for ''b'' = {{math|''e''}}, and [[dit (information)|dit]] (or digit) for ''b'' = 10.<ref>Schneider, T.D, [http://alum.mit.edu/www/toms/paper/primer/primer.pdf Information theory primer with an appendix on logarithms], National Cancer Institute, 14 April 2007.</ref>
| |
| | |
| In the case of ''p''(''x''<sub>''i''</sub>) = 0 for some ''i'', the value of the corresponding summand 0 log<sub>''b''</sub>(0) is taken to be 0, which is consistent with the well-known [[limit of a function|limit]]:
| |
| | |
| :<math>\lim_{p\to0+}p\log (p) = 0</math>.
| |
| | |
| ==Example==
| |
| [[File:Binary entropy plot.svg|thumbnail|right|200px|Entropy H(X) (i.e. the [[expected value|expected]] surprisal) of a coin flip, measured in bits, graphed versus the fairness of the coin Pr(X=1), where X=1 represents a result of heads.<br/><br/>Note that the maximum of the graph depends on the distribution. Here, at most 1 bit is required to communicate the outcome of a fair coin flip (2 possible values), but the result of a fair die (6 possible values) would require at most log<sub>2</sub>6 bits.]]
| |
| {{Main|Binary entropy function}}
| |
| {{Main|Bernoulli process}}
| |
| Consider tossing a coin with known, not necessarily fair, probabilities of coming up heads or tails; this is known as the [[Bernoulli process]].
| |
| | |
| The entropy of the unknown result of the next toss of the coin is maximized if the coin is fair (that is, if heads and tails both have equal probability 1/2). This is the situation of maximum uncertainty as it is most difficult to predict the outcome of the next toss; the result of each toss of the coin delivers one full [[bit]] of information.
| |
| | |
| However, if we know the coin is not fair, but comes up heads or tails with probabilities ''p'' and ''q'', where ''p'' ≠ ''q'', then there is less uncertainty. Every time it is tossed, one side is more likely to come up than the other. The reduced uncertainty is quantified in a lower entropy: on average each toss of the coin delivers less than one full bit of information.
| |
| | |
| The extreme case is that of a double-headed coin that never comes up tails, or a double-tailed coin that never results in a head. Then there is no uncertainty. The entropy is zero: each toss of the coin delivers no new information as the outcome of each coin toss is always certain. In this respect, entropy can be normalized by dividing it by information length. The measure is called [[metric entropy]] and allows to measure the randomness of the information.
| |
| | |
| ==Rationale==
| |
| As said, for a random variable ''X'' with ''n'' outcomes {''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>}, the Shannon entropy, a measure of uncertainty (see further below) and denoted by ''H''(''X''), is defined as
| |
| :{| style="width:50%" border="0"
| |
| | style="width:95%" |
| |
| <math> \displaystyle
| |
| H(X)= - \sum_{i=1}^np(x_i)\log_b p(x_i)
| |
| </math>
| |
| | style= | (1)
| |
| |}
| |
| where ''p''(''x''<sub>''i''</sub>) is the [[probability mass function]] of outcome ''x''<sub>''i''</sub>.
| |
| | |
| To understand the meaning of Eq. (1), first consider a set of ''n'' possible outcomes (events) {''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>}, with equal probability ''p''(''x<sub>i</sub>'') = 1/''n''. An example would be a fair [[Dice|die]] with ''n'' values, from 1 to ''n''. The ''uncertainty'' for such a set of ''n'' outcomes is defined by
| |
| :{| style="width:50%" border="0"
| |
| | style="width:95%" |
| |
| <math> \displaystyle
| |
| u = \log_b (n).
| |
| </math>
| |
| | style= | (2)
| |
| |}
| |
| | |
| For concreteness, consider the case in which the base of the logarithm is ''b'' = 2. To specify an outcome of a fair ''n''-sided die roll, we must specify one of the ''n'' values, which requires log<sub>2</sub>(''n'') bits i.e. the number of bits required to represent the outcomes in binary form. For our ''n'' sided die, this would be binary numbers {''000'', ..., ''(n-1)''}. Hence in this case, the uncertainty of an outcome is the number of bits needed to specify the outcome.
| |
| | |
| Intuitively, if we have two independent sources of uncertainty, the overall uncertainty should be the sum of the individual uncertainties. The logarithm captures this '''additivity''' characteristic for independent uncertainties. For example, consider appending to each value of the first die the value of a second die, which has ''m'' possible outcomes {''y''<sub>1</sub>, ..., ''y<sub>m</sub>''}. There are thus ''mn'' possible outcomes <math>\left\{ x_i y_j : i = 1 , \ldots , n , j = 1 , \ldots , m \right\}</math>. The uncertainty for such a set of ''mn'' outcomes is then
| |
| :{| style="width:100%" border="0"
| |
| |-
| |
| | style="width:95%" |
| |
| <math>
| |
| \displaystyle
| |
| u = \log_b (nm) = \log_b (n) + \log_b (m).
| |
| </math>
| |
| | style= | (3)
| |
| |}
| |
| Thus the uncertainty of playing with two dice is obtained by adding the uncertainty of the second die log<sub>''b''</sub>(''m'') to the uncertainty of the first die log<sub>''b''</sub>(''n''). In the case that ''b'' = 2 this means that to specify the outcome of an ''n''-sided die roll and an ''m''-sided die roll, we need to specify log<sub>2</sub>(''n'') + log<sub>2</sub>(''m'') = log<sub>2</sub>(''mn'') bits.
| |
| | |
| Now return to the case of playing with one die only (the first one). Since the probability of each event is 1/''n'', we can write
| |
| :<math> u_i = \log_b \left(\frac{1}{p(x_i)}\right) = - \log_b (p(x_i))=-\log_b\left(\frac{1}{n}\right), \ \forall i \in \{1, \ldots , n\}. </math>
| |
| | |
| In the case of a non-uniform probability mass function (or density in the case of continuous random variables), we let
| |
| :{| style="width:100%" border="0"
| |
| |-
| |
| | style="width:95%" |
| |
| <math> \displaystyle
| |
| u_i := - \log_b (p(x_i))
| |
| </math>
| |
| | style= | (4)
| |
| |}
| |
| which is also called a [[self-information|surprisal]]; the lower the probability ''p''(''x''<sub>''i''</sub>), i.e. ''p''(''x''<sub>''i''</sub>) → 0, the higher the uncertainty or the surprise, i.e. ''u''<sub>''i''</sub> → ∞, for the outcome ''x''<sub>''i''</sub>.
| |
| | |
| The average uncertainty <math>\langle u \rangle</math>, with <math>\langle \cdot \rangle</math> being the average operator, is obtained by
| |
| :{| style="width:100%" border="0"
| |
| |-
| |
| | style="width:95%" |
| |
| <math>
| |
| \displaystyle
| |
| \langle u \rangle
| |
| =
| |
| \sum_{i=1}^{n}
| |
| p(x_i)
| |
| u_i
| |
| =
| |
| -
| |
| \sum_{i=1}^{n}
| |
| p(x_i)
| |
| \log_b (p(x_i))
| |
| </math>
| |
| | style= | (5)
| |
| |}
| |
| and is used as the definition of the entropy ''H''(''X'') in Eq. (1). In his original formulation, Shannon proved that this definition (up to a multiplicative factor) was the only one that satisfied a few key [[Entropy (information theory)#Characterization|requirements]]. The above also explains why information ''entropy'' and information ''uncertainty'' can be used interchangeably.<ref>{{Cite journal| last=Jaynes | first=E.T. | url=http://bayes.wustl.edu/etj/articles/theory.1.pdf | title=Information Theory and Statistical Mechanics | journal=Physical Review | volume=106 | issue=4 | pages=620–630 |date=May 1957 | doi=10.1103/PhysRev.106.620 |bibcode = 1957PhRv..106..620J }}</ref>
| |
| In the case that ''b'' = 2, Eq. (1) measures the expected number of bits we need to specify the outcome of a random experiment.
| |
| | |
| One may also define the [[conditional entropy]] of two events ''X'' and ''Y'' taking values ''x<sub>i</sub>'' and ''y<sub>j</sub>'' respectively, as
| |
| | |
| :<math> H(X|Y)=\sum_{i,j}p(x_{i},y_{j})\log\frac{p(y_{j})}{p(x_{i},y_{j})}</math>
| |
| | |
| where ''p(x<sub>i</sub>,y<sub>j</sub>)'' is the probability that ''X=x<sub>i</sub>'' and ''Y=y<sub>j</sub>''. This quantity should be understood as the amount of randomness in the random variable ''X'' given that you know the value of ''Y''. For example, the entropy associated with a six-sided die is ''H(die)'', but if you were told that it had in fact landed on 1, 2, or 3, then its entropy would be equal to ''H (die: the die landed on 1, 2, or 3)''.
| |
| | |
| ==Aspects==
| |
| | |
| ===Relationship to thermodynamic entropy===
| |
| {{Main|Entropy in thermodynamics and information theory}}
| |
| The inspiration for adopting the word ''entropy'' in information theory came from the close resemblance between Shannon's formula and very similar known formulae from [[statistical mechanics]].
| |
| | |
| In [[statistical thermodynamics]] the most general formula for the thermodynamic [[entropy]] ''S'' of a [[thermodynamic system]] is the [[Gibbs entropy]],
| |
| :<math>S = - k_B \sum p_i \ln p_i \,</math>
| |
| where ''k''<sub>B</sub> is the [[Boltzmann constant]], and ''p<sub>i</sub>'' is the probability of a microstate. The Gibbs entropy was defined by [[J. Willard Gibbs]] in 1878 after earlier work by [[Ludwig Boltzmann|Boltzmann]] (1872).<ref>Compare: Boltzmann, Ludwig (1896, 1898). Vorlesungen über Gastheorie : 2 Volumes – Leipzig 1895/98 UB: O 5262-6. English version: Lectures on gas theory. Translated by Stephen G. Brush (1964) Berkeley: University of California Press; (1995) New York: Dover ISBN 0-486-68455-5</ref>
| |
| | |
| The Gibbs entropy translates over almost unchanged into the world of [[quantum physics]] to give the [[von Neumann entropy]], introduced by [[John von Neumann]] in 1927,
| |
| :<math>S = - k_B \,\,{\rm Tr}(\rho \ln \rho) \,</math>
| |
| where ρ is the [[density matrix]] of the quantum mechanical system and Tr is the [[Trace (linear algebra)|trace]].
| |
| | |
| At an everyday practical level the links between information entropy and thermodynamic entropy are not evident. Physicists and chemists are apt to be more interested in ''changes'' in entropy as a system spontaneously evolves away from its initial conditions, in accordance with the [[second law of thermodynamics]], rather than an unchanging probability distribution. And, as the minuteness of [[Boltzmann's constant]] ''k<sub>B</sub>'' indicates, the changes in ''S''/''k<sub>B</sub>'' for even tiny amounts of substances in chemical and physical processes represent amounts of entropy that are extremely large compared to anything in data compression or signal processing. Furthermore, in classical thermodynamics the entropy is defined in terms of macroscopic measurements and makes no reference to any probability distribution, which is central to the definition of information entropy.
| |
| | |
| At a multidisciplinary level, however, connections can be made between thermodynamic and informational entropy, although it took many years in the development of the theories of statistical mechanics and information theory to make the relationship fully apparent. In fact, in the view of [[Edwin Thompson Jaynes|Jaynes]] (1957), thermodynamic entropy, as explained by [[statistical mechanics]], should be seen as an ''application'' of Shannon's information theory: the thermodynamic entropy is interpreted as being proportional to the amount of further Shannon information needed to define the detailed microscopic state of the system, that remains uncommunicated by a description solely in terms of the macroscopic variables of classical thermodynamics, with the constant of proportionality being just the [[Boltzmann constant]]. For example, adding heat to a system increases its thermodynamic entropy because it increases the number of possible microscopic states for the system, thus making any complete state description longer. (See article: ''[[maximum entropy thermodynamics]]''). [[Maxwell's demon]] can (hypothetically) reduce the thermodynamic entropy of a system by using information about the states of individual molecules; but, as [[Rolf Landauer|Landauer]] (from 1961) and co-workers have shown, to function the demon himself must increase thermodynamic entropy in the process, by at least the amount of Shannon information he proposes to first acquire and store; and so the total thermodynamic entropy does not decrease (which resolves the paradox). [[Landauer's principle]] has implications on the amount of heat a computer must dissipate to process a given amount of information, though modern computers are nowhere near the efficiency limit.
| |
| | |
| ===Entropy as information content===
| |
| {{Main|Shannon's source coding theorem}}
| |
| Entropy is defined in the context of a probabilistic model. Independent fair coin flips have an entropy of 1 bit per flip. A source that always generates a long string of B's has an entropy of 0, since the next character will always be a 'B'.
| |
| | |
| The entropy rate of a data source means the average number of [[bit]]s per symbol needed to encode it. Shannon's experiments with human predictors show an information rate between 0.6 and 1.3 bits per character in English;<ref>{{cite web| url=http://marknelson.us/2006/08/24/the-hutter-prize/ | title=The Hutter Prize | accessdate=2008-11-27 | date=24 August 2006 | author=Mark Nelson}}</ref> the [[PPM compression algorithm]] can achieve a compression ratio of 1.5 bits per character in English text.
| |
| | |
| From the preceding example, note the following points:
| |
| | |
| # The amount of entropy is not always an integer number of bits.
| |
| # Many data bits may not convey information. For example, data structures often store information redundantly, or have identical sections regardless of the information in the data structure.
| |
| | |
| Shannon's definition of entropy, when applied to an information source, can determine the minimum channel capacity required to reliably transmit the source as encoded binary digits (see caveat below in italics). The formula can be derived by calculating the mathematical expectation of the ''amount of information'' contained in a digit from the information source. ''See also'' [[Shannon-Hartley theorem]].
| |
| | |
| Shannon's entropy measures the information contained in a message as opposed to the portion of the message that is determined (or predictable). ''Examples of the latter include redundancy in language structure or statistical properties relating to the occurrence frequencies of letter or word pairs, triplets etc.'' See [[Markov chain]].
| |
| | |
| ===Data compression===
| |
| {{Main|Data compression}}
| |
| Entropy effectively bounds the performance of the strongest lossless compression possible, which can be realized in theory by using the [[typical set]] or in practice using [[Huffman coding|Huffman]], [[LZW|Lempel-Ziv]] or [[arithmetic coding]]. The performance of existing data compression algorithms is often used as a rough estimate of the entropy of a block of data.<ref>T. Schürmann and P. Grassberger, [http://arxiv.org/abs/cond-mat/0203436 Entropy Estimation of Symbol Sequences], ''CHAOS'',Vol. 6, No. 3 (1996) 414–427</ref><ref>T. Schürmann, [http://arxiv.org/abs/cond-mat/0403192 Bias Analysis in Entropy Estimation] J. Phys. A: Math. Gen. 37 (2004) L295-L301.</ref> See also [[Kolmogorov complexity]]. In practice, compression algorithms deliberately include some judicious redundancy in the form of [[checksum]]s to protect against errors.
| |
| | |
| === World's technological capacity to store and communicate entropic information ===
| |
| A 2011 study in ''[[Science (journal)|Science]]'' estimates the world's technological capacity to store and communicate optimally compressed information normalized on the most effective compression algorithms available in the year 2007, therefore estimating the entropy of the technologically available sources.<ref name="HilbertLopez2011">[http://www.sciencemag.org/content/332/6025/60 "The World's Technological Capacity to Store, Communicate, and Compute Information"], Martin Hilbert and Priscila López (2011), [[Science (journal)]], 332(6025), 60–65; free access to the article through here: martinhilbert.net/WorldInfoCapacity.html</ref>
| |
| {| class="wikitable"
| |
| |+
| |
| All figures in entropically compressed [[exabytes]]
| |
| |-
| |
| ! Type of Information !! 1986 !! 2007 !! Increase
| |
| |-
| |
| | Storage || 2.6 || 295 || 113.5x
| |
| |-
| |
| | Broadcast || 432 || 1900 || 4.398x
| |
| |-
| |
| | Telecommunications || 0.281 || 65 || 231.3x
| |
| |}
| |
| The authors estimate humankind technological capacity to store information (fully entropically compressed) in 1986 and again in 2007. They break the information into three categories - To store information on a medium, To receive information through a one-way [[broadcast]] networks, to exchange information through two-way [[telecommunication]] networks <ref name="HilbertLopez2011"/>
| |
| | |
| ===Limitations of entropy as information content===
| |
| There are a number of entropy-related concepts that mathematically quantify information content in some way:
| |
| * the '''[[self-information]]''' of an individual message or symbol taken from a given probability distribution,
| |
| * the '''entropy''' of a given probability distribution of messages or symbols, and
| |
| * the '''[[entropy rate]]''' of a [[stochastic process]].
| |
| (The "rate of self-information" can also be defined for a particular sequence of messages or symbols generated by a given stochastic process: this will always be equal to the entropy rate in the case of a [[stationary process]].) Other [[quantities of information]] are also used to compare or relate different sources of information.
| |
| | |
| It is important not to confuse the above concepts. Often it is only clear from context which one is meant. For example, when someone says that the "entropy" of the English language is about 1 bit per character, they are actually modeling the English language as a stochastic process and talking about its entropy ''rate''.
| |
| | |
| <!-- This should still be reworded ... -->
| |
| Although entropy is often used as a characterization of the information content of a data source, this information content is not absolute: it depends crucially on the probabilistic model. A source that always generates the same symbol has an [[entropy rate]] of 0, but the definition of what a symbol is depends on the alphabet. Consider a source that produces the string ABABABABAB... in which A is always followed by B and vice versa. If the probabilistic model considers individual letters as [[Statistical independence|independent]], the entropy rate of the sequence is 1 bit per character. But if the sequence is considered as "AB AB AB AB AB..." with symbols as two-character blocks, then the entropy rate is 0 bits per character.
| |
| | |
| However, if we use very large blocks, then the estimate of per-character entropy rate may become artificially low. This is because in reality, the probability distribution of the sequence is not knowable exactly; it is only an estimate. For example, suppose one considers the text of every book ever published as a sequence, with each symbol being the text of a complete book. If there are ''N'' published books, and each book is only published once, the estimate of the probability of each book is 1/''N'', and the entropy (in bits) is −log<sub>2</sub>(1/''N'') = log<sub>2</sub>(''N''). As a practical code, this corresponds to assigning each book a [[ISBN|unique identifier]] and using it in place of the text of the book whenever one wants to refer to the book. This is enormously useful for talking about books, but it is not so useful for characterizing the information content of an individual book, or of language in general: it is not possible to reconstruct the book from its identifier without knowing the probability distribution, that is, the complete text of all the books. The key idea is that the complexity of the probabilistic model must be considered. [[Kolmogorov complexity]] is a theoretical generalization of this idea that allows the consideration of the information content of a sequence independent of any particular probability model; it considers the shortest [[Computer program|program]] for a [[universal computer]] that outputs the sequence. A code that achieves the entropy rate of a sequence for a given model, plus the codebook (i.e. the probabilistic model), is one such program, but it may not be the shortest.
| |
| | |
| For example, the Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, ... . Treating the sequence as a message and each number as a symbol, there are almost as many symbols as there are characters in the message, giving an entropy of approximately log<sub>2</sub>(''n''). So the first 128 symbols of the Fibonacci sequence has an entropy of approximately 7 bits/symbol. However, the sequence can be expressed using a formula [F(''n'') = F(''n''−1) + F(''n''−2) for ''n''={3,4,5,...}, F(1)=1, F(2)=1] and this formula has a much lower entropy and applies to any length of the Fibonacci sequence.
| |
| | |
| ===Limitations of entropy as a measure of unpredictability===
| |
| In [[cryptanalysis]], entropy is often roughly used as a measure of the unpredictability of a cryptographic key. For example, a 128-bit key that is randomly generated has 128 bits of entropy. It takes (on average) <math>2^{128-1}</math> guesses to break by brute force. If the key's first digit is 0, and the others random, then the entropy is 127 bits, and it takes (on average) <math>2^{127-1}</math> guesses.
| |
| | |
| However, entropy fails to capture the number of guesses required if the possible keys are not of equal probability.<ref>{{cite conference |first1=James |last1=Massey |year=1994 |title=Guessing and Entropy |booktitle=Proc. IEEE International Symposium on Information Theory |url=http://www.isiweb.ee.ethz.ch/archive/massey_pub/pdf/BI633.pdf |accessdate=December 31, 2013}}</ref><ref>{{cite conference |first1=David |last1=Malone|first2=Wayne |last2=Sullivan |year=2005 |title=Guesswork is not a Substitute for Entropy |booktitle=Proceedings of the Information Technology & Telecommunications Conference |url=http://www.maths.tcd.ie/~dwmalone/p/itt05.pdf |accessdate=December 31, 2013}}</ref> If the key is half the time "password" and half the time a true random 128-bit key, then the entropy is approximately 65 bits. Yet half the time the key may be guessed on the first try, if your first guess is "password", and on average, it takes around <math>2^{126}</math> guesses (not <math>2^{65-1}</math>) to break this password.
| |
| | |
| Similarly, consider a 1000000-digit binary [[one-time pad]]. If the pad has 1000000 bits of entropy, it is perfect. If the pad has 999999 bits of entropy, evenly distributed (each individual bit of the pad having 0.999999 bits of entropy) it may still be considered very good. But if the pad has 999999 bits of entropy, where the first digit is fixed and the remaining 999999 digits are perfectly random, then the first digit of the ciphertext will not be encrypted at all.
| |
| | |
| ===Data as a Markov process===
| |
| A common way to define entropy for text is based on the [[Markov model]] of text. For an order-0 source (each character is selected independent of the last characters), the binary entropy is:
| |
| | |
| :<math>H(\mathcal{S}) = - \sum p_i \log_2 p_i, \,\!</math>
| |
| | |
| where ''p''<sub>''i''</sub> is the probability of ''i''. For a first-order [[Markov source]] (one in which the probability of selecting a character is dependent only on the immediately preceding character), the '''[[entropy rate]]''' is:
| |
| | |
| :<math>H(\mathcal{S}) = - \sum_i p_i \sum_j \ p_i (j) \log_2 p_i (j), \,\!</math> {{citation needed|date=April 2013}}
| |
| | |
| where ''i'' is a '''state''' (certain preceding characters) and <math>p_i(j)</math> is the probability of ''j'' given ''i'' as the previous character.
| |
| | |
| For a second order Markov source, the entropy rate is
| |
| | |
| :<math>H(\mathcal{S}) = -\sum_i p_i \sum_j p_i(j) \sum_k p_{i,j}(k)\ \log_2 \ p_{i,j}(k). \,\!</math>
| |
| | |
| ===''b''-ary entropy===
| |
| In general the '''''b''-ary entropy''' of a source <math>\mathcal{S}</math> = (''S'',''P'') with [[source alphabet]] ''S'' = {''a''<sub>1</sub>, ..., ''a<sub>n</sub>''} and [[discrete probability distribution]] ''P'' = {''p''<sub>1</sub>, ..., ''p<sub>n</sub>''} where ''p<sub>i</sub>'' is the probability of ''a<sub>i</sub>'' (say ''p<sub>i</sub>'' = ''p''(''a<sub>i</sub>'')) is defined by:
| |
| | |
| :<math> H_b(\mathcal{S}) = - \sum_{i=1}^n p_i \log_b p_i, \,\!</math>
| |
| | |
| Note: the ''b'' in "''b''-ary entropy" is the number of different symbols of the ''ideal alphabet'' used as a standard yardstick to measure source alphabets. In information theory, two symbols are [[necessary and sufficient]] for an alphabet to encode information. Therefore, the default is to let ''b'' = 2 ("binary entropy"). Thus, the entropy of the source alphabet, with its given empiric probability distribution, is a number equal to the number (possibly fractional) of symbols of the "ideal alphabet", with an optimal probability distribution, necessary to encode for each symbol of the source alphabet. Also note that "optimal probability distribution" here means a [[Uniform distribution (discrete)|uniform distribution]]: a source alphabet with ''n'' symbols has the highest possible entropy (for an alphabet with ''n'' symbols) when the probability distribution of the alphabet is uniform. This optimal entropy turns out to be log<sub>''b''</sub>(''n'').
| |
| | |
| ==Efficiency==
| |
| A source alphabet with non-uniform distribution will have less entropy than if those symbols had uniform distribution (i.e. the "optimized alphabet"). This deficiency in entropy can be expressed as a ratio:
| |
| | |
| : <math>\mathrm{efficiency}(X) = -\sum_{i=1}^n \frac{p(x_i) \log_b (p(x_i))}{\log_b (n)}</math>
| |
| | |
| Efficiency has utility in quantifying the effective use of a communications channel. This formulation is also referred to as the normalized entropy, as the entropy is divided by the maximum entropy <math>{\log_b (n)}</math>.
| |
| | |
| ==Characterization==
| |
| Shannon entropy is [[characterization (mathematics)|characterized]] by a small number of criteria, listed below. Any definition of entropy satisfying these assumptions has the form
| |
| :<math>-K\sum_{i=1}^np_i\log (p_i)</math>
| |
| where ''K'' is a constant corresponding to a choice of measurement units.
| |
| | |
| In the following, ''p<sub>i</sub>'' = Pr (''X'' = ''x<sub>i</sub>'') and <math>H_n(p_1,\ldots,p_n)=H(X)</math>.
| |
| | |
| ===Continuity===
| |
| The measure should be [[continuous function|continuous]], so that changing the values of the probabilities by a very small amount should only change the entropy by a small amount.
| |
| | |
| ===Symmetry===
| |
| The measure should be unchanged if the outcomes ''x<sub>i</sub>'' are re-ordered.
| |
| :<math>H_n\left(p_1, p_2, \ldots \right) = H_n\left(p_2, p_1, \ldots \right)</math> etc.
| |
| | |
| ===Maximum===
| |
| The measure should be maximal if all the outcomes are equally likely (uncertainty is highest when all possible events are equiprobable).
| |
| :<math> H_n(p_1,\ldots,p_n) \le H_n\left(\frac{1}{n}, \ldots, \frac{1}{n}\right) = \log_b (n).</math>
| |
| | |
| For equiprobable events the entropy should increase with the number of outcomes.
| |
| :<math>H_n\bigg(\underbrace{\frac{1}{n}, \ldots, \frac{1}{n}}_{n}\bigg) = \log_b(n) < \log_b (n+1) = H_{n+1}\bigg(\underbrace{\frac{1}{n+1}, \ldots, \frac{1}{n+1}}_{n+1}\bigg).</math>
| |
| | |
| ===Additivity===
| |
| The amount of entropy should be independent of how the process is regarded as being divided into parts.
| |
| | |
| This last functional relationship characterizes the entropy of a system with sub-systems. It demands that the entropy of a system can be calculated from the entropies of its sub-systems if the interactions between the sub-systems are known.
| |
| | |
| Given an ensemble of ''n'' uniformly distributed elements that are divided into ''k'' boxes (sub-systems) with ''b''<sub>1</sub>, ... , ''b<sub>k</sub>'' elements each, the entropy of the whole ensemble should be equal to the sum of the entropy of the system of boxes and the individual entropies of the boxes, each weighted with the probability of being in that particular box.
| |
| | |
| For [[positive integers]] ''b<sub>i</sub>'' where ''b''<sub>1</sub> + ... + ''b<sub>k</sub>'' = ''n'',
| |
| :<math>H_n\left(\frac{1}{n}, \ldots, \frac{1}{n}\right) = H_k\left(\frac{b_1}{n}, \ldots, \frac{b_k}{n}\right) + \sum_{i=1}^k \frac{b_i}{n} \, H_{b_i}\left(\frac{1}{b_i}, \ldots, \frac{1}{b_i}\right).</math>
| |
| | |
| Choosing ''k'' = ''n'', ''b''<sub>1</sub> = ... = ''b<sub>n</sub>'' = 1 this implies that the entropy of a certain outcome is zero: ''H''<sub>1</sub>(1) = 0. This implies that the efficiency of a source alphabet with ''n'' symbols can be defined simply as being equal to its ''n''-ary entropy. See also [[Redundancy (information theory)]].
| |
| | |
| ==Further properties==
| |
| The Shannon entropy satisfies the following properties, for some of which it is useful to interpret entropy as the amount of information learned (or uncertainty eliminated) by revealing the value of a random variable ''X'':
| |
| | |
| * Adding or removing an event with probability zero does not contribute to the entropy:
| |
| ::<math>H_{n+1}(p_1,\ldots,p_n,0) = H_n(p_1,\ldots,p_n)</math>.
| |
| * It can be confirmed using the [[Jensen inequality]] that
| |
| ::<math>H(X) = \operatorname{E}\left[\log_b \left( \frac{1}{p(X)}\right) \right] \leq \log_b \left( \operatorname{E}\left[ \frac{1}{p(X)} \right] \right) = \log_b(n)</math>.
| |
| :This maximal entropy of log<sub>''b''</sub>(''n'') is effectively attained by a source alphabet having a uniform probability distribution: uncertainty is maximal when all possible events are equiprobable.
| |
| * The entropy or the amount of information revealed by evaluating (''X'',''Y'') (that is, evaluating ''X'' and ''Y'' simultaneously) is equal to the information revealed by conducting two consecutive experiments: first evaluating the value of ''Y'', then revealing the value of ''X'' given that you know the value of ''Y''. This may be written as
| |
| ::<math> H[(X,Y)]=H(X|Y)+H(Y)=H(Y|X)+H(X).</math>
| |
| * If ''Y=f(X)'' where ''f'' is deterministic, then ''H''(''f''(''X'')|''X'') = 0. Applying the previous formula to ''H''(''X'', ''f''(''X'')) yields
| |
| ::<math> H(X)+H(f(X)|X)=H(f(X))+H(X|f(X)),</math>
| |
| :so ''H''(''f''(''X'')) ≤ ''H''(''X''), thus the entropy of a variable can only decrease when the latter is passed through a deterministic function.
| |
| * If ''X'' and ''Y'' are two independent experiments, then knowing the value of ''Y'' doesn't influence our knowledge of the value of ''X'' (since the two don't influence each other by independence):
| |
| ::<math> H(X|Y)=H(X).</math>
| |
| * The entropy of two simultaneous events is no more than the sum of the entropies of each individual event, and are equal if the two events are independent. More specifically, if ''X'' and ''Y'' are two random variables on the same probability space, and ''(X,Y)'' denotes their Cartesian product, then
| |
| ::<math> H[(X,Y)]\leq H(X)+H(Y).</math>
| |
| Proving this mathematically follows easily from the previous two properties of entropy.
| |
| | |
| ==Extending discrete entropy to the continuous case: differential entropy==
| |
| {{Main|Differential entropy}}
| |
| | |
| The Shannon entropy is restricted to random variables taking discrete values. The corresponding formula for a continuous random variable with [[probability density function]] ''f(x)'' with finite or infinite support <math>\mathbb X</math> on the real line is defined by analogy, using the above form of the entropy as an expectation:
| |
| | |
| :<math>h[f] = \operatorname{E}[-\ln (f(x))] = -\int_\mathbb X f(x) \ln (f(x))\, dx.</math>
| |
| | |
| This formula is usually referred to as the '''continuous entropy''', or [[differential entropy]]. A precursor of the continuous entropy ''h''[''f''] is the expression for the functional ''H'' in the [[H-theorem]] of [[Boltzmann]].
| |
| | |
| Although the analogy between both functions is suggestive, the following question must be set: is the differential entropy a valid extension of the Shannon discrete entropy? Differential entropy lacks a number of properties that the Shannon discrete entropy has – it can even be negative – and thus corrections have been suggested, notably [[limiting density of discrete points]].
| |
| | |
| To answer this question, we must establish a connection between the two functions:
| |
| | |
| We wish to obtain a generally finite measure as the [[bin size]] goes to zero. In the discrete case, the bin size is the (implicit) width of each of the ''n'' (finite or infinite) bins whose probabilities are denoted by ''p<sub>n</sub>''. As we generalize to the continuous domain, we must make this width explicit.
| |
| | |
| To do this, start with a continuous function ''f'' discretized into bins of size <math>\Delta</math>.
| |
| <!-- Figure: Discretizing the function $ f$ into bins of width $ \Delta$ \includegraphics[width=\textwidth]{function-with-bins.eps} --><!-- The original article this figure came from is at http://planetmath.org/shannonsentropy but it is broken there too -->
| |
| By the mean-value theorem there exists a value ''x<sub>i</sub>'' in each bin such that
| |
| | |
| :<math>f(x_i) \Delta = \int_{i\Delta}^{(i+1)\Delta} f(x)\, dx</math>
| |
| | |
| and thus the integral of the function ''f'' can be approximated (in the Riemannian sense) by
| |
| | |
| :<math>\int_{-\infty}^{\infty} f(x)\, dx = \lim_{\Delta \to 0} \sum_{i = -\infty}^{\infty} f(x_i) \Delta</math>
| |
| | |
| where this limit and "bin size goes to zero" are equivalent.
| |
| | |
| We will denote
| |
| | |
| :<math>H^{\Delta} :=- \sum_{i=-\infty}^{\infty} f(x_i) \Delta \log \left( f(x_i) \Delta \right)</math>
| |
| | |
| and expanding the logarithm, we have
| |
| | |
| :<math>H^{\Delta} = - \sum_{i=-\infty}^{\infty} f(x_i) \Delta \log (f(x_i)) -\sum_{i=-\infty}^{\infty} f(x_i) \Delta \log (\Delta).</math>
| |
| | |
| As Δ → 0, we have
| |
| | |
| :<math>\begin{align}
| |
| \sum_{i=-\infty}^{\infty} f(x_i) \Delta &\to \int_{-\infty}^{\infty} f(x)\, dx = 1 \\
| |
| \sum_{i=-\infty}^{\infty} f(x_i) \Delta \log (f(x_i)) &\to \int_{-\infty}^{\infty} f(x) \log f(x)\, dx.
| |
| \end{align}</math>
| |
| | |
| But note that log(Δ) → −∞ as Δ → 0, therefore we need a special definition of the differential or continuous entropy:
| |
| | |
| :<math>h[f] = \lim_{\Delta \to 0} \left(H^{\Delta} + \log \Delta\right) = -\int_{-\infty}^{\infty} f(x) \log f(x)\,dx,</math>
| |
| | |
| which is, as said before, referred to as the '''differential entropy'''. This means that the differential entropy ''is not'' a limit of the Shannon entropy for ''n'' → ∞. Rather, it differs from the limit of the Shannon entropy by an infinite offset.
| |
| | |
| It turns out as a result that, unlike the Shannon entropy, the differential entropy is ''not'' in general a good measure of uncertainty or information. For example, the differential entropy can be negative; also it is not invariant under continuous co-ordinate transformations.
| |
| | |
| ==Relative entropy==
| |
| {{main|Generalized relative entropy}}
| |
| Another useful measure of entropy that works equally well in the discrete and the continuous case is the '''relative entropy''' of a distribution. It is defined as the [[Kullback-Leibler divergence]] from the distribution to a reference measure ''m'' as follows. Assume that a probability distribution ''p'' is [[absolutely continuous]] with respect to a measure ''m'', i.e. is of the form ''p''(''dx'') = ''f''(''x'')''m''(''dx'') for some non-negative ''m''-integrable function ''f'' with ''m''-integral 1, then the relative entropy can be defined as
| |
| :<math>D_{\mathrm{KL}}(p \| m ) = \int \log (f(x)) p(dx) = \int f(x)\log (f(x)) m(dx) .</math>
| |
| | |
| In this form the relative entropy generalises (up to change in sign) both the discrete entropy, where the measure ''m'' is the [[counting measure]], and the differential entropy, where the measure ''m'' is the [[Lebesgue measure]]. If the measure ''m'' is itself a probability distribution, the relative entropy is non-negative, and zero iff ''p'' = ''m'' as measures. It is defined for any measure space, hence coordinate independent and invariant under co-ordinate reparameterizations if one properly takes into account the transformation of the measure ''m''. The relative entropy, and implicitly entropy and differential entropy, do depend on the "reference" measure ''m''.
| |
| | |
| ==Use in combinatorics==
| |
| Entropy has become a useful quantity in [[combinatorics]].
| |
| | |
| ===Loomis-Whitney inequality===
| |
| A simple example of this is an alternate proof of the [[Loomis-Whitney inequality]]: for every subset ''A'' ⊆ '''Z'''<sup>''d''</sup>, we have
| |
| :<math> |A|^{d-1}\leq \prod_{i=1}^{d} |P_{i}(A)|</math>
| |
| where ''P<sub>i</sub>'' is the [[orthogonal projection]] in the ith coordinate:
| |
| :<math> P_{i}(A)=\{(x_{1}, ..., x_{i-1}, x_{i+1}, ..., x_{d}) : (x_{1}, ..., x_{d})\in A\}.</math>
| |
| | |
| The proof follows as a simple corollary of [[Shearer's inequality]]: if ''X''<sub>1</sub>, ..., ''X<sub>d</sub>'' are random variables and ''S''<sub>1</sub>, ..., ''S<sub>n</sub>'' are subsets of {1, ..., ''d''} such that every integer between ''1'' and ''d'' lies in exactly ''r'' of these subsets, then
| |
| :<math> H[(X_{1},...,X_{d})]\leq \frac{1}{r}\sum_{i=1}^{n}H[(X_{j})_{j\in S_{i}}]</math>
| |
| where <math> (X_{j})_{j\in S_{i}}</math> is the Cartesian product of random variables ''X<sub>j</sub>'' with indexes ''j'' in ''S<sub>i</sub>'' (so the dimension of this vector is equal to the size of ''S<sub>i</sub>'').
| |
| | |
| We sketch how Loomis-Whitney follows from this: Indeed, let ''X'' be a uniformly distributed random variable with values in ''A'' and so that each point in ''A'' occurs with equal probability. Then (by the further properties of entropy mentioned above) ''H''(''X'') = log|''A''|, where ''|A|'' denotes the cardinality of ''A''. Let ''S<sub>i</sub>'' = {1, 2, ..., ''i''−1, ''i''+1, ..., ''d''}. The range of <math>(X_{j})_{j\in S_{i}}</math> is contained in ''P<sub>i</sub>''(''A'') and hence <math> H[(X_{j})_{j\in S_{i}}]\leq \log |P_{i}(A)|</math>. Now use this to bound the right side of Shearer's inequality and exponentiate the opposite sides of the resulting inequality you obtain.
| |
| | |
| ===Approximation to binomial coefficient===
| |
| For integers 0 < ''k'' < ''n'' let ''q'' = ''k''/''n''. Then
| |
| :<math>\frac{2^{nH(q)}}{n+1} \leq \tbinom nk \leq 2^{nH(q)},</math>
| |
| where
| |
| :<math>H(q) = -q \log_2(q) - (1-q) \log_2(1-q).</math><ref>Aoki, New Approaches to Macroeconomic Modeling. page 43.</ref>
| |
| | |
| Here is a sketch proof. Note that <math>\tbinom nk q^{qn}(1-q)^{n-nq}</math> is one term of the expression
| |
| :<math>\sum_{i=0}^n \tbinom ni q^i(1-q)^{n-i} = (q + (1-q))^n = 1.</math>
| |
| Rearranging gives the upper bound. For the lower bound one first shows, using some algebra, that it is the largest term in the summation. But then,
| |
| :<math>\tbinom nk q^{qn}(1-q)^{n-nq} \geq \tfrac{1}{n+1}</math>
| |
| since there are ''n''+1 terms in the summation. Rearranging gives the lower bound.
| |
| | |
| A nice interpretation of this is that the number of binary strings of length ''n'' with exactly ''k'' many 1's is approximately <math>2^{nH(k/n)}</math>.<ref>Probability and Computing, M. Mitzenmacher and E. Upfal, Cambridge University Press</ref>
| |
| | |
| ==See also==
| |
| {{Portal|Statistics|Cryptography}}
| |
| *[[Conditional entropy]]
| |
| *[[Cross entropy]] – is a measure of the average number of bits needed to identify an event from a set of possibilities between two probability distributions
| |
| *[[Entropy (arrow of time)]]
| |
| *[[Entropy encoding]] – a coding scheme that assigns codes to symbols so as to match code lengths with the probabilities of the symbols.
| |
| *[[Entropy estimation]]
| |
| *[[Entropy power inequality]]
| |
| *[[Entropy rate]]
| |
| *[[Fisher information]]
| |
| *[[Hamming distance]]
| |
| *[[History of entropy]]
| |
| *[[History of information theory]]
| |
| *[[Joint entropy]] – is the measure how much entropy is contained in a joint system of two random variables.
| |
| *[[Kolmogorov-Sinai entropy]] in [[dynamical system]]s
| |
| *[[Levenshtein distance]]
| |
| *[[Mutual information]]
| |
| *[[Negentropy]]
| |
| *[[Perplexity]]
| |
| *[[Qualitative variation]] – other measures of [[statistical dispersion]] for [[nominal distributions]]
| |
| *[[Quantum relative entropy]] – a measure of distinguishability between two quantum states.
| |
| *[[Rényi entropy]] – a generalisation of Shannon entropy; it is one of a family of functionals for quantifying the diversity, uncertainty or randomness of a system.
| |
| *[[Shannon index]]
| |
| *[[Theil index]]
| |
| *[[Weighted entropy]]
| |
| | |
| ==References==
| |
| {{refimprove|date=April 2012}}
| |
| {{reflist|2}}
| |
| {{PlanetMath attribution|id=968|title=Shannon's entropy}}
| |
| | |
| ==Further reading==
| |
| <!-- * {{cite book|authors=|title=|publisher=|year=|isbn=|url=}} -->
| |
| | |
| * {{cite book|authors=Martin, Nathaniel F.G. & England, James W.|title=Mathematical Theory of Entropy|publisher=Cambridge University Press|year=2011|isbn=978-0-521-17738-2|url=http://books.google.com/books?id=_77lvx7y8joC}}
| |
| | |
| ==External links==
| |
| * {{springer|title=Entropy|id=p/e035740}}
| |
| * [http://pespmc1.vub.ac.be/ENTRINFO.html Introduction to entropy and information] on [[Principia Cybernetica Web]]
| |
| * ''[http://www.mdpi.com/journal/entropy Entropy]'' an interdisciplinary journal on all aspect of the entropy concept. Open access.
| |
| * [http://alum.mit.edu/www/toms/information.is.not.uncertainty.html Information is not entropy, information is not uncertainty !] – a discussion of the use of the terms "information" and "entropy".
| |
| * [http://alum.mit.edu/www/toms/bionet.info-theory.faq.html#Information.Equal.Entropy I'm Confused: How Could Information Equal Entropy?] – a similar discussion on the bionet.info-theory FAQ.
| |
| * [http://www.rheingold.com/texts/tft/6.html Description of information entropy from "Tools for Thought" by Howard Rheingold]
| |
| * [http://math.ucsd.edu/~crypto/java/ENTROPY/ A java applet representing Shannon's Experiment to Calculate the Entropy of English]
| |
| * [http://www.autonlab.org/tutorials/infogain.html Slides on information gain and entropy]
| |
| *[http://en.wikibooks.org/wiki/An_Intuitive_Guide_to_the_Concept_of_Entropy_Arising_in_Various_Sectors_of_Science ''An Intuitive Guide to the Concept of Entropy Arising in Various Sectors of Science''] – a wikibook on the interpretation of the concept of entropy.
| |
| * [http://www.shannonentropy.netmark.pl Calculator for Shannon entropy estimation and interpretation]
| |
| * [https://researchspace.auckland.ac.nz/handle/2292/3427 Network Event Detection With Entropy Measures], Dr. Raimund Eimann, University of Auckland, PDF; 5993 kB – a PhD thesis demonstrating how entropy measures may be used in network anomaly detection.
| |
| | |
| {{Compression Methods}}
| |
| | |
| {{DEFAULTSORT:Entropy (Information Theory)}}
| |
| [[Category:Entropy and information| ]]
| |
| [[Category:Information theory]]
| |
| [[Category:Statistical theory]]
| |
| [[Category:Randomness]]
| |