4889:
1249:. For example, consider the transmission of sequences comprising the 4 characters 'A', 'B', 'C', and 'D' over a binary channel. If all 4 letters are equally likely (25%), one cannot do better than using two bits to encode each letter. 'A' might code as '00', 'B' as '01', 'C' as '10', and 'D' as '11'. However, if the probabilities of each letter are unequal, say 'A' occurs with 70% probability, 'B' with 26%, and 'C' and 'D' with 2% each, one could assign variable length codes. In this case, 'A' would be coded as '0', 'B' as '10', 'C' as '110', and 'D' as '111'. With this representation, 70% of the time only one bit needs to be sent, 26% of the time two bits, and only 4% of the time 3 bits. On average, fewer than 2 bits are required since the entropy is lower (owing to the high prevalence of 'A' followed by 'B' – together 96% of characters). The calculation of the sum of probability-weighted log probabilities measures and captures this effect.
4282:
4884:{\displaystyle {\begin{aligned}&\operatorname {I} (p_{1}p_{2})&=\ &\operatorname {I} (p_{1})+\operatorname {I} (p_{2})&&\quad {\text{Starting from property 3}}\\&p_{2}\operatorname {I} '(p_{1}p_{2})&=\ &\operatorname {I} '(p_{1})&&\quad {\text{taking the derivative w.r.t}}\ p_{1}\\&\operatorname {I} '(p_{1}p_{2})+p_{1}p_{2}\operatorname {I} ''(p_{1}p_{2})&=\ &0&&\quad {\text{taking the derivative w.r.t}}\ p_{2}\\&\operatorname {I} '(u)+u\operatorname {I} ''(u)&=\ &0&&\quad {\text{introducing}}\,u=p_{1}p_{2}\\&(u\operatorname {I} '(u))'&=\ &0&&\quad {\text{combining terms into one}}\ \\&u\operatorname {I} '(u)-k&=\ &0&&\quad {\text{integrating w.r.t}}\ u,{\text{producing constant}}\,k\\\end{aligned}}}
14169:
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3155:
561:
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43:
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9034:. In practical use, this is generally not a problem, because one is usually only interested in compressing certain types of messages, such as a document in English, 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.
592:, and a receiver. The "fundamental problem of communication" – as expressed by Shannon – is for the receiver to be able to identify what data was generated by the source, based on the signal it receives through the channel. Shannon considered various ways to encode, compress, and transmit messages from a data source, and proved in his
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scheme is lossless – one in which you can always recover the entire original message by decompression – then a compressed message has the same quantity of information as the original but communicated in fewer characters. It has more information (higher entropy) per character. A compressed message has
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satisfying the above properties must be a constant multiple of
Shannon entropy, with a non-negative constant. Compared to the previously mentioned characterizations of entropy, this characterization focuses on the properties of entropy as a function of random variables (subadditivity and additivity),
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could take) have trivially low entropy and their sum would become big. But the key insight was showing a reduction in entropy by non negligible amounts as one expands H leading inturn to unbounded growth of a mathematical object over this random variable is equivalent to showing the unbounded growth
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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
9025:
than one bit of information per bit of message can be attained by employing a suitable coding scheme. The entropy of a message per bit multiplied by the length of that message is a measure of how much total information the message contains. Shannon's theorem also implies that no lossless compression
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The core idea of information theory is that the "informational value" of a communicated message depends on the degree to which the content of the message is surprising. If a highly likely event occurs, the message carries very little information. On the other hand, if a highly unlikely event occurs,
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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 –
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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
300:
quantifies the average level of uncertainty or information associated with the variable's potential states or possible outcomes. This measures the expected amount of information needed to describe the state of the variable, considering the distribution of probabilities across all potential states.
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using exclusive or. If the pad has 1,000,000 bits of entropy, it is perfect. If the pad has 999,999 bits of entropy, evenly distributed (each individual bit of the pad having 0.999999 bits of entropy) it may provide good security. But if the pad has 999,999 bits of entropy, where the first bit is
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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
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If very large blocks are used, the estimate of per-character entropy rate may become artificially low because the probability distribution of the sequence is not known exactly; it is only an estimate. If one considers the text of every book ever published as a sequence, with each symbol being the
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Uniform probability yields maximum uncertainty and therefore maximum entropy. Entropy, then, can only decrease from the value associated with uniform probability. 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
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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. Shannon's entropy measures the information contained in a message as opposed to the portion of the message that is
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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
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English text, treated as a string of characters, has fairly low entropy; i.e. it is fairly predictable. We can be fairly certain that, for example, 'e' will be far more common than 'z', that the combination 'qu' will be much more common than any other combination with a 'q' in it, and that the
8945:(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).
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2180:
8931:. Adding heat to a system increases its thermodynamic entropy because it increases the number of possible microscopic states of the system that are consistent with the measurable values of its macroscopic variables, making any complete state description longer. (See article:
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6255:{\displaystyle \mathrm {H} _{n}\left({\frac {1}{n}},\ldots ,{\frac {1}{n}}\right)=\mathrm {H} _{k}\left({\frac {b_{1}}{n}},\ldots ,{\frac {b_{k}}{n}}\right)+\sum _{i=1}^{k}{\frac {b_{i}}{n}}\,\mathrm {H} _{b_{i}}\left({\frac {1}{b_{i}}},\ldots ,{\frac {1}{b_{i}}}\right).}
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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. The minimum channel capacity can be realized in theory by using the
3501:, 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. For example, if
15080:
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
9043:
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.
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10454:{\displaystyle \eta (X)=-\sum _{i=1}^{n}{\frac {p(x_{i})\log _{b}(p(x_{i}))}{\log _{b}(n)}}=\sum _{i=1}^{n}{\frac {\log _{b}(p(x_{i})^{-p(x_{i})})}{\log _{b}(n)}}=\sum _{i=1}^{n}\log _{n}(p(x_{i})^{-p(x_{i})})=\log _{n}(\prod _{i=1}^{n}p(x_{i})^{-p(x_{i})}).}
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10013:
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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 one-way
1082:
Entropy measures the expected (i.e., average) amount of information conveyed by identifying the outcome of a random trial. This implies that rolling a die has higher entropy than tossing a coin because each outcome of a die toss has smaller probability
11589:{\displaystyle {\begin{aligned}\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{aligned}}}
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9278:. The first 128 symbols of the Fibonacci sequence has an entropy of approximately 7 bits/symbol, but the sequence can be expressed using a formula and this formula has a much lower entropy and applies to any length of the Fibonacci sequence.
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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. This problem may be illustrated by a change of units when
11144:
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Here, the entropy is at most 1 bit, and to communicate the outcome of a coin flip (2 possible values) will require an average of at most 1 bit (exactly 1 bit for a fair coin). The result of a fair die (6 possible values) would have entropy
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loss, that minimizes the average cross entropy between ground truth and predicted distributions. In general, cross entropy is a measure of the differences between two datasets similar to the KL divergence (also known as relative entropy).
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615:. The analogy results when the values of the random variable designate energies of microstates, so Gibbs's formula for the entropy is formally identical to Shannon's formula. Entropy has relevance to other areas of mathematics such as
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11923:
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The
Shannon entropy satisfies the following properties, for some of which it is useful to interpret entropy as the expected amount of information learned (or uncertainty eliminated) by revealing the value of a random variable
3470:{\displaystyle {\begin{aligned}\mathrm {H} (X)&=-\sum _{i=1}^{n}{p(x_{i})\log _{b}p(x_{i})}\\&=-\sum _{i=1}^{2}{{\frac {1}{2}}\log _{2}{\frac {1}{2}}}\\&=-\sum _{i=1}^{2}{{\frac {1}{2}}\cdot (-1)}=1.\end{aligned}}}
9265:
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
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times the
Shannon entropy), Boltzmann's equation results. In information theoretic terms, the information entropy of a system is the amount of "missing" information needed to determine a microstate, given the macrostate.
2006:
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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
481:
12945:
1229:, when the event outcome is known ahead of time, and the entropy is zero bits. When the entropy is zero bits, this is sometimes referred to as unity, where there is no uncertainty at all – no freedom of choice – no
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combination 'th' will be more common than 'z', 'q', or 'qu'. After the first few letters one can often guess the rest of the word. English text has between 0.6 and 1.3 bits of entropy per character of the message.
15115:"Translation of Ludwig Boltzmann's Paper "On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium""
13904:
8183:
6581:
568:— with two coins there are four possible outcomes, and two bits of entropy. Generally, information entropy is the average amount of information conveyed by an event, when considering all possible outcomes.
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9419:
3739:{\displaystyle {\begin{aligned}\mathrm {H} (X)&=-p\log _{2}(p)-q\log _{2}(q)\\&=-0.7\log _{2}(0.7)-0.3\log _{2}(0.3)\\&\approx -0.7\cdot (-0.515)-0.3\cdot (-1.737)\\&=0.8816<1.\end{aligned}}}
5921:{\displaystyle \mathrm {H} _{n}{\bigg (}\underbrace {{\frac {1}{n}},\ldots ,{\frac {1}{n}}} _{n}{\bigg )}<\mathrm {H} _{n+1}{\bigg (}\underbrace {{\frac {1}{n+1}},\ldots ,{\frac {1}{n+1}}} _{n+1}{\bigg )}.}
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with a non-uniform distribution will have less entropy than the same set with a uniform distribution (i.e. the "optimized alphabet"). This deficiency in entropy can be expressed as a ratio called efficiency:
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7396:
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9192:(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
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of a data source is the average number of bits 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; the
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8785:. In classical thermodynamics, 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.
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10900:
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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.
1518:
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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
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5214:) does not matter in the definition of entropy. Entropy only takes into account the probability of observing a specific event, so the information it encapsulates is information about the underlying
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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.
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distributions. The idea is that the distribution that best represents the current state of knowledge of a system is the one with the largest entropy, and is therefore suitable to be the prior.
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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
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1907:
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Intuitively the idea behind the proof was if there is low information in terms of the
Shannon entropy between consecutive random variables (here the random variable is defined using the
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5047:
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391:
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techniques arise largely from statistics and also information theory. In general, entropy is a measure of uncertainty and the objective of machine learning is to minimize uncertainty.
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be the winning number of a lottery provides very little information, because any particular chosen number will almost certainly not win. However, knowledge that a particular number
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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
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4228:
12397:
11835:
1213:
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and related quantities inherit simple relation, in turn. The measure theoretic definition in the previous section defined the entropy as a sum over expected surprisals
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501:
14124:. The information gain is used to identify which attributes of the dataset provide the most information and should be used to split the nodes of the tree optimally.
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9128:. A diversity index is a quantitative statistical measure of how many different types exist in a dataset, such as species in a community, accounting for ecological
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guesses to break by brute force. Entropy fails to capture the number of guesses required if the possible keys are not chosen uniformly. Instead, a measure called
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imposes a lower bound on the amount of heat a computer must generate to process a given amount of information, though modern computers are far less efficient.
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14922:
11313:{\displaystyle \mathrm {H} ^{\Delta }=-\sum _{i=-\infty }^{\infty }f(x_{i})\Delta \log(f(x_{i}))-\sum _{i=-\infty }^{\infty }f(x_{i})\Delta \log(\Delta ).}
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for an extremal partition. Here the logarithm is ad hoc and the entropy is not a measure in itself. At least in the information theory of a binary string,
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11817:. The argument of the logarithm must be dimensionless, otherwise it is improper, so that the differential entropy as given above will be improper. If
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for even tiny amounts of substances in chemical and physical processes represent amounts of entropy that are extremely large compared to anything in
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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,
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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.
564:
Two bits of entropy: In the case of two fair coin tosses, the information entropy in bits is the base-2 logarithm of the number of possible outcomes
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1006:
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to that of subsets of a universal set. Information is quantified as "dits" (distinctions), a measure on partitions. "Dits" can be converted into
14302:– a generalization of Shannon entropy; it is one of a family of functionals for quantifying the diversity, uncertainty or randomness of a system.
15301:"A tribute to Claude Shannon (1916–2001) and a plea for more rigorous use of species richness, species diversity and the 'Shannon–Wiener' Index"
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The Shannon entropy is restricted to random variables taking discrete values. The corresponding formula for a continuous random variable with
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60:
13805:
15610:
15368:
13287:. Now use this to bound the right side of Shearer's inequality and exponentiate the opposite sides of the resulting inequality you obtain.
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275:
14197:– is a measure of the average number of bits needed to identify an event from a set of possibilities between two probability distributions
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11992:, which would in general be infinite. This is expected: continuous variables would typically have infinite entropy when discretized. The
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is unmeasurable. For example, a 128-bit key that is uniformly and randomly generated has 128 bits of entropy. It also takes (on average)
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can (hypothetically) reduce the thermodynamic entropy of a system by using information about the states of individual molecules; but, as
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is the thermodynamic entropy of a particular macrostate (defined by thermodynamic parameters such as temperature, volume, energy, etc.),
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establishing that entropy should be a measure of how informative the average outcome of a variable is. For a continuous random variable,
107:
5470:{\displaystyle \mathrm {H} _{n}\left(p_{1},p_{2},\ldots p_{n}\right)=\mathrm {H} _{n}\left(p_{i_{1}},p_{i_{2}},\ldots ,p_{i_{n}}\right)}
14746:
11996:
is really a measure of how much easier a distribution is to describe than a distribution that is uniform over its quantization scheme.
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2175:{\displaystyle \mathrm {H} (X|Y)=-\sum _{x,y\in {\mathcal {X}}\times {\mathcal {Y}}}p_{X,Y}(x,y)\log {\frac {p_{X,Y}(x,y)}{p_{Y}(y)}},}
79:
16337:
12359:. And in an interval the sum over that interval could become arbitrary large. For example, a sequence of +1's (which are values of
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2475:
11749:{\displaystyle h=\lim _{\Delta \to 0}\left(\mathrm {H} ^{\Delta }+\log \Delta \right)=-\int _{-\infty }^{\infty }f(x)\log f(x)\,dx,}
8664:
7316:: it is attained by the uniform probability distribution. That is, uncertainty is maximal when all possible events are equiprobable:
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is the number of microstates (various combinations of particles in various energy states) that can yield the given macrostate, and
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The proof is quite involved and it brought together breakthroughs not just in novel use of Shannon Entropy, but also its used the
86:
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of an individual message or symbol taken from a given probability distribution (message or sequence seen as an individual event),
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Consider tossing a coin with known, not necessarily fair, probabilities of coming up heads or tails; this can be modelled as a
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simultaneously) is equal to the information revealed by conducting two consecutive experiments: first evaluating the value of
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8377:{\displaystyle \mathrm {H} (\lambda p_{1}+(1-\lambda )p_{2})\geq \lambda \mathrm {H} (p_{1})+(1-\lambda )\mathrm {H} (p_{2})}
15887:
7879:
5725:{\displaystyle \mathrm {H} _{n}(p_{1},\ldots ,p_{n})\leq \mathrm {H} _{n}\left({\frac {1}{n}},\ldots ,{\frac {1}{n}}\right)}
937:
93:
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11829:(i.e. "bin size") and therefore has the same units, then a modified differential entropy may be written in proper form as:
8632:
8534:
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5293:, so that changing the values of the probabilities by a very small amount should only change the entropy by a small amount.
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There are a number of entropy-related concepts that mathematically quantify information content of a sequence or message:
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fixed and the remaining 999,999 bits are perfectly random, the first bit of the ciphertext will not be encrypted at all.
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17:
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14104:, quantifies the expected information, or the reduction in entropy, from additionally knowing the value of an attribute
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The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation and Machine Learning
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of text. For an order-0 source (each character is selected independent of the last characters), the binary entropy is:
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compressed onto a perfectly noiseless channel. Shannon strengthened this result considerably for noisy channels in his
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9445:(one in which the probability of selecting a character is dependent only on the immediately preceding character), the
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11134:{\displaystyle \mathrm {H} ^{\Delta }:=-\sum _{i=-\infty }^{\infty }f(x_{i})\Delta \log \left(f(x_{i})\Delta \right)}
8501:
in information theory came from the close resemblance between Shannon's formula and very similar known formulae from
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Increasing number of outcomes: for equiprobable events, the entropy should increase with the number of outcomes i.e.
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9800:{\displaystyle \mathrm {H} ({\mathcal {S}})=-\sum _{i}p_{i}\sum _{j}p_{i}(j)\sum _{k}p_{i,j}(k)\ \log \ p_{i,j}(k).}
9328:
Other problems may arise from non-uniform distributions used in cryptography. For example, a 1,000,000-digit binary
17179:
17068:
11012:{\displaystyle \int _{-\infty }^{\infty }f(x)\,dx=\lim _{\Delta \to 0}\sum _{i=-\infty }^{\infty }f(x_{i})\Delta ,}
10468:. This formulation is also referred to as the normalized entropy, as the entropy is divided by the maximum entropy
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4085:
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win a lottery has high informational value because it communicates the occurrence of a very low probability event.
268:
14208:– a coding scheme that assigns codes to symbols so as to match code lengths with the probabilities of the symbols.
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12243:
In this form the relative entropy generalizes (up to change in sign) both the discrete entropy, where the measure
10008:{\displaystyle \eta (X)={\frac {H}{H_{max}}}=-\sum _{i=1}^{n}{\frac {p(x_{i})\log _{b}(p(x_{i}))}{\log _{b}(n)}}.}
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Information theory is useful to calculate the smallest amount of information required to convey a message, as in
14913:
12285:. The relative entropy, and (implicitly) entropy and differential entropy, do depend on the "reference" measure
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bit of information, which is approximately 0.693 nats or 0.301 decimal digits. Because of additivity,
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348:
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The entropy of two simultaneous events is no more than the sum of the entropies of each individual event i.e.,
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192:
64:
6407:
3831:: an increase in the probability of an event decreases the information from an observed event, and vice versa.
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Chakrabarti, C. G., and Indranil Chakrabarty. "Shannon entropy: axiomatic characterization and application."
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9290:, entropy is often roughly used as a measure of the unpredictability of a cryptographic key, though its real
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12407:
While the use of Shannon Entropy in the proof is novel it is likely to open new research in this direction.
11759:
which is, as said before, referred to as the differential entropy. This means that the differential entropy
8901:. When these probabilities are substituted into the above expression for the Gibbs entropy (or equivalently
7641:
6484:
Motivated by such relations, a plethora of related and competing quantities have been defined. For example,
17138:
16583:
16141:
15955:
15833:
14985:
14256:
14131:
12005:
10535:
8749:
2268:
1565:
1215:(about 1.58496) bits of information because it can have one of three values.) The minimum surprise is when
601:
244:
14676:
14611:
13119:
12953:
12372:
12303:
11937:
11770:. Rather, it differs from the limit of the Shannon entropy by an infinite offset (see also the article on
8802:
8431:
7695:
1323:
17184:
16131:
16126:
15903:
12010:
Another useful measure of entropy that works equally well in the discrete and the continuous case is the
8886:. It is assumed that each microstate is equally likely, so that the probability of a given microstate is
7068:
6908:
5483:
4954:
1800:
261:
249:
4065:
3958:
1371:
100:
17073:
17000:
16838:
16818:
16762:
16420:
16211:
16014:
15882:
14003:
algorithms use relative entropy to determine the decision rules that govern the data at each node. The
9218:
published books, and each book is only published once, the estimate of the probability of each book is
9148:
index with parameter equal to 1. The Shannon index is related to the proportional abundances of types.
9107:
8509:
2445:
1174:, for which one outcome is not expected over the other. In this case a coin flip has an entropy of one
612:
223:
12731:{\displaystyle P_{i}(A)=\{(x_{1},\ldots ,x_{i-1},x_{i+1},\ldots ,x_{d}):(x_{1},\ldots ,x_{d})\in A\}.}
8748:
in entropy as a system spontaneously evolves away from its initial conditions, in accordance with the
7570:{\displaystyle \mathrm {H} (X,Y)=\mathrm {H} (X|Y)+\mathrm {H} (Y)=\mathrm {H} (Y|X)+\mathrm {H} (X).}
5588:
2379:
17174:
17083:
17024:
16950:
16798:
16388:
16383:
16238:
16081:
15591:. Advances in Intelligent Systems and Computing. Vol. 652. Singapore: Springer. pp. 31–36.
14149:
9197:
8998:
8988:.) In practice, compression algorithms deliberately include some judicious redundancy in the form of
8191:
2748:
1715:
10471:
9818:
8788:
The connection between thermodynamics and what is now known as information theory was first made by
4260:
1982:
1958:
1299:
324:
17088:
16661:
16455:
16156:
14293:
14216:
14189:
12327:
5215:
3822:
1274:
596:
that the entropy represents an absolute mathematical limit on how well data from the source can be
31:
13429:{\displaystyle {\frac {2^{n\mathrm {H} (q)}}{n+1}}\leq {\tbinom {n}{k}}\leq 2^{n\mathrm {H} (q)},}
10551:
7117:
7045:
2421:
1525:
822:
is close to 0, the surprisal of the event is high. This relationship is described by the function
634:
17029:
16400:
16287:
16243:
16056:
16039:
16029:
14200:
12742:
8946:
8391:
6326:
5541:
4190:
3201:
53:
15584:
15357:
10702:
This is the differential entropy (or continuous entropy). A precursor of the continuous entropy
4277:
be the information function which one assumes to be twice continuously differentiable, one has:
1185:
16654:
16405:
16189:
16034:
15038:
Aczél, J.; Forte, B.; Ng, C. T. (1974). "Why the Shannon and Hartley entropies are 'natural'".
14284:
14000:
10692:{\displaystyle \mathrm {H} (X)=\mathbb {E} =-\int _{\mathbb {X} }f(x)\log f(x)\,\mathrm {d} x.}
10570:
on the real line is defined by analogy, using the above form of the entropy as an expectation:
8738:
7284:{\displaystyle \mathrm {H} _{n+1}(p_{1},\ldots ,p_{n},0)=\mathrm {H} _{n}(p_{1},\ldots ,p_{n})}
6831:{\displaystyle \mathrm {H} _{n+1}(p_{1},\ldots ,p_{n},0)=\mathrm {H} _{n}(p_{1},\ldots ,p_{n})}
6457:
1179:
16926:
15289:, 332(6025); free access to the article through here: martinhilbert.net/WorldInfoCapacity.html
14438:
9591:
2789:
689:
the message is much more informative. For instance, the knowledge that some particular number
17058:
15898:
14809:
14782:
14349:
14288:
14280:
14009:
12555:
12031:
11963:
11771:
10769:
10465:
9568:{\displaystyle \mathrm {H} ({\mathcal {S}})=-\sum _{i}p_{i}\sum _{j}\ p_{i}(j)\log p_{i}(j),}
9297:
9291:
9251:
8985:
8919:
8915:
8502:
7865:{\displaystyle \mathrm {H} (X)+\mathrm {H} (f(X)|X)=\mathrm {H} (f(X))+\mathrm {H} (X|f(X)),}
6489:
4895:
1701:
589:
486:
14730:
7586:
5090:
4164:
4132:
1765:
16742:
16204:
16166:
15987:
15312:
15159:
14265:
14251:
13063:
occurs with equal probability. Then (by the further properties of entropy mentioned above)
10529:
9031:
8522:
7933:, the entropy of a variable can only decrease when the latter is passed through a function.
5123:
5052:
1120:
1086:
880:
628:
506:
187:
167:
14665:
9124:
Entropy is one of several ways to measure biodiversity, and is applied in the form of the
9030:
messages. If some messages come out shorter, at least one must come out longer due to the
3105:
2639:
796:
767:
738:
8:
16973:
16864:
16823:
16808:
16777:
16772:
16681:
16588:
16521:
16490:
16475:
16258:
14183:
14145:
9137:
8651:
6686:
6586:
6401:
5290:
4988:
4237:
3807:
3479:
However, if we know the coin is not fair, but comes up heads or tails with probabilities
1912:
1842:
1230:
702:
172:
15316:
15163:
12269:
is itself a probability distribution, the relative entropy is non-negative, and zero if
10740:
goes to zero. In the discrete case, the bin size is the (implicit) width of each of the
10018:
Applying the basic properties of the logarithm, this quantity can also be expressed as:
2324:. This quantity should be understood as the remaining randomness in the random variable
17046:
17016:
16995:
16901:
16833:
16727:
16415:
16231:
16221:
16116:
16096:
16091:
15713:
15535:
15517:
15338:
15175:
15063:
15055:
14941:
14835:
14726:
14656:
14591:
14270:
14236:
14221:
14211:
14174:
14127:
14107:
14087:
14067:
14047:
12379:
12310:
11788:
It turns out as a result that, unlike the Shannon entropy, the differential entropy is
9259:
9193:
9185:
8938:
8928:
8883:
8850:
8753:
8613:
8226:
7621:
6322:
3872:
3129:
2966:
2946:
2922:
2821:
2657:
2544:
2455:
2347:
2327:
1938:
1918:
917:
718:
585:
304:
289:
182:
144:
27:
Expected amount of information needed to specify the output of a stochastic data source
16627:
12233:{\displaystyle D_{\mathrm {KL} }(p\|m)=\int \log(f(x))p(dx)=\int f(x)\log(f(x))m(dx).}
7114:
It is worth noting that if we drop the "small for small probabilities" property, then
16990:
16978:
16960:
16828:
16712:
16649:
16495:
16410:
16366:
16327:
16009:
15929:—repository of implementations of Shannon entropy in different programming languages.
15846:
15823:
15798:
15774:
15760:
15752:
15743:
15690:
15653:
15600:
15445:
15330:
15325:
15300:
15285:
15214:
15082:
15067:
14874:
14815:
14788:
14736:
14546:
14168:
14135:
10733:
To answer this question, a connection must be established between the two functions:
9039:
8981:
8782:
7159:
Adding or removing an event with probability zero does not contribute to the entropy:
5971:
5156:
5145:
3211:
3205:
2415:
536:
15539:
15342:
15249:
15179:
905:
is the only function that satisfies а specific set of conditions defined in section
16965:
16921:
16894:
16889:
16747:
16732:
16642:
16551:
16546:
16375:
16108:
16086:
15978:
15645:
15592:
15573:
Probability and Computing, M. Mitzenmacher and E. Upfal, Cambridge University Press
15527:
15435:
15320:
15274:"The World's Technological Capacity to Store, Communicate, and Compute Information"
15206:
15167:
15147:
15126:
15047:
15000:
14660:
14652:
14595:
14587:
14260:
14231:
14205:
13994:
12260:
12250:
11918:{\displaystyle \mathrm {H} =\int _{-\infty }^{\infty }f(x)\log(f(x)\,\Delta )\,dx,}
9255:
9162:
9133:
9129:
9017:
states a lossless compression scheme cannot compress messages, on average, to have
9005:
8962:
8789:
8778:
8655:
8640:
8636:
8468:
8464:
8220:
5131:
3170:
1553:
1279:
1246:
620:
554:
239:
197:
15587:. In Panigrahi, Bijaya Ketan; Hoda, M. N.; Sharma, Vinod; Goel, Shivendra (eds.).
15509:
14299:
3057:{\displaystyle \mathrm {H} _{\mu }(M)=\sup _{P\subseteq M}\mathrm {H} _{\mu }(P).}
16884:
16698:
16622:
16603:
16573:
16541:
16507:
16066:
16004:
15940:
15922:
15842:
15792:
15731:
15486:
15477:
15280:
14972:
14683:
14618:
14540:
12401:
12390:
9125:
9119:
8647:
7137:
6488:'s analysis of a "logic of partitions" defines a competing measure in structures
4053:{\displaystyle \operatorname {I} (p)=\log \left({\tfrac {1}{p}}\right)=-\log(p).}
2818:. (This is a relaxation of the usual conditions for a partition.) The entropy of
297:
15944:
an interdisciplinary journal on all aspects of the entropy concept. Open access.
15771:
Information Measures: Information and its Description in Science and Engineering
15596:
15394:
15273:
13789:{\displaystyle \sum _{i=0}^{n}{\tbinom {n}{i}}q^{i}(1-q)^{n-i}=(q+(1-q))^{n}=1.}
6499:
16676:
16470:
16199:
16194:
16051:
16024:
15996:
15811:
15633:
15194:
14634:
14569:
14310:
12313:(which is a useful mathematical function for studying distribution of primes)
9145:
8973:
8752:, rather than an unchanging probability distribution. As the minuteness of the
8734:
6493:
6485:
3167:
2373:
1545:
573:
550:
546:
532:
14696:
14600:
10515:, as indicated by the insensitivity within the final logarithm above thereto.
8927:
classical thermodynamics, with the constant of proportionality being just the
7065:
rather than the properties of entropy as a function of the probability vector
1687:{\displaystyle \mathrm {H} (X)=-\sum _{x\in {\mathcal {X}}}p(x)\log _{b}p(x),}
17168:
16983:
16931:
16598:
16593:
16568:
16500:
16121:
16019:
15815:
15735:
15657:
15649:
15465:
15402:
Proceedings of the Information Technology & Telecommunications Conference
15334:
15218:
14965:
14346:
This definition allows events with probability 0, resulting in the undefined
14316:
14226:
14194:
14153:
13925:
A nice interpretation of this is that the number of binary strings of length
12416:
9442:
9287:
9171:
8942:
8526:
5226:
Another characterization of entropy uses the following properties. We denote
2940:
616:
177:
15440:
15171:
15004:
14986:"Logical Information Theory: New Logical Foundations for Information Theory"
10757:. As the continuous domain is generalized, the width must be made explicit.
6838:, i.e., adding an outcome with probability zero does not change the entropy.
6504:
Another succinct axiomatic characterization of Shannon entropy was given by
503:
denotes the sum over the variable's possible values. The choice of base for
17104:
16071:
16046:
15947:
15926:
14326:
11607:, requires a special definition of the differential or continuous entropy:
10509:. Furthermore, the efficiency is indifferent to choice of (positive) base
9447:
9342:
9329:
9180:
9001:
can achieve a compression ratio of 1.5 bits per character in English text.
8993:
3885:
476:{\displaystyle \mathrm {H} (X):=-\sum _{x\in {\mathcal {X}}}p(x)\log p(x),}
202:
12940:{\displaystyle \mathrm {H} \leq {\frac {1}{r}}\sum _{i=1}^{n}\mathrm {H} }
901:, which gives 0 surprise when the probability of the event is 1. In fact,
17063:
16941:
16737:
16613:
16563:
14321:
12299:
9175:
of the symbols forming the message or sequence (seen as a set of events),
8969:
7136:
must be a non-negative linear combination of the Shannon entropy and the
6321:
The characterization here imposes an additive property with respect to a
5478:
5169:
of each other. For instance, in case of a fair coin toss, heads provides
14893:
3173:) of a coin flip, measured in bits, graphed versus the bias of the coin
17120:
16911:
16906:
16793:
16752:
16558:
15790:
15709:
15210:
15059:
14305:
14275:
12383:
10884:{\displaystyle f(x_{i})\Delta =\int _{i\Delta }^{(i+1)\Delta }f(x)\,dx}
5152:
5141:
2661:
1749:
1735:
1724:
560:
542:
152:
15933:
15131:
15114:
13899:{\displaystyle {\binom {n}{k}}q^{qn}(1-q)^{n-nq}\geq {\frac {1}{n+1}}}
6400:. Observe that a logarithm mediates between these two operations. The
3154:
15522:
15434:. Lecture Notes in Computer Science. Vol. 1758. pp. 62–77.
10719:
10715:
9325:
can be used to measure the effort required for a brute force attack.
9200:
are also used to compare or relate different sources of information.
9103:
8646:
The Gibbs entropy translates over almost unchanged into the world of
8472:
8178:{\displaystyle \mathrm {H} (X,Y)\leq \mathrm {H} (X)+\mathrm {H} (Y)}
6576:{\displaystyle \mathrm {H} (X,Y)\leq \mathrm {H} (X)+\mathrm {H} (Y)}
3800:. The amount of information acquired due to the observation of event
2449:
1705:
1262:
898:
524:
15531:
15051:
42:
17034:
16879:
16536:
15100:
Geometry of Quantum States: An Introduction to Quantum Entanglement
13546:{\displaystyle \mathrm {H} (q)=-q\log _{2}(q)-(1-q)\log _{2}(1-q).}
10737:
9414:{\displaystyle \mathrm {H} ({\mathcal {S}})=-\sum p_{i}\log p_{i},}
9049:
9021:
than one bit of information per bit of message, but that any value
8989:
597:
3879:
Given two independent events, if the first event can yield one of
16803:
16277:
16226:
15915:
14712:
9245:. As a practical code, this corresponds to assigning each book a
8513:
6676:{\displaystyle \mathrm {H} (X,Y)=\mathrm {H} (X)+\mathrm {H} (Y)}
3806:
follows from Shannon's solution of the fundamental properties of
2910:{\displaystyle \mathrm {H} _{\mu }(P)=\sum _{A\in P}h_{\mu }(A).}
7950:
are two independent random variables, then knowing the value of
7405:
The entropy or the amount of information revealed by evaluating
914:
Hence, we can define the information, or surprisal, of an event
16317:
7391:{\displaystyle \mathrm {H} (p_{1},\dots ,p_{n})\leq \log _{b}n}
15195:"Irreversibility and Heat Generation in the Computing Process"
14962:
International Journal of Mathematics and Mathematical Sciences
8723:{\displaystyle S=-k_{\text{B}}\,{\rm {Tr}}(\rho \ln \rho )\,,}
8185:, with equality if and only if the two events are independent.
5614:
should be maximal if all the outcomes are equally likely i.e.
527:, varies for different applications. Base 2 gives the unit of
17152:
16757:
16350:
16297:
15708:
This article incorporates material from Shannon's entropy on
14296:– a measure of distinguishability between two quantum states.
12302:
used entropy to make a useful connection trying to solve the
11019:
where this limit and "bin size goes to zero" are equivalent.
10746:(finite or infinite) bins whose probabilities are denoted by
10464:
Efficiency has utility in quantifying the effective use of a
6500:
Alternative characterization via additivity and subadditivity
3900:
equiprobable outcomes of the joint event. This means that if
1073:{\displaystyle I(E)=\log _{2}\left({\frac {1}{p(E)}}\right).}
624:
14868:
13045:
We sketch how Loomis–Whitney follows from this: Indeed, let
11928:
and the result will be the same for any choice of units for
6294:
this implies that the entropy of a certain outcome is zero:
16307:
16161:
16146:
16136:
14897:
14807:
13916:
terms in the summation. Rearranging gives the lower bound.
9246:
7962:(since the two don't influence each other by independence):
7026:{\displaystyle \lim _{q\to 0^{+}}\mathrm {H} _{2}(1-q,q)=0}
5970:
The rule of additivity has the following consequences: for
3218:
coin delivers one full bit of information. This is because
607:
Entropy in information theory is directly analogous to the
14518:, not allowing events with probability equal to exactly 0.
14044:, which is equal to the difference between the entropy of
13051:
be a uniformly distributed random variable with values in
10518:
6496:, to get the formulas for conditional entropy, and so on.
3837:: events that always occur do not communicate information.
1513:{\displaystyle \mathrm {H} (X)=\mathbb {E} =\mathbb {E} .}
16282:
16248:
13031:(so the dimension of this vector is equal to the size of
12085:-integral 1, then the relative entropy can be defined as
9281:
8977:
5936:
uniformly distributed elements that are partitioned into
5127:
1723:, and 10, and the corresponding units of entropy are the
1270:
1175:
793:
is close to 1, the surprisal of the event is low, but if
528:
15432:
International Workshop on Selected Areas in Cryptography
15365:
Proc. IEEE International Symposium on Information Theory
14836:
Information theory primer with an appendix on logarithms
14428:{\displaystyle \lim \limits _{x\rightarrow 0}x\log(x)=0}
13290:
12533:{\displaystyle |A|^{d-1}\leq \prod _{i=1}^{d}|P_{i}(A)|}
12427:
A simple example of this is an alternative proof of the
9341:
A common way to define entropy for text is based on the
9188:(message or sequence is seen as a succession of events).
8486:
6302:. This implies that the efficiency of a source set with
15683:
Rubinstein, Reuven Y.; Kroese, Dirk P. (9 March 2013).
14470:
equals 0 in this context. Alternatively one can define
588:
system composed of three elements: a source of data, a
15430:
Pliam, John (1999). "Selected Areas in Cryptography".
14732:
Information Theory, Inference, and Learning Algorithms
13682:
13574:
13373:
11777:
9639:
For a second order Markov source, the entropy rate is
7926:{\displaystyle \mathrm {H} (f(X))\leq \mathrm {H} (X)}
4263:
4011:
3875:
is the sum of the information learned from each event.
1282:
15757:
Information Theory, Inference and Learning Algorithms
14476:
14441:
14384:
14352:
14110:
14090:
14070:
14050:
14012:
13943:
13808:
13659:
13572:
13448:
13329:
13191:
13122:
12956:
12809:
12573:
12450:
12330:
12094:
11966:
11940:
11838:
11616:
11335:
11147:
11028:
10903:
10803:
10772:
10736:
In order to obtain a generally finite measure as the
10579:
10554:
10474:
10027:
9849:
9821:
9648:
9594:
9460:
9354:
9300:
9106:
networks, or to exchange information through two-way
8853:
8805:
8667:
8593:{\displaystyle S=-k_{\text{B}}\sum p_{i}\ln p_{i}\,,}
8537:
8434:
8394:
8255:
8229:
8194:
8120:
8100:{\displaystyle \mathrm {H} (X|Y)\leq \mathrm {H} (X)}
8057:
7974:
7882:
7750:
7698:
7644:
7624:
7589:
7459:
7328:
7171:
7120:
7071:
7048:
6960:
6911:
6898:{\displaystyle \mathrm {H} _{n}(p_{1},\ldots ,p_{n})}
6847:
6718:
6689:
6618:
6589:
6518:
6460:
6410:
6335:
6015:
5741:
5620:
5591:
5544:
5486:
5317:
5093:
5055:
5017:
4991:
4957:
4904:
4285:
4193:
4167:
4135:
4088:
4068:
3981:
3961:
3513:
3224:
3132:
3108:
3070:
2989:
2969:
2949:
2925:
2844:
2824:
2792:
2751:
2708:
2669:
2642:
2627:{\displaystyle h_{\mu }(A)=\mu (A)\sigma _{\mu }(A).}
2567:
2547:
2478:
2458:
2424:
2382:
2350:
2330:
2271:
2188:
2009:
1985:
1961:
1941:
1921:
1851:
1803:
1768:
1603:
1568:
1528:
1428:
1374:
1326:
1302:
1188:
1123:
1089:
1009:
940:
920:
883:
828:
799:
770:
741:
721:
637:
572:
The concept of information entropy was introduced by
509:
489:
399:
351:
327:
307:
15638:
IEEE Transactions on Systems Science and Cybernetics
14164:
9140:. Specifically, Shannon entropy is the logarithm of
9113:
6308:
symbols can be defined simply as being equal to its
870:{\displaystyle \log \left({\frac {1}{p(E)}}\right),}
15464:Indices of Qualitative Variation. AR Wilcox - 1967
15269:
15267:
14761:
12024:as follows. Assume that a probability distribution
2372:Entropy can be formally defined in the language of
67:. Unsourced material may be challenged and removed.
15791:Martin, Nathaniel F.G.; England, James W. (2011).
15725:
15585:"Comparative Analysis of Decision Tree Algorithms"
14510:
14462:
14427:
14370:
14116:
14096:
14076:
14056:
14036:
13978:
13898:
13788:
13644:{\displaystyle {\tbinom {n}{k}}q^{qn}(1-q)^{n-nq}}
13643:
13545:
13428:
13279:
13161:
12995:
12939:
12730:
12532:
12351:
12253:, and the differential entropy, where the measure
12232:
11984:
11952:
11917:
11748:
11588:
11312:
11133:
11011:
10883:
10778:
10691:
10562:
10501:
10453:
10007:
9831:
9810:
9799:
9616:
9567:
9413:
9313:
8859:
8836:
8722:
8592:
8452:
8420:
8376:
8235:
8211:
8177:
8099:
8020:{\displaystyle \mathrm {H} (X|Y)=\mathrm {H} (X).}
8019:
7925:
7864:
7730:
7684:
7630:
7610:
7569:
7390:
7283:
7128:
7103:
7056:
7025:
6943:
6897:
6830:
6701:
6675:
6601:
6575:
6473:
6446:
6392:
6329:is defined in terms of a multiplicative property,
6254:
5920:
5724:
5606:
5574:
5530:
5469:
5105:
5079:
5041:
5003:
4977:
4943:
4883:
4269:
4222:
4179:
4147:
4121:
4074:
4052:
3967:
3738:
3469:
3138:
3114:
3094:
3056:
2975:
2955:
2931:
2909:
2830:
2810:
2778:
2737:
2694:
2648:
2626:
2553:
2527:
2464:
2436:
2406:
2356:
2336:
2316:
2257:
2174:
1995:
1971:
1947:
1927:
1901:
1819:
1789:
1686:
1586:
1536:
1512:
1412:
1360:
1312:
1288:
1207:
1167:of landing on tails. The maximum surprise is when
1143:
1109:
1072:
995:
926:
889:
869:
814:
785:
756:
727:
672:
515:
495:
475:
385:
337:
313:
14915:An introduction to information theory and entropy
14697:"Entropy (for data science) Clearly Explained!!!"
13825:
13812:
10897:can be approximated (in the Riemannian sense) by
10786:. By the mean-value theorem there exists a value
5910:
5836:
5808:
5756:
3064:Finally, the entropy of the probability space is
735:is a function which increases as the probability
545:, and base 10 gives units of "dits", "bans", or "
17166:
15714:Creative Commons Attribution/Share-Alike License
15682:
15298:
15264:
15112:
13280:{\displaystyle \mathrm {H} \leq \log |P_{i}(A)|}
11633:
10945:
8493:Entropy in thermodynamics and information theory
7956:doesn't influence our knowledge of the value of
6962:
3015:
1853:
15564:Aoki, New Approaches to Macroeconomic Modeling.
15033:
15031:
15029:
14629:
14627:
14564:
14562:
14511:{\displaystyle p\colon {\mathcal {X}}\to (0,1]}
14152:often employs a standard loss function, called
8918:(1957), thermodynamic entropy, as explained by
8737:of the quantum mechanical system and Tr is the
8512:the most general formula for the thermodynamic
6393:{\displaystyle P(A\mid B)\cdot P(B)=P(A\cap B)}
5221:
4944:{\displaystyle \operatorname {I} (u)=k\log u+c}
2528:{\displaystyle \sigma _{\mu }(A)=-\ln \mu (A).}
15392:
15148:"Information Theory and Statistical Mechanics"
14545:(Third ed.). Academic Press. p. 51.
12393:in short intervals. Proving it also broke the
12384:averages of modulated multiplicative functions
6508:, Forte and Ng, via the following properties:
3912:bits are needed to encode the first value and
623:. The definition can be derived from a set of
549:". An equivalent definition of entropy is the
15963:
15299:Spellerberg, Ian F.; Fedor, Peter J. (2003).
14864:
14784:Fundamentals in Information Theory and Coding
14780:
13698:
13685:
13590:
13577:
13389:
13376:
13003:is the Cartesian product of random variables
12018:from the distribution to a reference measure
10760:To do this, start with a continuous function
10726:and corrections have been suggested, notably
9208:. Shannon himself used the term in this way.
4122:{\displaystyle \operatorname {I} (u)=k\log u}
3955:Shannon discovered that a suitable choice of
1902:{\displaystyle \lim _{p\to 0^{+}}p\log(p)=0.}
269:
15977:
15582:
15283:, Martin Hilbert and Priscila López (2011),
15037:
15026:
14862:
14860:
14858:
14856:
14854:
14852:
14850:
14848:
14846:
14844:
14624:
14559:
14538:
12722:
12596:
12422:
12116:
11934:. In fact, the limit of discrete entropy as
5569:
5545:
5525:
5487:
5218:, not the meaning of the events themselves.
5189:bits of information, which is approximately
3095:{\displaystyle \mathrm {H} _{\mu }(\Sigma )}
2695:{\displaystyle P\subseteq {\mathcal {P}}(X)}
15835:Information Theory: A Tutorial Introduction
15482:"A Magical Answer to an 80-Year-Old Puzzle"
15395:"Guesswork is not a Substitute for Entropy"
15243:
14838:, National Cancer Institute, 14 April 2007.
12741:The proof follows as a simple corollary of
9336:
6481:lends itself to practical interpretations.
5042:{\displaystyle \operatorname {I} (p)\geq 0}
15970:
15956:
15466:https://www.osti.gov/servlets/purl/4167340
15102:. Cambridge University Press. p. 301.
15097:
14946:: CS1 maint: location missing publisher (
14532:
13989:
9212:text of a complete book, and if there are
2258:{\displaystyle p_{X,Y}(x,y):=\mathbb {P} }
1597:The entropy can explicitly be written as:
1237:give entropies between zero and one bits.
386:{\displaystyle p\colon {\mathcal {X}}\to }
276:
262:
15689:. Springer Science & Business Media.
15521:
15476:
15439:
15324:
15130:
14841:
14719:
14664:
14599:
11905:
11898:
11736:
11572:
11423:
10934:
10874:
10677:
10641:
10598:
10556:
8716:
8687:
8586:
8467:(negentropy) function is convex, and its
8033:More generally, for any random variables
6583:for jointly distributed random variables
6176:
4971:
4873:
4710:
2738:{\displaystyle \mu (\mathop {\cup } P)=1}
2295:
2224:
1827:, the value of the corresponding summand
1530:
1476:
1447:
1391:
639:
127:Learn how and when to remove this message
15820:The Mathematical Theory of Communication
15740:Elements of Information Theory – 2nd Ed.
15583:Batra, Mridula; Agrawal, Rashmi (2018).
15192:
14983:
14639:"A Mathematical Theory of Communication"
14574:"A Mathematical Theory of Communication"
12788:} such that every integer between 1 and
12415:Entropy has become a useful quantity in
12014:of a distribution. It is defined as the
9151:
6447:{\displaystyle \mu (A)\cdot \ln \mu (A)}
3153:
631:is analogous to entropy. The definition
559:
15393:Malone, David; Sullivan, Wayne (2005).
15199:IBM Journal of Research and Development
15113:Sharp, Kim; Matschinsky, Franz (2015).
14869:Thomas M. Cover; Joy A. Thomas (1991).
14633:
14568:
12410:
12294:
10523:
10519:Entropy for continuous random variables
9048:All figures in entropically compressed
5210:of the events observed (the meaning of
3779:, first define an information function
3102:, that is, the entropy with respect to
14:
17167:
15631:
15546:from the original on 25 September 2023
15355:
15145:
14911:
14808:Han, Te Sun; Kobayashi, Kingo (2002).
14771:, Second edition, John Wiley and Sons.
14725:
13979:{\displaystyle 2^{n\mathrm {H} (k/n)}}
9282:Limitations of entropy in cryptography
8497:The inspiration for adopting the word
7685:{\displaystyle \mathrm {H} (f(X)|X)=0}
996:{\displaystyle I(E)=-\log _{2}(p(E)),}
578:A Mathematical Theory of Communication
15951:
15664:from the original on 16 December 2021
15613:from the original on 19 December 2022
15429:
15225:from the original on 15 December 2021
15014:from the original on 25 December 2022
14811:Mathematics of Information and Coding
14749:from the original on 17 February 2016
14340:
13291:Approximation to binomial coefficient
11141:and expanding the logarithm, we have
10708:is the expression for the functional
8487:Relationship to thermodynamic entropy
7298:The maximal entropy of an event with
7143:
4155:. Additionally, choosing a value for
4062:In fact, the only possible values of
3894:equiprobable outcomes then there are
2317:{\displaystyle p_{Y}(y)=\mathbb {P} }
1587:{\displaystyle \operatorname {I} (X)}
907:
15845:, University of Sheffield, England.
14539:Pathria, R. K.; Beale, Paul (2011).
13162:{\displaystyle (X_{j})_{j\in S_{i}}}
12996:{\displaystyle (X_{j})_{j\in S_{i}}}
11953:{\displaystyle N\rightarrow \infty }
8837:{\displaystyle S=k_{\text{B}}\ln W,}
8453:{\displaystyle 0\leq \lambda \leq 1}
7731:{\displaystyle \mathrm {H} (X,f(X))}
5300:should be unchanged if the outcomes
1361:{\displaystyle p:{\mathcal {X}}\to }
1160:of landing on heads and probability
1117:) than each outcome of a coin toss (
65:adding citations to reliable sources
36:
15632:Jaynes, Edwin T. (September 1968).
15507:
15374:from the original on 1 January 2014
14703:from the original on 5 October 2021
11999:
11994:limiting density of discrete points
11784:Limiting density of discrete points
11778:Limiting density of discrete points
11763:a limit of the Shannon entropy for
10728:limiting density of discrete points
9588:(certain preceding characters) and
8952:
8388:for all probability mass functions
7692:. Applying the previous formula to
7104:{\displaystyle p_{1},\ldots ,p_{n}}
6944:{\displaystyle p_{1},\ldots ,p_{n}}
5531:{\displaystyle \{i_{1},...,i_{n}\}}
4978:{\displaystyle k,c\in \mathbb {R} }
3752:
1820:{\displaystyle x\in {\mathcal {X}}}
208:Limiting density of discrete points
24:
15832:Stone, J. V. (2014), Chapter 1 of
15720:
15411:from the original on 15 April 2016
14657:10.1002/j.1538-7305.1948.tb00917.x
14592:10.1002/j.1538-7305.1948.tb01338.x
14485:
14247:Information fluctuation complexity
14142:Classification in machine learning
14005:information gain in decision trees
13953:
13816:
13689:
13581:
13450:
13408:
13380:
13341:
13193:
12891:
12811:
12104:
12101:
11947:
11899:
11860:
11855:
11840:
11701:
11696:
11674:
11660:
11655:
11637:
11537:
11532:
11483:
11459:
11454:
11406:
11401:
11383:
11359:
11354:
11301:
11289:
11265:
11260:
11209:
11185:
11180:
11155:
11150:
11123:
11090:
11066:
11061:
11036:
11031:
11003:
10979:
10974:
10949:
10917:
10912:
10857:
10837:
10823:
10773:
10679:
10581:
9824:
9659:
9650:
9471:
9462:
9365:
9356:
8693:
8690:
8354:
8315:
8257:
8196:
8162:
8145:
8122:
8084:
8059:
8001:
7976:
7910:
7884:
7829:
7803:
7769:
7752:
7700:
7646:
7551:
7526:
7509:
7484:
7461:
7330:
7236:
7174:
7122:
7050:
6986:
6905:is invariant under permutation of
6850:
6783:
6721:
6660:
6643:
6620:
6560:
6543:
6520:
6179:
6072:
6018:
5818:
5744:
5673:
5623:
5594:
5384:
5320:
5018:
4905:
4812:
4751:
4669:
4643:
4566:
4506:
4452:
4405:
4358:
4333:
4291:
4264:
4161:is equivalent to choosing a value
4089:
4075:{\displaystyle \operatorname {I} }
4069:
3982:
3968:{\displaystyle \operatorname {I} }
3962:
3519:
3230:
3086:
3073:
3032:
2992:
2847:
2678:
2431:
2392:
2067:
2057:
2011:
1988:
1964:
1812:
1637:
1605:
1569:
1454:
1430:
1413:{\displaystyle p(x):=\mathbb {P} }
1335:
1305:
1182:with equiprobable values contains
490:
433:
401:
360:
330:
25:
17206:
15857:
15508:Tao, Terence (28 February 2016).
14814:. American Mathematical Society.
9114:Entropy as a measure of diversity
8223:in the probability mass function
7441:given that you know the value of
5930:Additivity: given an ensemble of
2367:
1154:Consider a coin with probability
301:Given a discrete random variable
219:Asymptotic equipartition property
76:"Entropy" information theory
17144:
17143:
17134:
17133:
15326:10.1046/j.1466-822X.2003.00015.x
14984:Ellerman, David (October 2017).
14928:from the original on 4 June 2016
14167:
10548:with finite or infinite support
6954:Small for small probabilities:
5607:{\displaystyle \mathrm {H} _{n}}
3924:to encode the second, one needs
3888:outcomes and another has one of
2407:{\displaystyle (X,\Sigma ,\mu )}
1320:and is distributed according to
1296:, which takes values in the set
345:and is distributed according to
321:, which takes values in the set
151:
41:
15726:Textbooks on information theory
15676:
15625:
15576:
15567:
15558:
15510:"The Erdős discrepancy problem"
15501:
15470:
15458:
15423:
15386:
15349:
15305:Global Ecology and Biogeography
15292:
15237:
15186:
15139:
15106:
15091:
15074:
15040:Advances in Applied Probability
14977:
14954:
14905:
14887:
14828:
14064:and the conditional entropy of
9811:Efficiency (normalized entropy)
9225:, and the entropy (in bits) is
9015:Shannon's source coding theorem
8992:to protect against errors. The
8959:Shannon's source coding theorem
8212:{\displaystyle \mathrm {H} (p)}
7042:It was shown that any function
7037:
6316:Redundancy (information theory)
4853:
4794:
4704:
4618:
4481:
4383:
4270:{\textstyle \operatorname {I} }
3871:: the information learned from
2779:{\displaystyle \mu (A\cap B)=0}
1841:, which is consistent with the
683:
235:Shannon's source coding theorem
52:needs additional citations for
15797:. Cambridge University Press.
15794:Mathematical Theory of Entropy
15785:Entropy and Information Theory
15759:, Cambridge University Press,
15712:, which is licensed under the
15244:Mark Nelson (24 August 2006).
14873:. Hoboken, New Jersey: Wiley.
14871:Elements of Information Theory
14801:
14774:
14735:. Cambridge University Press.
14689:
14666:11858/00-001M-0000-002C-4317-B
14505:
14493:
14490:
14457:
14451:
14416:
14410:
14393:
14365:
14359:
14031:
14019:
13971:
13957:
13857:
13844:
13771:
13767:
13755:
13746:
13728:
13715:
13651:is one term of the expression
13623:
13610:
13537:
13525:
13509:
13497:
13491:
13485:
13460:
13454:
13418:
13412:
13351:
13345:
13273:
13269:
13263:
13249:
13236:
13214:
13200:
13197:
13137:
13123:
12971:
12957:
12934:
12912:
12898:
12895:
12853:
12850:
12818:
12815:
12713:
12681:
12675:
12599:
12590:
12584:
12526:
12522:
12516:
12502:
12461:
12452:
12346:
12334:
12224:
12215:
12209:
12206:
12200:
12194:
12185:
12179:
12167:
12158:
12152:
12149:
12143:
12137:
12122:
12110:
11979:
11973:
11944:
11902:
11895:
11889:
11883:
11874:
11868:
11733:
11727:
11715:
11709:
11640:
11626:
11620:
11569:
11563:
11551:
11545:
11521:
11514:
11511:
11498:
11492:
11480:
11467:
11420:
11414:
11390:
11380:
11367:
11304:
11298:
11286:
11273:
11240:
11237:
11224:
11218:
11206:
11193:
11120:
11107:
11087:
11074:
11000:
10987:
10952:
10931:
10925:
10871:
10865:
10854:
10842:
10820:
10807:
10766:discretized into bins of size
10674:
10668:
10656:
10650:
10626:
10623:
10617:
10602:
10591:
10585:
10502:{\displaystyle {\log _{b}(n)}}
10495:
10489:
10445:
10440:
10427:
10417:
10403:
10376:
10357:
10352:
10339:
10329:
10315:
10309:
10266:
10260:
10242:
10237:
10224:
10214:
10200:
10194:
10148:
10142:
10124:
10121:
10108:
10102:
10086:
10073:
10037:
10031:
9996:
9990:
9972:
9969:
9956:
9950:
9934:
9921:
9859:
9853:
9832:{\displaystyle {\mathcal {X}}}
9791:
9785:
9754:
9748:
9719:
9713:
9664:
9654:
9611:
9605:
9559:
9553:
9534:
9528:
9476:
9466:
9370:
9360:
8934:maximum entropy thermodynamics
8713:
8698:
8639:in 1878 after earlier work by
8371:
8358:
8350:
8338:
8332:
8319:
8305:
8292:
8280:
8261:
8206:
8200:
8172:
8166:
8155:
8149:
8138:
8126:
8094:
8088:
8077:
8070:
8063:
8011:
8005:
7994:
7987:
7980:
7920:
7914:
7903:
7900:
7894:
7888:
7856:
7853:
7847:
7840:
7833:
7822:
7819:
7813:
7807:
7796:
7789:
7785:
7779:
7773:
7762:
7756:
7725:
7722:
7716:
7704:
7673:
7666:
7662:
7656:
7650:
7605:
7599:
7561:
7555:
7544:
7537:
7530:
7519:
7513:
7502:
7495:
7488:
7477:
7465:
7435:, then revealing the value of
7366:
7334:
7278:
7246:
7228:
7190:
7014:
6996:
6969:
6892:
6860:
6825:
6793:
6775:
6737:
6670:
6664:
6653:
6647:
6636:
6624:
6570:
6564:
6553:
6547:
6536:
6524:
6441:
6435:
6420:
6414:
6387:
6375:
6366:
6360:
6351:
6339:
5665:
5633:
5074:
5062:
5030:
5024:
4917:
4911:
4828:
4822:
4771:
4767:
4761:
4744:
4685:
4679:
4659:
4653:
4599:
4576:
4539:
4516:
4475:
4462:
4438:
4415:
4377:
4364:
4352:
4339:
4320:
4297:
4244:by the above four properties.
4101:
4095:
4044:
4038:
3994:
3988:
3710:
3701:
3689:
3680:
3658:
3652:
3630:
3624:
3592:
3586:
3564:
3558:
3529:
3523:
3453:
3444:
3323:
3310:
3291:
3278:
3240:
3234:
3089:
3083:
3048:
3042:
3008:
3002:
2901:
2895:
2863:
2857:
2767:
2755:
2726:
2712:
2689:
2683:
2618:
2612:
2599:
2593:
2584:
2578:
2519:
2513:
2495:
2489:
2401:
2383:
2311:
2299:
2288:
2282:
2252:
2228:
2217:
2205:
2163:
2157:
2142:
2130:
2102:
2090:
2029:
2022:
2015:
1996:{\displaystyle {\mathcal {Y}}}
1972:{\displaystyle {\mathcal {X}}}
1890:
1884:
1860:
1778:
1772:
1678:
1672:
1653:
1647:
1615:
1609:
1581:
1575:
1504:
1501:
1495:
1480:
1469:
1466:
1460:
1451:
1440:
1434:
1407:
1395:
1384:
1378:
1355:
1343:
1340:
1313:{\displaystyle {\mathcal {X}}}
1265:, Shannon defined the entropy
1057:
1051:
1019:
1013:
987:
984:
978:
972:
950:
944:
854:
848:
809:
803:
780:
774:
751:
745:
667:
664:
658:
643:
580:", and is also referred to as
467:
461:
449:
443:
411:
405:
380:
368:
365:
338:{\displaystyle {\mathcal {X}}}
193:Conditional mutual information
13:
1:
15146:Jaynes, E. T. (15 May 1957).
14644:Bell System Technical Journal
14579:Bell System Technical Journal
14525:
14242:History of information theory
12352:{\displaystyle \lambda (n+H)}
11960:would also include a term of
10891:the integral of the function
5965:
5011:. Property 1 and 2 give that
3757:To understand the meaning of
3188:represents a result of heads.
1594:is itself a random variable.
1256:
584:. Shannon's theory defines a
15888:Resources in other libraries
15869:Entropy (information theory)
14132:principle of maximum entropy
13557:
12006:Generalized relative entropy
11823:is some "standard" value of
11810:will then have the units of
10563:{\displaystyle \mathbb {X} }
10536:probability density function
8750:second law of thermodynamics
7129:{\displaystyle \mathrm {H} }
7057:{\displaystyle \mathrm {H} }
5222:Alternative characterization
4248:
3149:
2437:{\displaystyle A\in \Sigma }
1537:{\displaystyle \mathbb {E} }
764:of an event decreases. When
673:{\displaystyle \mathbb {E} }
602:noisy-channel coding theorem
245:Noisy-channel coding theorem
7:
15904:Encyclopedia of Mathematics
15597:10.1007/978-981-10-6747-1_4
14964:2005. 17 (2005): 2847-2854
14435:and it can be assumed that
14386:
14160:
13087:denotes the cardinality of
12404:for this specific problem.
12016:Kullback–Leibler divergence
11799:is a dimensioned variable.
9636:as the previous character.
9108:telecommunications networks
8421:{\displaystyle p_{1},p_{2}}
5575:{\displaystyle \{1,...,n\}}
4621:taking the derivative w.r.t
4484:taking the derivative w.r.t
4223:{\displaystyle k=-1/\log x}
10:
17211:
17025:Compressed data structures
16347:RLE + BWT + MTF + Huffman
16015:Asymmetric numeral systems
15822:, Univ of Illinois Press.
15193:Landauer, R. (July 1961).
15098:Życzkowski, Karol (2006).
14912:Carter, Tom (March 2014).
14150:artificial neural networks
13937:many 1's is approximately
13057:and so that each point in
12034:with respect to a measure
12003:
11781:
10527:
9117:
8956:
8764:indicates, the changes in
8510:statistical thermodynamics
8490:
8481:
6683:when the random variables
3199:
2344:given the random variable
1240:
1208:{\displaystyle \log _{2}3}
613:statistical thermodynamics
29:
17129:
17113:
17097:
17015:
16940:
16872:
16863:
16786:
16720:
16711:
16612:
16529:
16520:
16436:
16384:Discrete cosine transform
16374:
16365:
16314:LZ77 + Huffman + context
16267:
16177:
16107:
15995:
15986:
15883:Resources in your library
15589:Nature Inspired Computing
14993:Logic Journal of the IGPL
12763:are random variables and
12429:Loomis–Whitney inequality
12423:Loomis–Whitney inequality
12373:Erdős discrepancy problem
12304:Erdős discrepancy problem
9198:quantities of information
8999:PPM compression algorithm
7447:. This may be written as:
6474:{\displaystyle \log _{2}}
5942:boxes (sub-systems) with
5311:are re-ordered. That is,
17089:Smallest grammar problem
15650:10.1109/TSSC.1968.300117
14463:{\displaystyle 0\log(0)}
14333:
14294:Quantum relative entropy
14257:Kolmogorov–Sinai entropy
14217:Entropy power inequality
14190:Entropy (thermodynamics)
9617:{\displaystyle p_{i}(j)}
9337:Data as a Markov process
8627:is the probability of a
5216:probability distribution
4797:combining terms into one
4386:Starting from property 3
3823:monotonically decreasing
3122:of the sigma-algebra of
2811:{\displaystyle A,B\in P}
1955:taking values from sets
1911:One may also define the
1275:discrete random variable
32:Entropy (disambiguation)
17180:Entropy and information
17030:Compressed suffix array
16579:Nyquist–Shannon theorem
15441:10.1007/3-540-46513-8_5
15172:10.1103/PhysRev.106.620
14371:{\displaystyle \log(0)}
14201:Entropy (arrow of time)
14130:models often apply the
14037:{\displaystyle IG(Y,X)}
13990:Use in machine learning
12800:of these subsets, then
12389:28 October 2023 at the
11985:{\displaystyle \log(N)}
10779:{\displaystyle \Delta }
9314:{\displaystyle 2^{127}}
8922:, should be seen as an
6327:conditional probability
6314:-ary entropy. See also
3202:Binary entropy function
1708:used. Common values of
1546:expected value operator
908:§ Characterization
680:generalizes the above.
496:{\displaystyle \Sigma }
250:Shannon–Hartley theorem
17195:Complex systems theory
17190:Statistical randomness
15742:, Wiley-Interscience,
15358:"Guessing and Entropy"
15356:Massey, James (1994).
14971:5 October 2021 at the
14781:Borda, Monica (2011).
14512:
14464:
14429:
14372:
14285:statistical dispersion
14118:
14098:
14078:
14058:
14038:
14001:Decision tree learning
13980:
13900:
13790:
13680:
13645:
13547:
13430:
13281:
13163:
12997:
12941:
12889:
12732:
12534:
12500:
12353:
12234:
12067:for some non-negative
12040:, i.e. is of the form
11986:
11954:
11919:
11750:
11590:
11463:
11363:
11314:
11269:
11189:
11135:
11070:
11013:
10983:
10885:
10797:in each bin such that
10780:
10693:
10564:
10503:
10455:
10399:
10295:
10177:
10066:
10009:
9914:
9833:
9801:
9624:is the probability of
9618:
9569:
9435:is the probability of
9415:
9315:
8861:
8838:
8724:
8594:
8454:
8422:
8378:
8237:
8213:
8179:
8101:
8021:
7927:
7866:
7732:
7686:
7632:
7612:
7611:{\displaystyle Y=f(X)}
7571:
7392:
7302:different outcomes is
7285:
7130:
7105:
7058:
7027:
6945:
6899:
6832:
6703:
6677:
6603:
6577:
6475:
6448:
6394:
6256:
6158:
5922:
5726:
5608:
5576:
5532:
5471:
5107:
5106:{\displaystyle k<0}
5081:
5043:
5005:
4979:
4945:
4898:leads to the solution
4885:
4271:
4238:base for the logarithm
4224:
4181:
4180:{\displaystyle x>1}
4149:
4148:{\displaystyle k<0}
4123:
4076:
4054:
3969:
3740:
3471:
3429:
3360:
3273:
3197:
3140:
3126:measurable subsets of
3116:
3096:
3058:
2977:
2957:
2933:
2911:
2832:
2812:
2780:
2739:
2696:
2650:
2628:
2555:
2529:
2466:
2438:
2408:
2358:
2338:
2318:
2259:
2176:
1997:
1973:
1949:
1929:
1903:
1821:
1791:
1790:{\displaystyle p(x)=0}
1688:
1588:
1538:
1514:
1414:
1362:
1314:
1290:
1269:(Greek capital letter
1209:
1145:
1111:
1074:
997:
928:
891:
871:
816:
787:
758:
729:
674:
569:
541:gives "natural units"
517:
497:
477:
387:
339:
315:
224:Rate–distortion theory
17059:Kolmogorov complexity
16927:Video characteristics
16304:LZ77 + Huffman + ANS
15634:"Prior Probabilities"
15005:10.1093/jigpal/jzx022
14542:Statistical Mechanics
14513:
14465:
14430:
14373:
14289:nominal distributions
14281:Qualitative variation
14119:
14099:
14079:
14059:
14039:
13981:
13901:
13791:
13660:
13646:
13548:
13431:
13282:
13164:
12998:
12942:
12869:
12733:
12556:orthogonal projection
12535:
12480:
12400:7 August 2023 at the
12354:
12235:
12073:-integrable function
12032:absolutely continuous
11987:
11955:
11920:
11772:information dimension
11751:
11591:
11440:
11340:
11315:
11246:
11166:
11136:
11047:
11014:
10960:
10886:
10781:
10694:
10565:
10504:
10466:communication channel
10456:
10379:
10275:
10157:
10046:
10010:
9894:
9834:
9802:
9619:
9570:
9416:
9316:
9252:Kolmogorov complexity
9152:Entropy of a sequence
8986:Kolmogorov complexity
8972:or in practice using
8920:statistical mechanics
8862:
8839:
8792:and expressed by his
8725:
8595:
8503:statistical mechanics
8455:
8423:
8379:
8238:
8214:
8180:
8102:
8022:
7928:
7867:
7733:
7687:
7633:
7613:
7572:
7417:(that is, evaluating
7393:
7286:
7131:
7106:
7059:
7028:
6946:
6900:
6833:
6704:
6678:
6604:
6578:
6476:
6449:
6395:
6257:
6138:
5923:
5727:
5609:
5577:
5533:
5472:
5108:
5082:
5080:{\displaystyle p\in }
5044:
5006:
4980:
4946:
4896:differential equation
4886:
4272:
4225:
4182:
4150:
4124:
4077:
4055:
3970:
3783:in terms of an event
3741:
3472:
3409:
3340:
3253:
3157:
3141:
3117:
3097:
3059:
2978:
2958:
2934:
2912:
2833:
2813:
2781:
2740:
2697:
2651:
2629:
2556:
2530:
2467:
2439:
2409:
2359:
2339:
2319:
2260:
2177:
1998:
1974:
1950:
1930:
1904:
1822:
1792:
1689:
1589:
1539:
1515:
1415:
1363:
1315:
1291:
1263:Boltzmann's Η-theorem
1210:
1146:
1144:{\displaystyle p=1/2}
1112:
1110:{\displaystyle p=1/6}
1075:
998:
929:
892:
890:{\displaystyle \log }
872:
817:
788:
759:
730:
675:
594:source coding theorem
590:communication channel
563:
518:
516:{\displaystyle \log }
498:
478:
388:
340:
316:
17149:Compression software
16743:Compression artifact
16699:Psychoacoustic model
15783:Gray, R. M. (2011),
15279:27 July 2013 at the
14769:Applied Cryptography
14617:20 June 2014 at the
14474:
14439:
14382:
14350:
14283:– other measures of
14266:Levenshtein distance
14252:Information geometry
14108:
14088:
14068:
14048:
14010:
13941:
13806:
13657:
13570:
13446:
13327:
13189:
13120:
12954:
12807:
12743:Shearer's inequality
12571:
12448:
12411:Use in combinatorics
12328:
12295:Use in number theory
12092:
11964:
11938:
11836:
11614:
11333:
11145:
11026:
10901:
10801:
10770:
10577:
10552:
10530:Differential entropy
10524:Differential entropy
10472:
10025:
9847:
9819:
9646:
9592:
9458:
9441:. For a first-order
9352:
9298:
9032:pigeonhole principle
8947:Landauer's principle
8851:
8803:
8665:
8535:
8523:thermodynamic system
8432:
8392:
8253:
8227:
8192:
8118:
8055:
7972:
7880:
7748:
7696:
7642:
7638:is a function, then
7622:
7587:
7457:
7326:
7169:
7118:
7069:
7046:
6958:
6909:
6845:
6716:
6687:
6616:
6587:
6516:
6458:
6408:
6333:
6013:
5739:
5618:
5589:
5542:
5484:
5315:
5124:units of information
5091:
5053:
5015:
4989:
4955:
4902:
4283:
4261:
4191:
4165:
4133:
4086:
4066:
3979:
3959:
3511:
3222:
3130:
3115:{\displaystyle \mu }
3106:
3068:
2987:
2967:
2947:
2923:
2842:
2822:
2790:
2749:
2706:
2667:
2649:{\displaystyle \mu }
2640:
2565:
2545:
2476:
2456:
2422:
2380:
2348:
2328:
2269:
2186:
2007:
1983:
1959:
1939:
1919:
1849:
1801:
1766:
1601:
1566:
1526:
1426:
1372:
1324:
1300:
1280:
1186:
1121:
1087:
1007:
938:
918:
881:
826:
815:{\displaystyle p(E)}
797:
786:{\displaystyle p(E)}
768:
757:{\displaystyle p(E)}
739:
719:
635:
629:differential entropy
507:
487:
397:
349:
325:
305:
188:Directed information
168:Differential entropy
61:improve this article
30:For other uses, see
17139:Compression formats
16778:Texture compression
16773:Standard test image
16589:Silence compression
15939:31 May 2016 at the
15921:4 June 2016 at the
15841:3 June 2016 at the
15317:2003GloEB..12..177S
15164:1957PhRv..106..620J
14682:10 May 2013 at the
14184:Approximate entropy
14146:logistic regression
12431:: for every subset
11864:
11705:
11541:
11410:
10921:
10861:
9056:Type of Information
9052:
9026:scheme can shorten
8652:von Neumann entropy
6702:{\displaystyle X,Y}
6602:{\displaystyle X,Y}
6402:conditional entropy
5004:{\displaystyle c=0}
4985:. Property 2 gives
4240:. Thus, entropy is
4236:corresponds to the
1913:conditional entropy
1554:information content
703:information content
576:in his 1948 paper "
173:Conditional entropy
18:Information entropy
17185:Information theory
17047:Information theory
16902:Display resolution
16728:Chroma subsampling
16117:Byte pair encoding
16062:Shannon–Fano–Elias
15769:Arndt, C. (2004),
15480:(1 October 2015).
15246:"The Hutter Prize"
15211:10.1147/rd.53.0183
14727:MacKay, David J.C.
14635:Shannon, Claude E.
14601:10338.dmlcz/101429
14570:Shannon, Claude E.
14508:
14460:
14425:
14400:
14368:
14271:Mutual information
14237:History of entropy
14222:Fisher information
14212:Entropy estimation
14175:Mathematics portal
14128:Bayesian inference
14114:
14094:
14074:
14054:
14034:
13976:
13896:
13786:
13703:
13641:
13595:
13543:
13426:
13394:
13277:
13159:
12993:
12937:
12728:
12530:
12380:Liouville function
12349:
12311:Liouville function
12230:
11982:
11950:
11915:
11847:
11746:
11688:
11647:
11586:
11584:
11524:
11393:
11310:
11131:
11009:
10959:
10904:
10881:
10829:
10776:
10689:
10560:
10499:
10451:
10005:
9829:
9797:
9731:
9702:
9682:
9614:
9565:
9514:
9494:
9411:
9311:
9260:universal computer
9194:stationary process
9186:stochastic process
9089:Telecommunications
9047:
8929:Boltzmann constant
8884:Boltzmann constant
8857:
8834:
8754:Boltzmann constant
8720:
8614:Boltzmann constant
8590:
8450:
8418:
8374:
8233:
8209:
8175:
8097:
8017:
7923:
7862:
7728:
7682:
7628:
7608:
7567:
7388:
7281:
7144:Further properties
7126:
7101:
7054:
7023:
6983:
6941:
6895:
6828:
6699:
6673:
6599:
6573:
6471:
6444:
6390:
6323:partition of a set
6252:
5918:
5907:
5894:
5805:
5798:
5722:
5604:
5572:
5528:
5467:
5167:constant multiples
5103:
5077:
5039:
5001:
4975:
4941:
4881:
4879:
4870:producing constant
4267:
4220:
4177:
4145:
4119:
4072:
4050:
4020:
3965:
3873:independent events
3736:
3734:
3467:
3465:
3198:
3136:
3112:
3092:
3054:
3029:
2973:
2953:
2929:
2907:
2884:
2828:
2808:
2776:
2735:
2692:
2646:
2624:
2551:
2525:
2462:
2434:
2404:
2354:
2334:
2314:
2255:
2172:
2073:
2003:respectively, as:
1993:
1969:
1945:
1925:
1899:
1874:
1817:
1787:
1684:
1643:
1584:
1534:
1510:
1410:
1358:
1310:
1286:
1233:. Other values of
1205:
1178:. (Similarly, one
1141:
1107:
1070:
993:
924:
887:
867:
812:
783:
754:
725:
670:
586:data communication
570:
513:
493:
473:
439:
383:
335:
311:
290:information theory
183:Mutual information
145:Information theory
17162:
17161:
17011:
17010:
16961:Deblocking filter
16859:
16858:
16707:
16706:
16516:
16515:
16361:
16360:
15864:Library resources
15804:978-0-521-17738-2
15779:978-3-540-40855-0
15765:978-0-521-64298-9
15748:978-0-471-24195-9
15696:978-1-4757-4321-0
15606:978-981-10-6747-1
15514:Discrete Analysis
15451:978-3-540-67185-5
15132:10.3390/e17041971
14880:978-0-471-24195-9
14821:978-0-8218-4256-0
14794:978-3-642-20346-6
14385:
14261:dynamical systems
14136:prior probability
14117:{\displaystyle X}
14097:{\displaystyle X}
14077:{\displaystyle Y}
14057:{\displaystyle Y}
13921:
13920:
13894:
13823:
13696:
13588:
13387:
13367:
12867:
12263:. If the measure
11632:
10944:
10270:
10152:
10000:
9886:
9768:
9759:
9722:
9693:
9673:
9517:
9505:
9485:
9247:unique identifier
9099:
9098:
8982:arithmetic coding
8860:{\displaystyle S}
8819:
8783:signal processing
8684:
8554:
8463:Accordingly, the
8236:{\displaystyle p}
7631:{\displaystyle f}
6961:
6325:. Meanwhile, the
6242:
6216:
6174:
6128:
6102:
6060:
6041:
5972:positive integers
5888:
5861:
5843:
5841:
5792:
5773:
5763:
5761:
5715:
5696:
5157:decimal logarithm
5146:natural logarithm
5118:
5117:
4871:
4861:
4857:
4856:integrating w.r.t
4844:
4802:
4798:
4785:
4708:
4695:
4626:
4622:
4609:
4489:
4485:
4448:
4387:
4330:
4019:
3789:with probability
3439:
3393:
3370:
3212:Bernoulli process
3206:Bernoulli process
3139:{\displaystyle X}
3014:
2976:{\displaystyle M}
2963:. The entropy of
2956:{\displaystyle X}
2932:{\displaystyle M}
2869:
2831:{\displaystyle P}
2786:for all distinct
2554:{\displaystyle A}
2465:{\displaystyle A}
2416:probability space
2357:{\displaystyle Y}
2337:{\displaystyle X}
2167:
2038:
1948:{\displaystyle Y}
1928:{\displaystyle X}
1915:of two variables
1852:
1624:
1061:
1003:or equivalently,
927:{\displaystyle E}
858:
728:{\displaystyle E}
713:self-information,
420:
393:, the entropy is
314:{\displaystyle X}
286:
285:
137:
136:
129:
111:
16:(Redirected from
17202:
17175:Data compression
17147:
17146:
17137:
17136:
16966:Lapped transform
16870:
16869:
16748:Image resolution
16733:Coding tree unit
16718:
16717:
16527:
16526:
16372:
16371:
15993:
15992:
15979:Data compression
15972:
15965:
15958:
15949:
15948:
15912:
15808:
15701:
15700:
15680:
15674:
15673:
15671:
15669:
15629:
15623:
15622:
15620:
15618:
15580:
15574:
15571:
15565:
15562:
15556:
15555:
15553:
15551:
15525:
15505:
15499:
15498:
15496:
15494:
15478:Klarreich, Erica
15474:
15468:
15462:
15456:
15455:
15443:
15427:
15421:
15420:
15418:
15416:
15410:
15399:
15390:
15384:
15383:
15381:
15379:
15373:
15362:
15353:
15347:
15346:
15328:
15296:
15290:
15271:
15262:
15261:
15259:
15257:
15248:. Archived from
15241:
15235:
15234:
15232:
15230:
15190:
15184:
15183:
15143:
15137:
15136:
15134:
15110:
15104:
15103:
15095:
15089:
15078:
15072:
15071:
15035:
15024:
15023:
15021:
15019:
15013:
14990:
14981:
14975:
14958:
14952:
14951:
14945:
14937:
14935:
14933:
14927:
14920:
14909:
14903:
14891:
14885:
14884:
14866:
14839:
14834:Schneider, T.D,
14832:
14826:
14825:
14805:
14799:
14798:
14778:
14772:
14765:
14759:
14758:
14756:
14754:
14723:
14717:
14716:
14710:
14708:
14693:
14687:
14675:, archived from
14670:
14668:
14637:(October 1948).
14631:
14622:
14610:, archived from
14605:
14603:
14566:
14557:
14556:
14536:
14519:
14517:
14515:
14514:
14509:
14489:
14488:
14469:
14467:
14466:
14461:
14434:
14432:
14431:
14426:
14399:
14377:
14375:
14374:
14369:
14344:
14232:Hamming distance
14206:Entropy encoding
14177:
14172:
14171:
14123:
14121:
14120:
14115:
14103:
14101:
14100:
14095:
14083:
14081:
14080:
14075:
14063:
14061:
14060:
14055:
14043:
14041:
14040:
14035:
13995:Machine learning
13985:
13983:
13982:
13977:
13975:
13974:
13967:
13956:
13936:
13930:
13915:
13909:since there are
13905:
13903:
13902:
13897:
13895:
13893:
13879:
13874:
13873:
13843:
13842:
13830:
13829:
13828:
13815:
13795:
13793:
13792:
13787:
13779:
13778:
13742:
13741:
13714:
13713:
13704:
13702:
13701:
13688:
13679:
13674:
13650:
13648:
13647:
13642:
13640:
13639:
13609:
13608:
13596:
13594:
13593:
13580:
13558:
13552:
13550:
13549:
13544:
13521:
13520:
13481:
13480:
13453:
13435:
13433:
13432:
13427:
13422:
13421:
13411:
13395:
13393:
13392:
13379:
13368:
13366:
13355:
13354:
13344:
13331:
13319:
13305:
13286:
13284:
13283:
13278:
13276:
13262:
13261:
13252:
13235:
13234:
13233:
13232:
13212:
13211:
13196:
13184:
13169:is contained in
13168:
13166:
13165:
13160:
13158:
13157:
13156:
13155:
13135:
13134:
13116:}. The range of
13115:
13092:
13086:
13084:
13076:
13074:
13062:
13056:
13050:
13041:
13030:
13019:
13013:
13002:
13000:
12999:
12994:
12992:
12991:
12990:
12989:
12969:
12968:
12946:
12944:
12943:
12938:
12933:
12932:
12931:
12930:
12910:
12909:
12894:
12888:
12883:
12868:
12860:
12849:
12848:
12830:
12829:
12814:
12799:
12794:lies in exactly
12793:
12787:
12780:
12762:
12737:
12735:
12734:
12729:
12712:
12711:
12693:
12692:
12674:
12673:
12655:
12654:
12636:
12635:
12611:
12610:
12583:
12582:
12563:
12553:
12539:
12537:
12536:
12531:
12529:
12515:
12514:
12505:
12499:
12494:
12476:
12475:
12464:
12455:
12440:
12395:"parity barrier"
12369:
12358:
12356:
12355:
12350:
12323:
12290:
12284:
12278:
12268:
12261:Lebesgue measure
12258:
12251:counting measure
12248:
12239:
12237:
12236:
12231:
12109:
12108:
12107:
12084:
12078:
12072:
12066:
12039:
12029:
12023:
12012:relative entropy
12000:Relative entropy
11991:
11989:
11988:
11983:
11959:
11957:
11956:
11951:
11933:
11924:
11922:
11921:
11916:
11863:
11858:
11843:
11828:
11822:
11816:
11809:
11798:
11769:
11755:
11753:
11752:
11747:
11704:
11699:
11681:
11677:
11664:
11663:
11658:
11646:
11606:
11602:
11595:
11593:
11592:
11587:
11585:
11540:
11535:
11510:
11509:
11479:
11478:
11462:
11457:
11409:
11404:
11379:
11378:
11362:
11357:
11325:
11319:
11317:
11316:
11311:
11285:
11284:
11268:
11263:
11236:
11235:
11205:
11204:
11188:
11183:
11159:
11158:
11153:
11140:
11138:
11137:
11132:
11130:
11126:
11119:
11118:
11086:
11085:
11069:
11064:
11040:
11039:
11034:
11018:
11016:
11015:
11010:
10999:
10998:
10982:
10977:
10958:
10920:
10915:
10896:
10890:
10888:
10887:
10882:
10860:
10840:
10819:
10818:
10796:
10785:
10783:
10782:
10777:
10765:
10756:
10745:
10713:
10707:
10698:
10696:
10695:
10690:
10682:
10646:
10645:
10644:
10601:
10584:
10569:
10567:
10566:
10561:
10559:
10547:
10514:
10508:
10506:
10505:
10500:
10498:
10485:
10484:
10460:
10458:
10457:
10452:
10444:
10443:
10439:
10438:
10415:
10414:
10398:
10393:
10372:
10371:
10356:
10355:
10351:
10350:
10327:
10326:
10305:
10304:
10294:
10289:
10271:
10269:
10256:
10255:
10245:
10241:
10240:
10236:
10235:
10212:
10211:
10190:
10189:
10179:
10176:
10171:
10153:
10151:
10138:
10137:
10127:
10120:
10119:
10098:
10097:
10085:
10084:
10068:
10065:
10060:
10014:
10012:
10011:
10006:
10001:
9999:
9986:
9985:
9975:
9968:
9967:
9946:
9945:
9933:
9932:
9916:
9913:
9908:
9887:
9885:
9884:
9866:
9838:
9836:
9835:
9830:
9828:
9827:
9806:
9804:
9803:
9798:
9784:
9783:
9766:
9757:
9747:
9746:
9730:
9712:
9711:
9701:
9692:
9691:
9681:
9663:
9662:
9653:
9635:
9629:
9623:
9621:
9620:
9615:
9604:
9603:
9583:
9574:
9572:
9571:
9566:
9552:
9551:
9527:
9526:
9515:
9513:
9504:
9503:
9493:
9475:
9474:
9465:
9440:
9434:
9420:
9418:
9417:
9412:
9407:
9406:
9391:
9390:
9369:
9368:
9359:
9320:
9318:
9317:
9312:
9310:
9309:
9277:
9244:
9224:
9217:
9163:self-information
9143:
9053:
9046:
9037:A 2011 study in
8963:Data compression
8953:Data compression
8909:
8900:
8881:
8872:
8866:
8864:
8863:
8858:
8843:
8841:
8840:
8835:
8821:
8820:
8817:
8790:Ludwig Boltzmann
8779:data compression
8776:
8763:
8729:
8727:
8726:
8721:
8697:
8696:
8686:
8685:
8682:
8656:John von Neumann
8637:J. Willard Gibbs
8626:
8611:
8599:
8597:
8596:
8591:
8585:
8584:
8569:
8568:
8556:
8555:
8552:
8520:
8469:convex conjugate
8465:negative entropy
8459:
8457:
8456:
8451:
8427:
8425:
8424:
8419:
8417:
8416:
8404:
8403:
8383:
8381:
8380:
8375:
8370:
8369:
8357:
8331:
8330:
8318:
8304:
8303:
8276:
8275:
8260:
8242:
8240:
8239:
8234:
8218:
8216:
8215:
8210:
8199:
8184:
8182:
8181:
8176:
8165:
8148:
8125:
8106:
8104:
8103:
8098:
8087:
8073:
8062:
8044:
8038:
8026:
8024:
8023:
8018:
8004:
7990:
7979:
7961:
7955:
7949:
7943:
7932:
7930:
7929:
7924:
7913:
7887:
7871:
7869:
7868:
7863:
7843:
7832:
7806:
7792:
7772:
7755:
7737:
7735:
7734:
7729:
7703:
7691:
7689:
7688:
7683:
7669:
7649:
7637:
7635:
7634:
7629:
7617:
7615:
7614:
7609:
7576:
7574:
7573:
7568:
7554:
7540:
7529:
7512:
7498:
7487:
7464:
7446:
7440:
7434:
7428:
7422:
7416:
7397:
7395:
7394:
7389:
7381:
7380:
7365:
7364:
7346:
7345:
7333:
7315:
7290:
7288:
7287:
7282:
7277:
7276:
7258:
7257:
7245:
7244:
7239:
7221:
7220:
7202:
7201:
7189:
7188:
7177:
7154:
7135:
7133:
7132:
7127:
7125:
7110:
7108:
7107:
7102:
7100:
7099:
7081:
7080:
7063:
7061:
7060:
7055:
7053:
7032:
7030:
7029:
7024:
6995:
6994:
6989:
6982:
6981:
6980:
6950:
6948:
6947:
6942:
6940:
6939:
6921:
6920:
6904:
6902:
6901:
6896:
6891:
6890:
6872:
6871:
6859:
6858:
6853:
6837:
6835:
6834:
6829:
6824:
6823:
6805:
6804:
6792:
6791:
6786:
6768:
6767:
6749:
6748:
6736:
6735:
6724:
6712:Expansibility:
6709:are independent.
6708:
6706:
6705:
6700:
6682:
6680:
6679:
6674:
6663:
6646:
6623:
6608:
6606:
6605:
6600:
6582:
6580:
6579:
6574:
6563:
6546:
6523:
6512:Subadditivity:
6480:
6478:
6477:
6472:
6470:
6469:
6453:
6451:
6450:
6445:
6399:
6397:
6396:
6391:
6313:
6307:
6301:
6293:
6274:
6261:
6259:
6258:
6253:
6248:
6244:
6243:
6241:
6240:
6228:
6217:
6215:
6214:
6202:
6195:
6194:
6193:
6192:
6182:
6175:
6170:
6169:
6160:
6157:
6152:
6134:
6130:
6129:
6124:
6123:
6114:
6103:
6098:
6097:
6088:
6081:
6080:
6075:
6066:
6062:
6061:
6053:
6042:
6034:
6027:
6026:
6021:
6005:
5983:
5959:
5941:
5935:
5927:
5925:
5924:
5919:
5914:
5913:
5906:
5895:
5890:
5889:
5887:
5873:
5862:
5860:
5846:
5840:
5839:
5833:
5832:
5821:
5812:
5811:
5804:
5799:
5794:
5793:
5785:
5774:
5766:
5760:
5759:
5753:
5752:
5747:
5731:
5729:
5728:
5723:
5721:
5717:
5716:
5708:
5697:
5689:
5682:
5681:
5676:
5664:
5663:
5645:
5644:
5632:
5631:
5626:
5613:
5611:
5610:
5605:
5603:
5602:
5597:
5581:
5579:
5578:
5573:
5537:
5535:
5534:
5529:
5524:
5523:
5499:
5498:
5476:
5474:
5473:
5468:
5466:
5462:
5461:
5460:
5459:
5458:
5435:
5434:
5433:
5432:
5415:
5414:
5413:
5412:
5393:
5392:
5387:
5378:
5374:
5373:
5372:
5357:
5356:
5344:
5343:
5329:
5328:
5323:
5310:
5299:
5288:
5280:
5250:
5203:decimal digits.
5202:
5195:
5188:
5182:
5176:
5164:
5150:
5139:
5132:binary logarithm
5112:
5110:
5109:
5104:
5086:
5084:
5083:
5078:
5048:
5046:
5045:
5040:
5010:
5008:
5007:
5002:
4984:
4982:
4981:
4976:
4974:
4950:
4948:
4947:
4942:
4890:
4888:
4887:
4882:
4880:
4872:
4869:
4859:
4858:
4855:
4851:
4842:
4818:
4806:
4800:
4799:
4796:
4792:
4783:
4777:
4757:
4740:
4736:
4735:
4726:
4725:
4709:
4706:
4702:
4693:
4675:
4649:
4640:
4636:
4635:
4624:
4623:
4620:
4616:
4607:
4598:
4597:
4588:
4587:
4572:
4564:
4563:
4554:
4553:
4538:
4537:
4528:
4527:
4512:
4503:
4499:
4498:
4487:
4486:
4483:
4479:
4474:
4473:
4458:
4446:
4437:
4436:
4427:
4426:
4411:
4403:
4402:
4392:
4388:
4385:
4381:
4376:
4375:
4351:
4350:
4328:
4319:
4318:
4309:
4308:
4289:
4276:
4274:
4273:
4268:
4249:
4235:
4229:
4227:
4226:
4221:
4210:
4186:
4184:
4183:
4178:
4160:
4154:
4152:
4151:
4146:
4128:
4126:
4125:
4120:
4081:
4079:
4078:
4073:
4059:
4057:
4056:
4051:
4025:
4021:
4012:
3974:
3972:
3971:
3966:
3952:to encode both.
3951:
3923:
3911:
3899:
3893:
3884:
3870:
3836:
3830:
3820:
3805:
3799:
3788:
3782:
3778:
3753:Characterization
3745:
3743:
3742:
3737:
3735:
3716:
3664:
3648:
3647:
3620:
3619:
3598:
3582:
3581:
3554:
3553:
3522:
3506:
3500:
3490:
3484:
3476:
3474:
3473:
3468:
3466:
3456:
3440:
3432:
3428:
3423:
3399:
3395:
3394:
3386:
3381:
3380:
3371:
3363:
3359:
3354:
3330:
3326:
3322:
3321:
3303:
3302:
3290:
3289:
3272:
3267:
3233:
3187:
3180:
3165:
3145:
3143:
3142:
3137:
3121:
3119:
3118:
3113:
3101:
3099:
3098:
3093:
3082:
3081:
3076:
3063:
3061:
3060:
3055:
3041:
3040:
3035:
3028:
3001:
3000:
2995:
2982:
2980:
2979:
2974:
2962:
2960:
2959:
2954:
2938:
2936:
2935:
2930:
2916:
2914:
2913:
2908:
2894:
2893:
2883:
2856:
2855:
2850:
2837:
2835:
2834:
2829:
2817:
2815:
2814:
2809:
2785:
2783:
2782:
2777:
2744:
2742:
2741:
2736:
2719:
2701:
2699:
2698:
2693:
2682:
2681:
2655:
2653:
2652:
2647:
2633:
2631:
2630:
2625:
2611:
2610:
2577:
2576:
2560:
2558:
2557:
2552:
2534:
2532:
2531:
2526:
2488:
2487:
2471:
2469:
2468:
2463:
2443:
2441:
2440:
2435:
2413:
2411:
2410:
2405:
2376:as follows: Let
2363:
2361:
2360:
2355:
2343:
2341:
2340:
2335:
2323:
2321:
2320:
2315:
2298:
2281:
2280:
2264:
2262:
2261:
2256:
2227:
2204:
2203:
2181:
2179:
2178:
2173:
2168:
2166:
2156:
2155:
2145:
2129:
2128:
2112:
2089:
2088:
2072:
2071:
2070:
2061:
2060:
2025:
2014:
2002:
2000:
1999:
1994:
1992:
1991:
1978:
1976:
1975:
1970:
1968:
1967:
1954:
1952:
1951:
1946:
1934:
1932:
1931:
1926:
1908:
1906:
1905:
1900:
1873:
1872:
1871:
1840:
1836:
1826:
1824:
1823:
1818:
1816:
1815:
1796:
1794:
1793:
1788:
1758:
1747:
1733:
1721:
1713:
1699:
1693:
1691:
1690:
1685:
1665:
1664:
1642:
1641:
1640:
1608:
1593:
1591:
1590:
1585:
1561:
1551:
1543:
1541:
1540:
1535:
1533:
1519:
1517:
1516:
1511:
1479:
1450:
1433:
1419:
1417:
1416:
1411:
1394:
1367:
1365:
1364:
1359:
1339:
1338:
1319:
1317:
1316:
1311:
1309:
1308:
1295:
1293:
1292:
1287:
1268:
1247:data compression
1228:
1221:
1214:
1212:
1211:
1206:
1198:
1197:
1173:
1166:
1159:
1150:
1148:
1147:
1142:
1137:
1116:
1114:
1113:
1108:
1103:
1079:
1077:
1076:
1071:
1066:
1062:
1060:
1043:
1034:
1033:
1002:
1000:
999:
994:
968:
967:
933:
931:
930:
925:
904:
896:
894:
893:
888:
876:
874:
873:
868:
863:
859:
857:
840:
821:
819:
818:
813:
792:
790:
789:
784:
763:
761:
760:
755:
734:
732:
731:
726:
707:also called the
679:
677:
676:
671:
642:
621:machine learning
567:
555:self-information
522:
520:
519:
514:
502:
500:
499:
494:
482:
480:
479:
474:
438:
437:
436:
404:
392:
390:
389:
384:
364:
363:
344:
342:
341:
336:
334:
333:
320:
318:
317:
312:
278:
271:
264:
240:Channel capacity
198:Relative entropy
155:
141:
140:
132:
125:
121:
118:
112:
110:
69:
45:
37:
21:
17210:
17209:
17205:
17204:
17203:
17201:
17200:
17199:
17165:
17164:
17163:
17158:
17125:
17109:
17093:
17074:Rate–distortion
17007:
16936:
16855:
16782:
16703:
16608:
16604:Sub-band coding
16512:
16437:Predictive type
16432:
16357:
16324:LZSS + Huffman
16274:LZ77 + Huffman
16263:
16173:
16109:Dictionary type
16103:
16005:Adaptive coding
15982:
15976:
15941:Wayback Machine
15923:Wayback Machine
15897:
15894:
15893:
15892:
15872:
15871:
15867:
15860:
15843:Wayback Machine
15805:
15728:
15723:
15721:Further reading
15705:
15704:
15697:
15681:
15677:
15667:
15665:
15630:
15626:
15616:
15614:
15607:
15581:
15577:
15572:
15568:
15563:
15559:
15549:
15547:
15532:10.19086/da.609
15506:
15502:
15492:
15490:
15487:Quanta Magazine
15475:
15471:
15463:
15459:
15452:
15428:
15424:
15414:
15412:
15408:
15397:
15391:
15387:
15377:
15375:
15371:
15360:
15354:
15350:
15297:
15293:
15281:Wayback Machine
15272:
15265:
15255:
15253:
15252:on 1 March 2018
15242:
15238:
15228:
15226:
15191:
15187:
15152:Physical Review
15144:
15140:
15111:
15107:
15096:
15092:
15079:
15075:
15052:10.2307/1426210
15036:
15027:
15017:
15015:
15011:
14988:
14982:
14978:
14973:Wayback Machine
14959:
14955:
14939:
14938:
14931:
14929:
14925:
14918:
14910:
14906:
14892:
14888:
14881:
14867:
14842:
14833:
14829:
14822:
14806:
14802:
14795:
14779:
14775:
14766:
14762:
14752:
14750:
14743:
14724:
14720:
14706:
14704:
14695:
14694:
14690:
14684:Wayback Machine
14632:
14625:
14619:Wayback Machine
14567:
14560:
14553:
14537:
14533:
14528:
14523:
14522:
14484:
14483:
14475:
14472:
14471:
14440:
14437:
14436:
14389:
14383:
14380:
14379:
14351:
14348:
14347:
14345:
14341:
14336:
14331:
14173:
14166:
14163:
14109:
14106:
14105:
14089:
14086:
14085:
14069:
14066:
14065:
14049:
14046:
14045:
14011:
14008:
14007:
13992:
13963:
13952:
13948:
13944:
13942:
13939:
13938:
13932:
13926:
13910:
13883:
13878:
13860:
13856:
13835:
13831:
13824:
13811:
13810:
13809:
13807:
13804:
13803:
13774:
13770:
13731:
13727:
13709:
13705:
13697:
13684:
13683:
13681:
13675:
13664:
13658:
13655:
13654:
13626:
13622:
13601:
13597:
13589:
13576:
13575:
13573:
13571:
13568:
13567:
13561:Proof (sketch)
13516:
13512:
13476:
13472:
13449:
13447:
13444:
13443:
13407:
13403:
13399:
13388:
13375:
13374:
13372:
13356:
13340:
13336:
13332:
13330:
13328:
13325:
13324:
13307:
13296:
13293:
13272:
13257:
13253:
13248:
13228:
13224:
13217:
13213:
13207:
13203:
13192:
13190:
13187:
13186:
13178:
13170:
13151:
13147:
13140:
13136:
13130:
13126:
13121:
13118:
13117:
13102:
13094:
13088:
13080:
13078:
13070:
13064:
13058:
13052:
13046:
13040:
13032:
13029:
13021:
13015:
13012:
13004:
12985:
12981:
12974:
12970:
12964:
12960:
12955:
12952:
12951:
12926:
12922:
12915:
12911:
12905:
12901:
12890:
12884:
12873:
12859:
12844:
12840:
12825:
12821:
12810:
12808:
12805:
12804:
12795:
12789:
12782:
12781:are subsets of
12779:
12770:
12764:
12761:
12752:
12746:
12707:
12703:
12688:
12684:
12669:
12665:
12644:
12640:
12625:
12621:
12606:
12602:
12578:
12574:
12572:
12569:
12568:
12564:th coordinate:
12559:
12552:
12544:
12525:
12510:
12506:
12501:
12495:
12484:
12465:
12460:
12459:
12451:
12449:
12446:
12445:
12432:
12425:
12413:
12402:Wayback Machine
12391:Wayback Machine
12368:
12360:
12329:
12326:
12325:
12322:
12314:
12297:
12286:
12280:
12270:
12264:
12254:
12244:
12100:
12099:
12095:
12093:
12090:
12089:
12080:
12074:
12068:
12041:
12035:
12025:
12019:
12008:
12002:
11965:
11962:
11961:
11939:
11936:
11935:
11929:
11859:
11851:
11839:
11837:
11834:
11833:
11824:
11818:
11811:
11800:
11794:
11786:
11780:
11764:
11700:
11692:
11659:
11654:
11653:
11652:
11648:
11636:
11615:
11612:
11611:
11604:
11600:
11583:
11582:
11536:
11528:
11517:
11505:
11501:
11474:
11470:
11458:
11444:
11437:
11436:
11405:
11397:
11386:
11374:
11370:
11358:
11344:
11336:
11334:
11331:
11330:
11323:
11280:
11276:
11264:
11250:
11231:
11227:
11200:
11196:
11184:
11170:
11154:
11149:
11148:
11146:
11143:
11142:
11114:
11110:
11103:
11099:
11081:
11077:
11065:
11051:
11035:
11030:
11029:
11027:
11024:
11023:
11022:We will denote
10994:
10990:
10978:
10964:
10948:
10916:
10908:
10902:
10899:
10898:
10892:
10841:
10833:
10814:
10810:
10802:
10799:
10798:
10795:
10787:
10771:
10768:
10767:
10761:
10755:
10747:
10741:
10709:
10703:
10678:
10640:
10639:
10635:
10597:
10580:
10578:
10575:
10574:
10555:
10553:
10550:
10549:
10538:
10532:
10526:
10521:
10510:
10480:
10476:
10475:
10473:
10470:
10469:
10434:
10430:
10420:
10416:
10410:
10406:
10394:
10383:
10367:
10363:
10346:
10342:
10332:
10328:
10322:
10318:
10300:
10296:
10290:
10279:
10251:
10247:
10246:
10231:
10227:
10217:
10213:
10207:
10203:
10185:
10181:
10180:
10178:
10172:
10161:
10133:
10129:
10128:
10115:
10111:
10093:
10089:
10080:
10076:
10069:
10067:
10061:
10050:
10026:
10023:
10022:
9981:
9977:
9976:
9963:
9959:
9941:
9937:
9928:
9924:
9917:
9915:
9909:
9898:
9874:
9870:
9865:
9848:
9845:
9844:
9823:
9822:
9820:
9817:
9816:
9813:
9773:
9769:
9736:
9732:
9726:
9707:
9703:
9697:
9687:
9683:
9677:
9658:
9657:
9649:
9647:
9644:
9643:
9631:
9625:
9599:
9595:
9593:
9590:
9589:
9579:
9547:
9543:
9522:
9518:
9509:
9499:
9495:
9489:
9470:
9469:
9461:
9459:
9456:
9455:
9436:
9433:
9425:
9402:
9398:
9386:
9382:
9364:
9363:
9355:
9353:
9350:
9349:
9339:
9305:
9301:
9299:
9296:
9295:
9284:
9271:
9267:
9238:
9230:
9226:
9219:
9213:
9154:
9141:
9122:
9120:Diversity index
9116:
8965:
8957:Main articles:
8955:
8939:Maxwell's demon
8914:In the view of
8908:
8902:
8895:
8887:
8880:
8874:
8868:
8852:
8849:
8848:
8816:
8812:
8804:
8801:
8800:
8775:
8765:
8762:
8756:
8733:where ρ is the
8689:
8688:
8681:
8677:
8666:
8663:
8662:
8648:quantum physics
8635:was defined by
8625:
8617:
8610:
8604:
8580:
8576:
8564:
8560:
8551:
8547:
8536:
8533:
8532:
8516:
8495:
8489:
8484:
8433:
8430:
8429:
8412:
8408:
8399:
8395:
8393:
8390:
8389:
8365:
8361:
8353:
8326:
8322:
8314:
8299:
8295:
8271:
8267:
8256:
8254:
8251:
8250:
8228:
8225:
8224:
8195:
8193:
8190:
8189:
8161:
8144:
8121:
8119:
8116:
8115:
8083:
8069:
8058:
8056:
8053:
8052:
8040:
8034:
8000:
7986:
7975:
7973:
7970:
7969:
7957:
7951:
7945:
7939:
7909:
7883:
7881:
7878:
7877:
7839:
7828:
7802:
7788:
7768:
7751:
7749:
7746:
7745:
7699:
7697:
7694:
7693:
7665:
7645:
7643:
7640:
7639:
7623:
7620:
7619:
7588:
7585:
7584:
7550:
7536:
7525:
7508:
7494:
7483:
7460:
7458:
7455:
7454:
7442:
7436:
7430:
7424:
7418:
7406:
7376:
7372:
7360:
7356:
7341:
7337:
7329:
7327:
7324:
7323:
7309:
7303:
7272:
7268:
7253:
7249:
7240:
7235:
7234:
7216:
7212:
7197:
7193:
7178:
7173:
7172:
7170:
7167:
7166:
7150:
7146:
7138:Hartley entropy
7121:
7119:
7116:
7115:
7095:
7091:
7076:
7072:
7070:
7067:
7066:
7049:
7047:
7044:
7043:
7040:
6990:
6985:
6984:
6976:
6972:
6965:
6959:
6956:
6955:
6935:
6931:
6916:
6912:
6910:
6907:
6906:
6886:
6882:
6867:
6863:
6854:
6849:
6848:
6846:
6843:
6842:
6819:
6815:
6800:
6796:
6787:
6782:
6781:
6763:
6759:
6744:
6740:
6725:
6720:
6719:
6717:
6714:
6713:
6688:
6685:
6684:
6659:
6642:
6619:
6617:
6614:
6613:
6588:
6585:
6584:
6559:
6542:
6519:
6517:
6514:
6513:
6502:
6465:
6461:
6459:
6456:
6455:
6409:
6406:
6405:
6334:
6331:
6330:
6309:
6303:
6299:
6295:
6291:
6282:
6276:
6266:
6236:
6232:
6227:
6210:
6206:
6201:
6200:
6196:
6188:
6184:
6183:
6178:
6177:
6165:
6161:
6159:
6153:
6142:
6119:
6115:
6113:
6093:
6089:
6087:
6086:
6082:
6076:
6071:
6070:
6052:
6033:
6032:
6028:
6022:
6017:
6016:
6014:
6011:
6010:
6000:
5991:
5985:
5982:
5974:
5968:
5958:
5949:
5943:
5937:
5931:
5909:
5908:
5896:
5877:
5872:
5850:
5845:
5844:
5842:
5835:
5834:
5822:
5817:
5816:
5807:
5806:
5800:
5784:
5765:
5764:
5762:
5755:
5754:
5748:
5743:
5742:
5740:
5737:
5736:
5707:
5688:
5687:
5683:
5677:
5672:
5671:
5659:
5655:
5640:
5636:
5627:
5622:
5621:
5619:
5616:
5615:
5598:
5593:
5592:
5590:
5587:
5586:
5543:
5540:
5539:
5519:
5515:
5494:
5490:
5485:
5482:
5481:
5454:
5450:
5449:
5445:
5428:
5424:
5423:
5419:
5408:
5404:
5403:
5399:
5398:
5394:
5388:
5383:
5382:
5368:
5364:
5352:
5348:
5339:
5335:
5334:
5330:
5324:
5319:
5318:
5316:
5313:
5312:
5309:
5301:
5297:
5286:
5274:
5265:
5258:
5252:
5248:
5235:
5227:
5224:
5197:
5190:
5184:
5183:tosses provide
5178:
5174:
5170:
5165:and so on) are
5163:
5159:
5148:
5138:
5134:
5092:
5089:
5088:
5054:
5051:
5050:
5016:
5013:
5012:
4990:
4987:
4986:
4970:
4956:
4953:
4952:
4903:
4900:
4899:
4878:
4877:
4868:
4854:
4850:
4845:
4837:
4811:
4804:
4803:
4795:
4791:
4786:
4778:
4770:
4750:
4738:
4737:
4731:
4727:
4721:
4717:
4705:
4701:
4696:
4688:
4668:
4642:
4638:
4637:
4631:
4627:
4619:
4615:
4610:
4602:
4593:
4589:
4583:
4579:
4565:
4559:
4555:
4549:
4545:
4533:
4529:
4523:
4519:
4505:
4501:
4500:
4494:
4490:
4482:
4478:
4469:
4465:
4451:
4449:
4441:
4432:
4428:
4422:
4418:
4404:
4398:
4394:
4390:
4389:
4384:
4380:
4371:
4367:
4346:
4342:
4331:
4323:
4314:
4310:
4304:
4300:
4286:
4284:
4281:
4280:
4262:
4259:
4258:
4231:
4206:
4192:
4189:
4188:
4166:
4163:
4162:
4156:
4134:
4131:
4130:
4087:
4084:
4083:
4067:
4064:
4063:
4010:
4006:
3980:
3977:
3976:
3960:
3957:
3956:
3945:
3937:
3929:
3925:
3917:
3913:
3905:
3901:
3895:
3889:
3880:
3868:
3861:
3854:
3847:
3840:
3834:
3826:
3814:
3801:
3798:
3790:
3784:
3780:
3776:
3767:
3758:
3755:
3733:
3732:
3714:
3713:
3662:
3661:
3643:
3639:
3615:
3611:
3596:
3595:
3577:
3573:
3549:
3545:
3532:
3518:
3514:
3512:
3509:
3508:
3502:
3492:
3486:
3480:
3464:
3463:
3431:
3430:
3424:
3413:
3397:
3396:
3385:
3376:
3372:
3362:
3361:
3355:
3344:
3328:
3327:
3317:
3313:
3298:
3294:
3285:
3281:
3274:
3268:
3257:
3243:
3229:
3225:
3223:
3220:
3219:
3208:
3200:Main articles:
3195:
3190:
3189:
3182:
3174:
3159:
3152:
3131:
3128:
3127:
3107:
3104:
3103:
3077:
3072:
3071:
3069:
3066:
3065:
3036:
3031:
3030:
3018:
2996:
2991:
2990:
2988:
2985:
2984:
2968:
2965:
2964:
2948:
2945:
2944:
2924:
2921:
2920:
2889:
2885:
2873:
2851:
2846:
2845:
2843:
2840:
2839:
2823:
2820:
2819:
2791:
2788:
2787:
2750:
2747:
2746:
2715:
2707:
2704:
2703:
2677:
2676:
2668:
2665:
2664:
2641:
2638:
2637:
2606:
2602:
2572:
2568:
2566:
2563:
2562:
2546:
2543:
2542:
2483:
2479:
2477:
2474:
2473:
2457:
2454:
2453:
2423:
2420:
2419:
2381:
2378:
2377:
2370:
2349:
2346:
2345:
2329:
2326:
2325:
2294:
2276:
2272:
2270:
2267:
2266:
2223:
2193:
2189:
2187:
2184:
2183:
2151:
2147:
2146:
2118:
2114:
2113:
2111:
2078:
2074:
2066:
2065:
2056:
2055:
2042:
2021:
2010:
2008:
2005:
2004:
1987:
1986:
1984:
1981:
1980:
1963:
1962:
1960:
1957:
1956:
1940:
1937:
1936:
1920:
1917:
1916:
1867:
1863:
1856:
1850:
1847:
1846:
1838:
1837:is taken to be
1834:
1828:
1811:
1810:
1802:
1799:
1798:
1767:
1764:
1763:
1762:In the case of
1753:
1739:
1728:
1717:
1716:Euler's number
1709:
1695:
1660:
1656:
1636:
1635:
1628:
1604:
1602:
1599:
1598:
1567:
1564:
1563:
1557:
1549:
1529:
1527:
1524:
1523:
1475:
1446:
1429:
1427:
1424:
1423:
1390:
1373:
1370:
1369:
1334:
1333:
1325:
1322:
1321:
1304:
1303:
1301:
1298:
1297:
1281:
1278:
1277:
1266:
1259:
1243:
1223:
1216:
1193:
1189:
1187:
1184:
1183:
1168:
1161:
1155:
1133:
1122:
1119:
1118:
1099:
1088:
1085:
1084:
1047:
1042:
1038:
1029:
1025:
1008:
1005:
1004:
963:
959:
939:
936:
935:
919:
916:
915:
902:
882:
879:
878:
844:
839:
835:
827:
824:
823:
798:
795:
794:
769:
766:
765:
740:
737:
736:
720:
717:
716:
686:
638:
636:
633:
632:
582:Shannon entropy
565:
557:of a variable.
535:"), while base
508:
505:
504:
488:
485:
484:
432:
431:
424:
400:
398:
395:
394:
359:
358:
350:
347:
346:
329:
328:
326:
323:
322:
306:
303:
302:
298:random variable
282:
133:
122:
116:
113:
70:
68:
58:
46:
35:
28:
23:
22:
15:
12:
11:
5:
17208:
17198:
17197:
17192:
17187:
17182:
17177:
17160:
17159:
17157:
17156:
17141:
17130:
17127:
17126:
17124:
17123:
17117:
17115:
17111:
17110:
17108:
17107:
17101:
17099:
17095:
17094:
17092:
17091:
17086:
17081:
17076:
17071:
17066:
17061:
17056:
17055:
17054:
17044:
17039:
17038:
17037:
17032:
17021:
17019:
17013:
17012:
17009:
17008:
17006:
17005:
17004:
17003:
16998:
16988:
16987:
16986:
16981:
16976:
16968:
16963:
16958:
16953:
16947:
16945:
16938:
16937:
16935:
16934:
16929:
16924:
16919:
16914:
16909:
16904:
16899:
16898:
16897:
16892:
16887:
16876:
16874:
16867:
16861:
16860:
16857:
16856:
16854:
16853:
16852:
16851:
16846:
16841:
16836:
16826:
16821:
16816:
16811:
16806:
16801:
16796:
16790:
16788:
16784:
16783:
16781:
16780:
16775:
16770:
16765:
16760:
16755:
16750:
16745:
16740:
16735:
16730:
16724:
16722:
16715:
16709:
16708:
16705:
16704:
16702:
16701:
16696:
16691:
16690:
16689:
16684:
16679:
16674:
16669:
16659:
16658:
16657:
16647:
16646:
16645:
16640:
16630:
16625:
16619:
16617:
16610:
16609:
16607:
16606:
16601:
16596:
16591:
16586:
16581:
16576:
16571:
16566:
16561:
16556:
16555:
16554:
16549:
16544:
16533:
16531:
16524:
16518:
16517:
16514:
16513:
16511:
16510:
16508:Psychoacoustic
16505:
16504:
16503:
16498:
16493:
16485:
16484:
16483:
16478:
16473:
16468:
16463:
16453:
16452:
16451:
16440:
16438:
16434:
16433:
16431:
16430:
16429:
16428:
16423:
16418:
16408:
16403:
16398:
16397:
16396:
16391:
16380:
16378:
16376:Transform type
16369:
16363:
16362:
16359:
16358:
16356:
16355:
16354:
16353:
16345:
16344:
16343:
16340:
16332:
16331:
16330:
16322:
16321:
16320:
16312:
16311:
16310:
16302:
16301:
16300:
16292:
16291:
16290:
16285:
16280:
16271:
16269:
16265:
16264:
16262:
16261:
16256:
16251:
16246:
16241:
16236:
16235:
16234:
16229:
16219:
16214:
16209:
16208:
16207:
16197:
16192:
16187:
16181:
16179:
16175:
16174:
16172:
16171:
16170:
16169:
16164:
16159:
16154:
16149:
16144:
16139:
16134:
16129:
16119:
16113:
16111:
16105:
16104:
16102:
16101:
16100:
16099:
16094:
16089:
16084:
16074:
16069:
16064:
16059:
16054:
16049:
16044:
16043:
16042:
16037:
16032:
16022:
16017:
16012:
16007:
16001:
15999:
15990:
15984:
15983:
15975:
15974:
15967:
15960:
15952:
15946:
15945:
15930:
15913:
15891:
15890:
15885:
15880:
15874:
15873:
15862:
15861:
15859:
15858:External links
15856:
15855:
15854:
15851:978-0956372857
15830:
15809:
15803:
15788:
15781:
15767:
15753:MacKay, D.J.C.
15750:
15727:
15724:
15722:
15719:
15703:
15702:
15695:
15675:
15644:(3): 227–241.
15624:
15605:
15575:
15566:
15557:
15500:
15469:
15457:
15450:
15422:
15385:
15348:
15311:(3): 177–179.
15291:
15263:
15236:
15205:(3): 183–191.
15185:
15158:(4): 620–630.
15138:
15105:
15090:
15073:
15046:(1): 131–146.
15025:
14999:(5): 806–835.
14976:
14953:
14904:
14886:
14879:
14840:
14827:
14820:
14800:
14793:
14773:
14760:
14741:
14718:
14688:
14651:(4): 623–656.
14623:
14586:(3): 379–423.
14558:
14552:978-0123821881
14551:
14530:
14529:
14527:
14524:
14521:
14520:
14507:
14504:
14501:
14498:
14495:
14492:
14487:
14482:
14479:
14459:
14456:
14453:
14450:
14447:
14444:
14424:
14421:
14418:
14415:
14412:
14409:
14406:
14403:
14398:
14395:
14392:
14388:
14367:
14364:
14361:
14358:
14355:
14338:
14337:
14335:
14332:
14330:
14329:
14324:
14319:
14314:
14311:Sample entropy
14308:
14303:
14297:
14291:
14278:
14273:
14268:
14263:
14254:
14249:
14244:
14239:
14234:
14229:
14224:
14219:
14214:
14209:
14203:
14198:
14192:
14187:
14180:
14179:
14178:
14162:
14159:
14113:
14093:
14073:
14053:
14033:
14030:
14027:
14024:
14021:
14018:
14015:
13991:
13988:
13973:
13970:
13966:
13962:
13959:
13955:
13951:
13947:
13923:
13922:
13919:
13918:
13907:
13906:
13892:
13889:
13886:
13882:
13877:
13872:
13869:
13866:
13863:
13859:
13855:
13852:
13849:
13846:
13841:
13838:
13834:
13827:
13822:
13819:
13814:
13797:
13796:
13785:
13782:
13777:
13773:
13769:
13766:
13763:
13760:
13757:
13754:
13751:
13748:
13745:
13740:
13737:
13734:
13730:
13726:
13723:
13720:
13717:
13712:
13708:
13700:
13695:
13692:
13687:
13678:
13673:
13670:
13667:
13663:
13638:
13635:
13632:
13629:
13625:
13621:
13618:
13615:
13612:
13607:
13604:
13600:
13592:
13587:
13584:
13579:
13563:
13562:
13554:
13553:
13542:
13539:
13536:
13533:
13530:
13527:
13524:
13519:
13515:
13511:
13508:
13505:
13502:
13499:
13496:
13493:
13490:
13487:
13484:
13479:
13475:
13471:
13468:
13465:
13462:
13459:
13456:
13452:
13437:
13436:
13425:
13420:
13417:
13414:
13410:
13406:
13402:
13398:
13391:
13386:
13383:
13378:
13371:
13365:
13362:
13359:
13353:
13350:
13347:
13343:
13339:
13335:
13292:
13289:
13275:
13271:
13268:
13265:
13260:
13256:
13251:
13247:
13244:
13241:
13238:
13231:
13227:
13223:
13220:
13216:
13210:
13206:
13202:
13199:
13195:
13174:
13154:
13150:
13146:
13143:
13139:
13133:
13129:
13125:
13103:= {1, 2, ...,
13098:
13036:
13025:
13008:
12988:
12984:
12980:
12977:
12973:
12967:
12963:
12959:
12948:
12947:
12936:
12929:
12925:
12921:
12918:
12914:
12908:
12904:
12900:
12897:
12893:
12887:
12882:
12879:
12876:
12872:
12866:
12863:
12858:
12855:
12852:
12847:
12843:
12839:
12836:
12833:
12828:
12824:
12820:
12817:
12813:
12775:
12768:
12757:
12750:
12739:
12738:
12727:
12724:
12721:
12718:
12715:
12710:
12706:
12702:
12699:
12696:
12691:
12687:
12683:
12680:
12677:
12672:
12668:
12664:
12661:
12658:
12653:
12650:
12647:
12643:
12639:
12634:
12631:
12628:
12624:
12620:
12617:
12614:
12609:
12605:
12601:
12598:
12595:
12592:
12589:
12586:
12581:
12577:
12548:
12541:
12540:
12528:
12524:
12521:
12518:
12513:
12509:
12504:
12498:
12493:
12490:
12487:
12483:
12479:
12474:
12471:
12468:
12463:
12458:
12454:
12424:
12421:
12412:
12409:
12364:
12348:
12345:
12342:
12339:
12336:
12333:
12318:
12296:
12293:
12241:
12240:
12229:
12226:
12223:
12220:
12217:
12214:
12211:
12208:
12205:
12202:
12199:
12196:
12193:
12190:
12187:
12184:
12181:
12178:
12175:
12172:
12169:
12166:
12163:
12160:
12157:
12154:
12151:
12148:
12145:
12142:
12139:
12136:
12133:
12130:
12127:
12124:
12121:
12118:
12115:
12112:
12106:
12103:
12098:
12004:Main article:
12001:
11998:
11981:
11978:
11975:
11972:
11969:
11949:
11946:
11943:
11926:
11925:
11914:
11911:
11908:
11904:
11901:
11897:
11894:
11891:
11888:
11885:
11882:
11879:
11876:
11873:
11870:
11867:
11862:
11857:
11854:
11850:
11846:
11842:
11782:Main article:
11779:
11776:
11757:
11756:
11745:
11742:
11739:
11735:
11732:
11729:
11726:
11723:
11720:
11717:
11714:
11711:
11708:
11703:
11698:
11695:
11691:
11687:
11684:
11680:
11676:
11673:
11670:
11667:
11662:
11657:
11651:
11645:
11642:
11639:
11635:
11631:
11628:
11625:
11622:
11619:
11597:
11596:
11581:
11578:
11575:
11571:
11568:
11565:
11562:
11559:
11556:
11553:
11550:
11547:
11544:
11539:
11534:
11531:
11527:
11523:
11520:
11518:
11516:
11513:
11508:
11504:
11500:
11497:
11494:
11491:
11488:
11485:
11482:
11477:
11473:
11469:
11466:
11461:
11456:
11453:
11450:
11447:
11443:
11439:
11438:
11435:
11432:
11429:
11426:
11422:
11419:
11416:
11413:
11408:
11403:
11400:
11396:
11392:
11389:
11387:
11385:
11382:
11377:
11373:
11369:
11366:
11361:
11356:
11353:
11350:
11347:
11343:
11339:
11338:
11309:
11306:
11303:
11300:
11297:
11294:
11291:
11288:
11283:
11279:
11275:
11272:
11267:
11262:
11259:
11256:
11253:
11249:
11245:
11242:
11239:
11234:
11230:
11226:
11223:
11220:
11217:
11214:
11211:
11208:
11203:
11199:
11195:
11192:
11187:
11182:
11179:
11176:
11173:
11169:
11165:
11162:
11157:
11152:
11129:
11125:
11122:
11117:
11113:
11109:
11106:
11102:
11098:
11095:
11092:
11089:
11084:
11080:
11076:
11073:
11068:
11063:
11060:
11057:
11054:
11050:
11046:
11043:
11038:
11033:
11008:
11005:
11002:
10997:
10993:
10989:
10986:
10981:
10976:
10973:
10970:
10967:
10963:
10957:
10954:
10951:
10947:
10943:
10940:
10937:
10933:
10930:
10927:
10924:
10919:
10914:
10911:
10907:
10880:
10877:
10873:
10870:
10867:
10864:
10859:
10856:
10853:
10850:
10847:
10844:
10839:
10836:
10832:
10828:
10825:
10822:
10817:
10813:
10809:
10806:
10791:
10775:
10751:
10700:
10699:
10688:
10685:
10681:
10676:
10673:
10670:
10667:
10664:
10661:
10658:
10655:
10652:
10649:
10643:
10638:
10634:
10631:
10628:
10625:
10622:
10619:
10616:
10613:
10610:
10607:
10604:
10600:
10596:
10593:
10590:
10587:
10583:
10558:
10528:Main article:
10525:
10522:
10520:
10517:
10497:
10494:
10491:
10488:
10483:
10479:
10462:
10461:
10450:
10447:
10442:
10437:
10433:
10429:
10426:
10423:
10419:
10413:
10409:
10405:
10402:
10397:
10392:
10389:
10386:
10382:
10378:
10375:
10370:
10366:
10362:
10359:
10354:
10349:
10345:
10341:
10338:
10335:
10331:
10325:
10321:
10317:
10314:
10311:
10308:
10303:
10299:
10293:
10288:
10285:
10282:
10278:
10274:
10268:
10265:
10262:
10259:
10254:
10250:
10244:
10239:
10234:
10230:
10226:
10223:
10220:
10216:
10210:
10206:
10202:
10199:
10196:
10193:
10188:
10184:
10175:
10170:
10167:
10164:
10160:
10156:
10150:
10147:
10144:
10141:
10136:
10132:
10126:
10123:
10118:
10114:
10110:
10107:
10104:
10101:
10096:
10092:
10088:
10083:
10079:
10075:
10072:
10064:
10059:
10056:
10053:
10049:
10045:
10042:
10039:
10036:
10033:
10030:
10016:
10015:
10004:
9998:
9995:
9992:
9989:
9984:
9980:
9974:
9971:
9966:
9962:
9958:
9955:
9952:
9949:
9944:
9940:
9936:
9931:
9927:
9923:
9920:
9912:
9907:
9904:
9901:
9897:
9893:
9890:
9883:
9880:
9877:
9873:
9869:
9864:
9861:
9858:
9855:
9852:
9826:
9812:
9809:
9808:
9807:
9796:
9793:
9790:
9787:
9782:
9779:
9776:
9772:
9765:
9762:
9756:
9753:
9750:
9745:
9742:
9739:
9735:
9729:
9725:
9721:
9718:
9715:
9710:
9706:
9700:
9696:
9690:
9686:
9680:
9676:
9672:
9669:
9666:
9661:
9656:
9652:
9613:
9610:
9607:
9602:
9598:
9576:
9575:
9564:
9561:
9558:
9555:
9550:
9546:
9542:
9539:
9536:
9533:
9530:
9525:
9521:
9512:
9508:
9502:
9498:
9492:
9488:
9484:
9481:
9478:
9473:
9468:
9464:
9429:
9422:
9421:
9410:
9405:
9401:
9397:
9394:
9389:
9385:
9381:
9378:
9375:
9372:
9367:
9362:
9358:
9338:
9335:
9308:
9304:
9283:
9280:
9269:
9236:
9228:
9190:
9189:
9176:
9167:
9153:
9150:
9146:true diversity
9118:Main article:
9115:
9112:
9097:
9096:
9093:
9090:
9086:
9085:
9082:
9079:
9075:
9074:
9071:
9068:
9064:
9063:
9060:
9057:
8954:
8951:
8906:
8891:
8878:
8856:
8845:
8844:
8833:
8830:
8827:
8824:
8815:
8811:
8808:
8773:
8760:
8735:density matrix
8731:
8730:
8719:
8715:
8712:
8709:
8706:
8703:
8700:
8695:
8692:
8680:
8676:
8673:
8670:
8654:introduced by
8621:
8608:
8601:
8600:
8589:
8583:
8579:
8575:
8572:
8567:
8563:
8559:
8550:
8546:
8543:
8540:
8491:Main article:
8488:
8485:
8483:
8480:
8479:
8478:
8477:
8476:
8449:
8446:
8443:
8440:
8437:
8415:
8411:
8407:
8402:
8398:
8386:
8385:
8384:
8373:
8368:
8364:
8360:
8356:
8352:
8349:
8346:
8343:
8340:
8337:
8334:
8329:
8325:
8321:
8317:
8313:
8310:
8307:
8302:
8298:
8294:
8291:
8288:
8285:
8282:
8279:
8274:
8270:
8266:
8263:
8259:
8245:
8244:
8232:
8208:
8205:
8202:
8198:
8186:
8174:
8171:
8168:
8164:
8160:
8157:
8154:
8151:
8147:
8143:
8140:
8137:
8134:
8131:
8128:
8124:
8111:
8110:
8109:
8108:
8096:
8093:
8090:
8086:
8082:
8079:
8076:
8072:
8068:
8065:
8061:
8047:
8046:
8030:
8029:
8028:
8027:
8016:
8013:
8010:
8007:
8003:
7999:
7996:
7993:
7989:
7985:
7982:
7978:
7964:
7963:
7935:
7934:
7922:
7919:
7916:
7912:
7908:
7905:
7902:
7899:
7896:
7893:
7890:
7886:
7874:
7873:
7872:
7861:
7858:
7855:
7852:
7849:
7846:
7842:
7838:
7835:
7831:
7827:
7824:
7821:
7818:
7815:
7812:
7809:
7805:
7801:
7798:
7795:
7791:
7787:
7784:
7781:
7778:
7775:
7771:
7767:
7764:
7761:
7758:
7754:
7740:
7739:
7727:
7724:
7721:
7718:
7715:
7712:
7709:
7706:
7702:
7681:
7678:
7675:
7672:
7668:
7664:
7661:
7658:
7655:
7652:
7648:
7627:
7607:
7604:
7601:
7598:
7595:
7592:
7580:
7579:
7578:
7577:
7566:
7563:
7560:
7557:
7553:
7549:
7546:
7543:
7539:
7535:
7532:
7528:
7524:
7521:
7518:
7515:
7511:
7507:
7504:
7501:
7497:
7493:
7490:
7486:
7482:
7479:
7476:
7473:
7470:
7467:
7463:
7449:
7448:
7402:
7401:
7400:
7399:
7387:
7384:
7379:
7375:
7371:
7368:
7363:
7359:
7355:
7352:
7349:
7344:
7340:
7336:
7332:
7318:
7317:
7305:
7295:
7294:
7293:
7292:
7280:
7275:
7271:
7267:
7264:
7261:
7256:
7252:
7248:
7243:
7238:
7233:
7230:
7227:
7224:
7219:
7215:
7211:
7208:
7205:
7200:
7196:
7192:
7187:
7184:
7181:
7176:
7161:
7160:
7145:
7142:
7124:
7098:
7094:
7090:
7087:
7084:
7079:
7075:
7052:
7039:
7036:
7035:
7034:
7022:
7019:
7016:
7013:
7010:
7007:
7004:
7001:
6998:
6993:
6988:
6979:
6975:
6971:
6968:
6964:
6952:
6938:
6934:
6930:
6927:
6924:
6919:
6915:
6894:
6889:
6885:
6881:
6878:
6875:
6870:
6866:
6862:
6857:
6852:
6839:
6827:
6822:
6818:
6814:
6811:
6808:
6803:
6799:
6795:
6790:
6785:
6780:
6777:
6774:
6771:
6766:
6762:
6758:
6755:
6752:
6747:
6743:
6739:
6734:
6731:
6728:
6723:
6710:
6698:
6695:
6692:
6672:
6669:
6666:
6662:
6658:
6655:
6652:
6649:
6645:
6641:
6638:
6635:
6632:
6629:
6626:
6622:
6610:
6598:
6595:
6592:
6572:
6569:
6566:
6562:
6558:
6555:
6552:
6549:
6545:
6541:
6538:
6535:
6532:
6529:
6526:
6522:
6501:
6498:
6494:Shannon's bits
6486:David Ellerman
6468:
6464:
6443:
6440:
6437:
6434:
6431:
6428:
6425:
6422:
6419:
6416:
6413:
6389:
6386:
6383:
6380:
6377:
6374:
6371:
6368:
6365:
6362:
6359:
6356:
6353:
6350:
6347:
6344:
6341:
6338:
6297:
6287:
6280:
6263:
6262:
6251:
6247:
6239:
6235:
6231:
6226:
6223:
6220:
6213:
6209:
6205:
6199:
6191:
6187:
6181:
6173:
6168:
6164:
6156:
6151:
6148:
6145:
6141:
6137:
6133:
6127:
6122:
6118:
6112:
6109:
6106:
6101:
6096:
6092:
6085:
6079:
6074:
6069:
6065:
6059:
6056:
6051:
6048:
6045:
6040:
6037:
6031:
6025:
6020:
5996:
5989:
5978:
5967:
5964:
5963:
5962:
5954:
5947:
5928:
5917:
5912:
5905:
5902:
5899:
5893:
5886:
5883:
5880:
5876:
5871:
5868:
5865:
5859:
5856:
5853:
5849:
5838:
5831:
5828:
5825:
5820:
5815:
5810:
5803:
5797:
5791:
5788:
5783:
5780:
5777:
5772:
5769:
5758:
5751:
5746:
5733:
5720:
5714:
5711:
5706:
5703:
5700:
5695:
5692:
5686:
5680:
5675:
5670:
5667:
5662:
5658:
5654:
5651:
5648:
5643:
5639:
5635:
5630:
5625:
5601:
5596:
5583:
5571:
5568:
5565:
5562:
5559:
5556:
5553:
5550:
5547:
5527:
5522:
5518:
5514:
5511:
5508:
5505:
5502:
5497:
5493:
5489:
5465:
5457:
5453:
5448:
5444:
5441:
5438:
5431:
5427:
5422:
5418:
5411:
5407:
5402:
5397:
5391:
5386:
5381:
5377:
5371:
5367:
5363:
5360:
5355:
5351:
5347:
5342:
5338:
5333:
5327:
5322:
5305:
5294:
5270:
5263:
5254:
5244:
5231:
5223:
5220:
5172:
5161:
5136:
5122:The different
5120:
5119:
5116:
5115:
5102:
5099:
5096:
5076:
5073:
5070:
5067:
5064:
5061:
5058:
5038:
5035:
5032:
5029:
5026:
5023:
5020:
5000:
4997:
4994:
4973:
4969:
4966:
4963:
4960:
4940:
4937:
4934:
4931:
4928:
4925:
4922:
4919:
4916:
4913:
4910:
4907:
4892:
4891:
4876:
4867:
4864:
4852:
4849:
4846:
4841:
4838:
4836:
4833:
4830:
4827:
4824:
4821:
4817:
4814:
4810:
4807:
4805:
4793:
4790:
4787:
4782:
4779:
4776:
4773:
4769:
4766:
4763:
4760:
4756:
4753:
4749:
4746:
4743:
4741:
4739:
4734:
4730:
4724:
4720:
4716:
4713:
4703:
4700:
4697:
4692:
4689:
4687:
4684:
4681:
4678:
4674:
4671:
4667:
4664:
4661:
4658:
4655:
4652:
4648:
4645:
4641:
4639:
4634:
4630:
4617:
4614:
4611:
4606:
4603:
4601:
4596:
4592:
4586:
4582:
4578:
4575:
4571:
4568:
4562:
4558:
4552:
4548:
4544:
4541:
4536:
4532:
4526:
4522:
4518:
4515:
4511:
4508:
4504:
4502:
4497:
4493:
4480:
4477:
4472:
4468:
4464:
4461:
4457:
4454:
4450:
4445:
4442:
4440:
4435:
4431:
4425:
4421:
4417:
4414:
4410:
4407:
4401:
4397:
4393:
4391:
4382:
4379:
4374:
4370:
4366:
4363:
4360:
4357:
4354:
4349:
4345:
4341:
4338:
4335:
4332:
4327:
4324:
4322:
4317:
4313:
4307:
4303:
4299:
4296:
4293:
4290:
4288:
4266:
4254:
4253:
4219:
4216:
4213:
4209:
4205:
4202:
4199:
4196:
4176:
4173:
4170:
4144:
4141:
4138:
4118:
4115:
4112:
4109:
4106:
4103:
4100:
4097:
4094:
4091:
4071:
4049:
4046:
4043:
4040:
4037:
4034:
4031:
4028:
4024:
4018:
4015:
4009:
4005:
4002:
3999:
3996:
3993:
3990:
3987:
3984:
3964:
3943:
3935:
3927:
3915:
3903:
3877:
3876:
3866:
3859:
3852:
3845:
3838:
3832:
3794:
3772:
3763:
3754:
3751:
3731:
3728:
3725:
3722:
3719:
3717:
3715:
3712:
3709:
3706:
3703:
3700:
3697:
3694:
3691:
3688:
3685:
3682:
3679:
3676:
3673:
3670:
3667:
3665:
3663:
3660:
3657:
3654:
3651:
3646:
3642:
3638:
3635:
3632:
3629:
3626:
3623:
3618:
3614:
3610:
3607:
3604:
3601:
3599:
3597:
3594:
3591:
3588:
3585:
3580:
3576:
3572:
3569:
3566:
3563:
3560:
3557:
3552:
3548:
3544:
3541:
3538:
3535:
3533:
3531:
3528:
3525:
3521:
3517:
3516:
3462:
3459:
3455:
3452:
3449:
3446:
3443:
3438:
3435:
3427:
3422:
3419:
3416:
3412:
3408:
3405:
3402:
3400:
3398:
3392:
3389:
3384:
3379:
3375:
3369:
3366:
3358:
3353:
3350:
3347:
3343:
3339:
3336:
3333:
3331:
3329:
3325:
3320:
3316:
3312:
3309:
3306:
3301:
3297:
3293:
3288:
3284:
3280:
3277:
3271:
3266:
3263:
3260:
3256:
3252:
3249:
3246:
3244:
3242:
3239:
3236:
3232:
3228:
3227:
3193:
3151:
3148:
3135:
3111:
3091:
3088:
3085:
3080:
3075:
3053:
3050:
3047:
3044:
3039:
3034:
3027:
3024:
3021:
3017:
3013:
3010:
3007:
3004:
2999:
2994:
2972:
2952:
2928:
2906:
2903:
2900:
2897:
2892:
2888:
2882:
2879:
2876:
2872:
2868:
2865:
2862:
2859:
2854:
2849:
2827:
2807:
2804:
2801:
2798:
2795:
2775:
2772:
2769:
2766:
2763:
2760:
2757:
2754:
2734:
2731:
2728:
2725:
2722:
2718:
2714:
2711:
2691:
2688:
2685:
2680:
2675:
2672:
2645:
2623:
2620:
2617:
2614:
2609:
2605:
2601:
2598:
2595:
2592:
2589:
2586:
2583:
2580:
2575:
2571:
2550:
2524:
2521:
2518:
2515:
2512:
2509:
2506:
2503:
2500:
2497:
2494:
2491:
2486:
2482:
2461:
2433:
2430:
2427:
2403:
2400:
2397:
2394:
2391:
2388:
2385:
2374:measure theory
2369:
2368:Measure theory
2366:
2353:
2333:
2313:
2310:
2307:
2304:
2301:
2297:
2293:
2290:
2287:
2284:
2279:
2275:
2254:
2251:
2248:
2245:
2242:
2239:
2236:
2233:
2230:
2226:
2222:
2219:
2216:
2213:
2210:
2207:
2202:
2199:
2196:
2192:
2171:
2165:
2162:
2159:
2154:
2150:
2144:
2141:
2138:
2135:
2132:
2127:
2124:
2121:
2117:
2110:
2107:
2104:
2101:
2098:
2095:
2092:
2087:
2084:
2081:
2077:
2069:
2064:
2059:
2054:
2051:
2048:
2045:
2041:
2037:
2034:
2031:
2028:
2024:
2020:
2017:
2013:
1990:
1966:
1944:
1924:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1870:
1866:
1862:
1859:
1855:
1830:
1814:
1809:
1806:
1786:
1783:
1780:
1777:
1774:
1771:
1683:
1680:
1677:
1674:
1671:
1668:
1663:
1659:
1655:
1652:
1649:
1646:
1639:
1634:
1631:
1627:
1623:
1620:
1617:
1614:
1611:
1607:
1583:
1580:
1577:
1574:
1571:
1532:
1509:
1506:
1503:
1500:
1497:
1494:
1491:
1488:
1485:
1482:
1478:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1449:
1445:
1442:
1439:
1436:
1432:
1409:
1406:
1403:
1400:
1397:
1393:
1389:
1386:
1383:
1380:
1377:
1357:
1354:
1351:
1348:
1345:
1342:
1337:
1332:
1329:
1307:
1289:{\textstyle X}
1285:
1258:
1255:
1242:
1239:
1204:
1201:
1196:
1192:
1140:
1136:
1132:
1129:
1126:
1106:
1102:
1098:
1095:
1092:
1069:
1065:
1059:
1056:
1053:
1050:
1046:
1041:
1037:
1032:
1028:
1024:
1021:
1018:
1015:
1012:
992:
989:
986:
983:
980:
977:
974:
971:
966:
962:
958:
955:
952:
949:
946:
943:
923:
886:
866:
862:
856:
853:
850:
847:
843:
838:
834:
831:
811:
808:
805:
802:
782:
779:
776:
773:
753:
750:
747:
744:
724:
685:
682:
669:
666:
663:
660:
657:
654:
651:
648:
645:
641:
574:Claude Shannon
551:expected value
512:
492:
472:
469:
466:
463:
460:
457:
454:
451:
448:
445:
442:
435:
430:
427:
423:
419:
416:
413:
410:
407:
403:
382:
379:
376:
373:
370:
367:
362:
357:
354:
332:
310:
284:
283:
281:
280:
273:
266:
258:
255:
254:
253:
252:
247:
242:
237:
229:
228:
227:
226:
221:
213:
212:
211:
210:
205:
200:
195:
190:
185:
180:
175:
170:
165:
157:
156:
148:
147:
135:
134:
49:
47:
40:
26:
9:
6:
4:
3:
2:
17207:
17196:
17193:
17191:
17188:
17186:
17183:
17181:
17178:
17176:
17173:
17172:
17170:
17154:
17150:
17142:
17140:
17132:
17131:
17128:
17122:
17119:
17118:
17116:
17112:
17106:
17103:
17102:
17100:
17096:
17090:
17087:
17085:
17082:
17080:
17077:
17075:
17072:
17070:
17067:
17065:
17062:
17060:
17057:
17053:
17050:
17049:
17048:
17045:
17043:
17040:
17036:
17033:
17031:
17028:
17027:
17026:
17023:
17022:
17020:
17018:
17014:
17002:
16999:
16997:
16994:
16993:
16992:
16989:
16985:
16982:
16980:
16977:
16975:
16972:
16971:
16969:
16967:
16964:
16962:
16959:
16957:
16954:
16952:
16949:
16948:
16946:
16943:
16939:
16933:
16932:Video quality
16930:
16928:
16925:
16923:
16920:
16918:
16915:
16913:
16910:
16908:
16905:
16903:
16900:
16896:
16893:
16891:
16888:
16886:
16883:
16882:
16881:
16878:
16877:
16875:
16871:
16868:
16866:
16862:
16850:
16847:
16845:
16842:
16840:
16837:
16835:
16832:
16831:
16830:
16827:
16825:
16822:
16820:
16817:
16815:
16812:
16810:
16807:
16805:
16802:
16800:
16797:
16795:
16792:
16791:
16789:
16785:
16779:
16776:
16774:
16771:
16769:
16766:
16764:
16761:
16759:
16756:
16754:
16751:
16749:
16746:
16744:
16741:
16739:
16736:
16734:
16731:
16729:
16726:
16725:
16723:
16719:
16716:
16714:
16710:
16700:
16697:
16695:
16692:
16688:
16685:
16683:
16680:
16678:
16675:
16673:
16670:
16668:
16665:
16664:
16663:
16660:
16656:
16653:
16652:
16651:
16648:
16644:
16641:
16639:
16636:
16635:
16634:
16631:
16629:
16626:
16624:
16621:
16620:
16618:
16615:
16611:
16605:
16602:
16600:
16599:Speech coding
16597:
16595:
16594:Sound quality
16592:
16590:
16587:
16585:
16582:
16580:
16577:
16575:
16572:
16570:
16569:Dynamic range
16567:
16565:
16562:
16560:
16557:
16553:
16550:
16548:
16545:
16543:
16540:
16539:
16538:
16535:
16534:
16532:
16528:
16525:
16523:
16519:
16509:
16506:
16502:
16499:
16497:
16494:
16492:
16489:
16488:
16486:
16482:
16479:
16477:
16474:
16472:
16469:
16467:
16464:
16462:
16459:
16458:
16457:
16454:
16450:
16447:
16446:
16445:
16442:
16441:
16439:
16435:
16427:
16424:
16422:
16419:
16417:
16414:
16413:
16412:
16409:
16407:
16404:
16402:
16399:
16395:
16392:
16390:
16387:
16386:
16385:
16382:
16381:
16379:
16377:
16373:
16370:
16368:
16364:
16352:
16349:
16348:
16346:
16341:
16339:
16336:
16335:
16334:LZ77 + Range
16333:
16329:
16326:
16325:
16323:
16319:
16316:
16315:
16313:
16309:
16306:
16305:
16303:
16299:
16296:
16295:
16293:
16289:
16286:
16284:
16281:
16279:
16276:
16275:
16273:
16272:
16270:
16266:
16260:
16257:
16255:
16252:
16250:
16247:
16245:
16242:
16240:
16237:
16233:
16230:
16228:
16225:
16224:
16223:
16220:
16218:
16215:
16213:
16210:
16206:
16203:
16202:
16201:
16198:
16196:
16193:
16191:
16188:
16186:
16183:
16182:
16180:
16176:
16168:
16165:
16163:
16160:
16158:
16155:
16153:
16150:
16148:
16145:
16143:
16140:
16138:
16135:
16133:
16130:
16128:
16125:
16124:
16123:
16120:
16118:
16115:
16114:
16112:
16110:
16106:
16098:
16095:
16093:
16090:
16088:
16085:
16083:
16080:
16079:
16078:
16075:
16073:
16070:
16068:
16065:
16063:
16060:
16058:
16055:
16053:
16050:
16048:
16045:
16041:
16038:
16036:
16033:
16031:
16028:
16027:
16026:
16023:
16021:
16018:
16016:
16013:
16011:
16008:
16006:
16003:
16002:
16000:
15998:
15994:
15991:
15989:
15985:
15980:
15973:
15968:
15966:
15961:
15959:
15954:
15953:
15950:
15943:
15942:
15938:
15935:
15931:
15928:
15924:
15920:
15917:
15914:
15910:
15906:
15905:
15900:
15896:
15895:
15889:
15886:
15884:
15881:
15879:
15876:
15875:
15870:
15865:
15852:
15848:
15844:
15840:
15837:
15836:
15831:
15829:
15828:0-252-72548-4
15825:
15821:
15817:
15813:
15812:Shannon, C.E.
15810:
15806:
15800:
15796:
15795:
15789:
15786:
15782:
15780:
15776:
15772:
15768:
15766:
15762:
15758:
15754:
15751:
15749:
15745:
15741:
15737:
15733:
15730:
15729:
15718:
15717:
15715:
15711:
15698:
15692:
15688:
15687:
15679:
15663:
15659:
15655:
15651:
15647:
15643:
15639:
15635:
15628:
15612:
15608:
15602:
15598:
15594:
15590:
15586:
15579:
15570:
15561:
15545:
15541:
15537:
15533:
15529:
15524:
15519:
15515:
15511:
15504:
15489:
15488:
15483:
15479:
15473:
15467:
15461:
15453:
15447:
15442:
15437:
15433:
15426:
15407:
15403:
15396:
15389:
15370:
15366:
15359:
15352:
15344:
15340:
15336:
15332:
15327:
15322:
15318:
15314:
15310:
15306:
15302:
15295:
15288:
15287:
15282:
15278:
15275:
15270:
15268:
15251:
15247:
15240:
15224:
15220:
15216:
15212:
15208:
15204:
15200:
15196:
15189:
15181:
15177:
15173:
15169:
15165:
15161:
15157:
15153:
15149:
15142:
15133:
15128:
15125:: 1971–2009.
15124:
15120:
15116:
15109:
15101:
15094:
15088:
15087:0-486-68455-5
15084:
15077:
15069:
15065:
15061:
15057:
15053:
15049:
15045:
15041:
15034:
15032:
15030:
15010:
15006:
15002:
14998:
14994:
14987:
14980:
14974:
14970:
14967:
14963:
14957:
14949:
14943:
14924:
14917:
14916:
14908:
14902:
14900:
14895:
14890:
14882:
14876:
14872:
14865:
14863:
14861:
14859:
14857:
14855:
14853:
14851:
14849:
14847:
14845:
14837:
14831:
14823:
14817:
14813:
14812:
14804:
14796:
14790:
14786:
14785:
14777:
14770:
14767:Schneier, B:
14764:
14748:
14744:
14742:0-521-64298-1
14738:
14734:
14733:
14728:
14722:
14714:
14702:
14698:
14692:
14685:
14681:
14678:
14674:
14667:
14662:
14658:
14654:
14650:
14646:
14645:
14640:
14636:
14630:
14628:
14620:
14616:
14613:
14609:
14602:
14597:
14593:
14589:
14585:
14581:
14580:
14575:
14572:(July 1948).
14571:
14565:
14563:
14554:
14548:
14544:
14543:
14535:
14531:
14502:
14499:
14496:
14480:
14477:
14454:
14448:
14445:
14442:
14422:
14419:
14413:
14407:
14404:
14401:
14396:
14390:
14362:
14356:
14353:
14343:
14339:
14328:
14325:
14323:
14320:
14318:
14317:Shannon index
14315:
14312:
14309:
14307:
14304:
14301:
14300:Rényi entropy
14298:
14295:
14292:
14290:
14286:
14282:
14279:
14277:
14274:
14272:
14269:
14267:
14264:
14262:
14258:
14255:
14253:
14250:
14248:
14245:
14243:
14240:
14238:
14235:
14233:
14230:
14228:
14227:Graph entropy
14225:
14223:
14220:
14218:
14215:
14213:
14210:
14207:
14204:
14202:
14199:
14196:
14195:Cross entropy
14193:
14191:
14188:
14185:
14182:
14181:
14176:
14170:
14165:
14158:
14155:
14154:cross-entropy
14151:
14147:
14144:performed by
14143:
14139:
14137:
14133:
14129:
14125:
14111:
14091:
14071:
14051:
14028:
14025:
14022:
14016:
14013:
14006:
14002:
13998:
13996:
13987:
13968:
13964:
13960:
13949:
13945:
13935:
13931:with exactly
13929:
13917:
13913:
13890:
13887:
13884:
13880:
13875:
13870:
13867:
13864:
13861:
13853:
13850:
13847:
13839:
13836:
13832:
13820:
13817:
13802:
13801:
13800:
13783:
13780:
13775:
13764:
13761:
13758:
13752:
13749:
13743:
13738:
13735:
13732:
13724:
13721:
13718:
13710:
13706:
13693:
13690:
13676:
13671:
13668:
13665:
13661:
13653:
13652:
13636:
13633:
13630:
13627:
13619:
13616:
13613:
13605:
13602:
13598:
13585:
13582:
13565:
13564:
13560:
13559:
13556:
13555:
13540:
13534:
13531:
13528:
13522:
13517:
13513:
13506:
13503:
13500:
13494:
13488:
13482:
13477:
13473:
13469:
13466:
13463:
13457:
13442:
13441:
13440:
13423:
13415:
13404:
13400:
13396:
13384:
13381:
13369:
13363:
13360:
13357:
13348:
13337:
13333:
13323:
13322:
13321:
13318:
13314:
13310:
13304:
13300:
13295:For integers
13288:
13266:
13258:
13254:
13245:
13242:
13239:
13229:
13225:
13221:
13218:
13208:
13204:
13182:
13177:
13173:
13152:
13148:
13144:
13141:
13131:
13127:
13114:
13110:
13106:
13101:
13097:
13091:
13083:
13073:
13069:) = log|
13068:
13061:
13055:
13049:
13043:
13039:
13035:
13028:
13024:
13018:
13014:with indexes
13011:
13007:
12986:
12982:
12978:
12975:
12965:
12961:
12927:
12923:
12919:
12916:
12906:
12902:
12885:
12880:
12877:
12874:
12870:
12864:
12861:
12856:
12845:
12841:
12837:
12834:
12831:
12826:
12822:
12803:
12802:
12801:
12798:
12792:
12786:
12778:
12774:
12767:
12760:
12756:
12749:
12744:
12725:
12719:
12716:
12708:
12704:
12700:
12697:
12694:
12689:
12685:
12678:
12670:
12666:
12662:
12659:
12656:
12651:
12648:
12645:
12641:
12637:
12632:
12629:
12626:
12622:
12618:
12615:
12612:
12607:
12603:
12593:
12587:
12579:
12575:
12567:
12566:
12565:
12562:
12557:
12551:
12547:
12519:
12511:
12507:
12496:
12491:
12488:
12485:
12481:
12477:
12472:
12469:
12466:
12456:
12444:
12443:
12442:
12439:
12435:
12430:
12420:
12418:
12417:combinatorics
12408:
12405:
12403:
12399:
12396:
12392:
12388:
12385:
12381:
12376:
12374:
12367:
12363:
12343:
12340:
12337:
12331:
12321:
12317:
12312:
12307:
12305:
12301:
12292:
12289:
12283:
12277:
12273:
12267:
12262:
12257:
12252:
12247:
12227:
12221:
12218:
12212:
12203:
12197:
12191:
12188:
12182:
12176:
12173:
12170:
12164:
12161:
12155:
12146:
12140:
12134:
12131:
12128:
12125:
12119:
12113:
12096:
12088:
12087:
12086:
12083:
12077:
12071:
12064:
12060:
12056:
12052:
12048:
12044:
12038:
12033:
12028:
12022:
12017:
12013:
12007:
11997:
11995:
11976:
11970:
11967:
11941:
11932:
11912:
11909:
11906:
11892:
11886:
11880:
11877:
11871:
11865:
11852:
11848:
11844:
11832:
11831:
11830:
11827:
11821:
11815:
11807:
11803:
11797:
11791:
11785:
11775:
11773:
11767:
11762:
11743:
11740:
11737:
11730:
11724:
11721:
11718:
11712:
11706:
11693:
11689:
11685:
11682:
11678:
11671:
11668:
11665:
11649:
11643:
11629:
11623:
11617:
11610:
11609:
11608:
11579:
11576:
11573:
11566:
11560:
11557:
11554:
11548:
11542:
11529:
11525:
11519:
11506:
11502:
11495:
11489:
11486:
11475:
11471:
11464:
11451:
11448:
11445:
11441:
11433:
11430:
11427:
11424:
11417:
11411:
11398:
11394:
11388:
11375:
11371:
11364:
11351:
11348:
11345:
11341:
11329:
11328:
11327:
11320:
11307:
11295:
11292:
11281:
11277:
11270:
11257:
11254:
11251:
11247:
11243:
11232:
11228:
11221:
11215:
11212:
11201:
11197:
11190:
11177:
11174:
11171:
11167:
11163:
11160:
11127:
11115:
11111:
11104:
11100:
11096:
11093:
11082:
11078:
11071:
11058:
11055:
11052:
11048:
11044:
11041:
11020:
11006:
10995:
10991:
10984:
10971:
10968:
10965:
10961:
10955:
10941:
10938:
10935:
10928:
10922:
10909:
10905:
10895:
10878:
10875:
10868:
10862:
10851:
10848:
10845:
10834:
10830:
10826:
10815:
10811:
10804:
10794:
10790:
10764:
10758:
10754:
10750:
10744:
10739:
10734:
10731:
10729:
10723:
10721:
10717:
10712:
10706:
10686:
10683:
10671:
10665:
10662:
10659:
10653:
10647:
10636:
10632:
10629:
10620:
10614:
10611:
10608:
10605:
10594:
10588:
10573:
10572:
10571:
10545:
10541:
10537:
10531:
10516:
10513:
10492:
10486:
10481:
10477:
10467:
10448:
10435:
10431:
10424:
10421:
10411:
10407:
10400:
10395:
10390:
10387:
10384:
10380:
10373:
10368:
10364:
10360:
10347:
10343:
10336:
10333:
10323:
10319:
10312:
10306:
10301:
10297:
10291:
10286:
10283:
10280:
10276:
10272:
10263:
10257:
10252:
10248:
10232:
10228:
10221:
10218:
10208:
10204:
10197:
10191:
10186:
10182:
10173:
10168:
10165:
10162:
10158:
10154:
10145:
10139:
10134:
10130:
10116:
10112:
10105:
10099:
10094:
10090:
10081:
10077:
10070:
10062:
10057:
10054:
10051:
10047:
10043:
10040:
10034:
10028:
10021:
10020:
10019:
10002:
9993:
9987:
9982:
9978:
9964:
9960:
9953:
9947:
9942:
9938:
9929:
9925:
9918:
9910:
9905:
9902:
9899:
9895:
9891:
9888:
9881:
9878:
9875:
9871:
9867:
9862:
9856:
9850:
9843:
9842:
9841:
9815:A source set
9794:
9788:
9780:
9777:
9774:
9770:
9763:
9760:
9751:
9743:
9740:
9737:
9733:
9727:
9723:
9716:
9708:
9704:
9698:
9694:
9688:
9684:
9678:
9674:
9670:
9667:
9642:
9641:
9640:
9637:
9634:
9628:
9608:
9600:
9596:
9587:
9582:
9562:
9556:
9548:
9544:
9540:
9537:
9531:
9523:
9519:
9510:
9506:
9500:
9496:
9490:
9486:
9482:
9479:
9454:
9453:
9452:
9450:
9449:
9444:
9443:Markov source
9439:
9432:
9428:
9408:
9403:
9399:
9395:
9392:
9387:
9383:
9379:
9376:
9373:
9348:
9347:
9346:
9344:
9334:
9331:
9326:
9324:
9306:
9302:
9293:
9289:
9288:cryptanalysis
9279:
9275:
9263:
9261:
9257:
9253:
9248:
9242:
9234:
9223:
9216:
9209:
9207:
9201:
9199:
9195:
9187:
9183:
9182:
9177:
9174:
9173:
9172:joint entropy
9168:
9165:
9164:
9159:
9158:
9157:
9149:
9147:
9139:
9135:
9131:
9127:
9126:Shannon index
9121:
9111:
9109:
9105:
9094:
9091:
9088:
9087:
9083:
9080:
9077:
9076:
9072:
9069:
9066:
9065:
9061:
9058:
9055:
9054:
9051:
9045:
9042:
9041:
9035:
9033:
9029:
9024:
9020:
9016:
9012:
9007:
9002:
9000:
8995:
8991:
8987:
8983:
8979:
8975:
8971:
8964:
8960:
8950:
8948:
8944:
8940:
8936:
8935:
8930:
8925:
8921:
8917:
8912:
8905:
8899:
8894:
8890:
8885:
8877:
8871:
8854:
8831:
8828:
8825:
8822:
8813:
8809:
8806:
8799:
8798:
8797:
8795:
8791:
8786:
8784:
8780:
8772:
8768:
8759:
8755:
8751:
8747:
8742:
8740:
8736:
8717:
8710:
8707:
8704:
8701:
8678:
8674:
8671:
8668:
8661:
8660:
8659:
8657:
8653:
8649:
8644:
8642:
8638:
8634:
8633:Gibbs entropy
8630:
8624:
8620:
8615:
8607:
8587:
8581:
8577:
8573:
8570:
8565:
8561:
8557:
8548:
8544:
8541:
8538:
8531:
8530:
8529:
8528:
8527:Gibbs entropy
8524:
8519:
8515:
8511:
8506:
8504:
8500:
8494:
8474:
8470:
8466:
8462:
8461:
8447:
8444:
8441:
8438:
8435:
8413:
8409:
8405:
8400:
8396:
8387:
8366:
8362:
8347:
8344:
8341:
8335:
8327:
8323:
8311:
8308:
8300:
8296:
8289:
8286:
8283:
8277:
8272:
8268:
8264:
8249:
8248:
8247:
8246:
8230:
8222:
8203:
8187:
8169:
8158:
8152:
8141:
8135:
8132:
8129:
8113:
8112:
8091:
8080:
8074:
8066:
8051:
8050:
8049:
8048:
8043:
8037:
8032:
8031:
8014:
8008:
7997:
7991:
7983:
7968:
7967:
7966:
7965:
7960:
7954:
7948:
7942:
7937:
7936:
7917:
7906:
7897:
7891:
7875:
7859:
7850:
7844:
7836:
7825:
7816:
7810:
7799:
7793:
7782:
7776:
7765:
7759:
7744:
7743:
7742:
7741:
7719:
7713:
7710:
7707:
7679:
7676:
7670:
7659:
7653:
7625:
7602:
7596:
7593:
7590:
7582:
7581:
7564:
7558:
7547:
7541:
7533:
7522:
7516:
7505:
7499:
7491:
7480:
7474:
7471:
7468:
7453:
7452:
7451:
7450:
7445:
7439:
7433:
7427:
7421:
7414:
7410:
7404:
7403:
7385:
7382:
7377:
7373:
7369:
7361:
7357:
7353:
7350:
7347:
7342:
7338:
7322:
7321:
7320:
7319:
7313:
7308:
7301:
7297:
7296:
7273:
7269:
7265:
7262:
7259:
7254:
7250:
7241:
7231:
7225:
7222:
7217:
7213:
7209:
7206:
7203:
7198:
7194:
7185:
7182:
7179:
7165:
7164:
7163:
7162:
7158:
7157:
7156:
7153:
7141:
7139:
7112:
7096:
7092:
7088:
7085:
7082:
7077:
7073:
7020:
7017:
7011:
7008:
7005:
7002:
6999:
6991:
6977:
6973:
6966:
6953:
6936:
6932:
6928:
6925:
6922:
6917:
6913:
6887:
6883:
6879:
6876:
6873:
6868:
6864:
6855:
6840:
6820:
6816:
6812:
6809:
6806:
6801:
6797:
6788:
6778:
6772:
6769:
6764:
6760:
6756:
6753:
6750:
6745:
6741:
6732:
6729:
6726:
6711:
6696:
6693:
6690:
6667:
6656:
6650:
6639:
6633:
6630:
6627:
6612:Additivity:
6611:
6596:
6593:
6590:
6567:
6556:
6550:
6539:
6533:
6530:
6527:
6511:
6510:
6509:
6507:
6497:
6495:
6491:
6487:
6482:
6466:
6462:
6438:
6432:
6429:
6426:
6423:
6417:
6411:
6403:
6384:
6381:
6378:
6372:
6369:
6363:
6357:
6354:
6348:
6345:
6342:
6336:
6328:
6324:
6319:
6317:
6312:
6306:
6290:
6286:
6279:
6273:
6269:
6249:
6245:
6237:
6233:
6229:
6224:
6221:
6218:
6211:
6207:
6203:
6197:
6189:
6185:
6171:
6166:
6162:
6154:
6149:
6146:
6143:
6139:
6135:
6131:
6125:
6120:
6116:
6110:
6107:
6104:
6099:
6094:
6090:
6083:
6077:
6067:
6063:
6057:
6054:
6049:
6046:
6043:
6038:
6035:
6029:
6023:
6009:
6008:
6007:
6004:
5999:
5995:
5988:
5981:
5977:
5973:
5957:
5953:
5946:
5940:
5934:
5929:
5915:
5903:
5900:
5897:
5891:
5884:
5881:
5878:
5874:
5869:
5866:
5863:
5857:
5854:
5851:
5847:
5829:
5826:
5823:
5813:
5801:
5795:
5789:
5786:
5781:
5778:
5775:
5770:
5767:
5749:
5734:
5718:
5712:
5709:
5704:
5701:
5698:
5693:
5690:
5684:
5678:
5668:
5660:
5656:
5652:
5649:
5646:
5641:
5637:
5628:
5599:
5584:
5566:
5563:
5560:
5557:
5554:
5551:
5548:
5520:
5516:
5512:
5509:
5506:
5503:
5500:
5495:
5491:
5480:
5463:
5455:
5451:
5446:
5442:
5439:
5436:
5429:
5425:
5420:
5416:
5409:
5405:
5400:
5395:
5389:
5379:
5375:
5369:
5365:
5361:
5358:
5353:
5349:
5345:
5340:
5336:
5331:
5325:
5308:
5304:
5295:
5292:
5284:
5283:
5282:
5278:
5273:
5269:
5262:
5257:
5247:
5243:
5239:
5234:
5230:
5219:
5217:
5213:
5209:
5204:
5201:
5194:
5187:
5181:
5168:
5158:
5154:
5147:
5143:
5133:
5129:
5125:
5114:
5100:
5097:
5094:
5071:
5068:
5065:
5059:
5056:
5036:
5033:
5027:
5021:
4998:
4995:
4992:
4967:
4964:
4961:
4958:
4938:
4935:
4932:
4929:
4926:
4923:
4920:
4914:
4908:
4897:
4874:
4865:
4862:
4847:
4839:
4834:
4831:
4825:
4819:
4815:
4808:
4788:
4780:
4774:
4764:
4758:
4754:
4747:
4742:
4732:
4728:
4722:
4718:
4714:
4711:
4698:
4690:
4682:
4676:
4672:
4665:
4662:
4656:
4650:
4646:
4632:
4628:
4612:
4604:
4594:
4590:
4584:
4580:
4573:
4569:
4560:
4556:
4550:
4546:
4542:
4534:
4530:
4524:
4520:
4513:
4509:
4495:
4491:
4470:
4466:
4459:
4455:
4443:
4433:
4429:
4423:
4419:
4412:
4408:
4399:
4395:
4372:
4368:
4361:
4355:
4347:
4343:
4336:
4325:
4315:
4311:
4305:
4301:
4294:
4279:
4278:
4256:
4255:
4251:
4250:
4247:
4246:
4245:
4243:
4242:characterized
4239:
4234:
4217:
4214:
4211:
4207:
4203:
4200:
4197:
4194:
4174:
4171:
4168:
4159:
4142:
4139:
4136:
4116:
4113:
4110:
4107:
4104:
4098:
4092:
4060:
4047:
4041:
4035:
4032:
4029:
4026:
4022:
4016:
4013:
4007:
4003:
4000:
3997:
3991:
3985:
3975:is given by:
3953:
3949:
3941:
3933:
3921:
3909:
3898:
3892:
3887:
3883:
3874:
3865:
3858:
3851:
3844:
3839:
3833:
3829:
3824:
3818:
3813:
3812:
3811:
3809:
3804:
3797:
3793:
3787:
3775:
3771:
3766:
3762:
3750:
3746:
3729:
3726:
3723:
3720:
3718:
3707:
3704:
3698:
3695:
3692:
3686:
3683:
3677:
3674:
3671:
3668:
3666:
3655:
3649:
3644:
3640:
3636:
3633:
3627:
3621:
3616:
3612:
3608:
3605:
3602:
3600:
3589:
3583:
3578:
3574:
3570:
3567:
3561:
3555:
3550:
3546:
3542:
3539:
3536:
3534:
3526:
3505:
3499:
3495:
3489:
3483:
3477:
3460:
3457:
3450:
3447:
3441:
3436:
3433:
3425:
3420:
3417:
3414:
3410:
3406:
3403:
3401:
3390:
3387:
3382:
3377:
3373:
3367:
3364:
3356:
3351:
3348:
3345:
3341:
3337:
3334:
3332:
3318:
3314:
3307:
3304:
3299:
3295:
3286:
3282:
3275:
3269:
3264:
3261:
3258:
3254:
3250:
3247:
3245:
3237:
3215:
3213:
3207:
3203:
3185:
3178:
3172:
3169:
3163:
3156:
3147:
3133:
3125:
3109:
3078:
3051:
3045:
3037:
3025:
3022:
3019:
3011:
3005:
2997:
2970:
2950:
2942:
2941:sigma-algebra
2926:
2917:
2904:
2898:
2890:
2886:
2880:
2877:
2874:
2870:
2866:
2860:
2852:
2825:
2805:
2802:
2799:
2796:
2793:
2773:
2770:
2764:
2761:
2758:
2752:
2732:
2729:
2723:
2720:
2716:
2709:
2686:
2673:
2670:
2663:
2659:
2643:
2634:
2621:
2615:
2607:
2603:
2596:
2590:
2587:
2581:
2573:
2569:
2548:
2541:surprisal of
2540:
2535:
2522:
2516:
2510:
2507:
2504:
2501:
2498:
2492:
2484:
2480:
2459:
2451:
2447:
2428:
2425:
2417:
2398:
2395:
2389:
2386:
2375:
2365:
2351:
2331:
2308:
2305:
2302:
2291:
2285:
2277:
2273:
2249:
2246:
2243:
2240:
2237:
2234:
2231:
2220:
2214:
2211:
2208:
2200:
2197:
2194:
2190:
2169:
2160:
2152:
2148:
2139:
2136:
2133:
2125:
2122:
2119:
2115:
2108:
2105:
2099:
2096:
2093:
2085:
2082:
2079:
2075:
2062:
2052:
2049:
2046:
2043:
2039:
2035:
2032:
2026:
2018:
1942:
1922:
1914:
1909:
1896:
1893:
1887:
1881:
1878:
1875:
1868:
1864:
1857:
1844:
1833:
1807:
1804:
1784:
1781:
1775:
1769:
1760:
1756:
1751:
1746:
1742:
1737:
1731:
1726:
1722:
1720:
1712:
1707:
1703:
1698:
1681:
1675:
1669:
1666:
1661:
1657:
1650:
1644:
1632:
1629:
1625:
1621:
1618:
1612:
1595:
1578:
1572:
1560:
1555:
1547:
1520:
1507:
1498:
1492:
1489:
1486:
1483:
1472:
1463:
1457:
1443:
1437:
1421:
1404:
1401:
1398:
1387:
1381:
1375:
1352:
1349:
1346:
1330:
1327:
1283:
1276:
1272:
1264:
1254:
1250:
1248:
1238:
1236:
1232:
1226:
1219:
1202:
1199:
1194:
1190:
1181:
1177:
1171:
1165:
1158:
1152:
1138:
1134:
1130:
1127:
1124:
1104:
1100:
1096:
1093:
1090:
1080:
1067:
1063:
1054:
1048:
1044:
1039:
1035:
1030:
1026:
1022:
1016:
1010:
990:
981:
975:
969:
964:
960:
956:
953:
947:
941:
921:
912:
910:
909:
900:
884:
864:
860:
851:
845:
841:
836:
832:
829:
806:
800:
777:
771:
748:
742:
722:
714:
710:
706:
704:
698:
696:
692:
681:
661:
655:
652:
649:
646:
630:
626:
622:
618:
617:combinatorics
614:
610:
605:
603:
599:
595:
591:
587:
583:
579:
575:
562:
558:
556:
552:
548:
544:
540:
539:
534:
530:
526:
510:
470:
464:
458:
455:
452:
446:
440:
428:
425:
421:
417:
414:
408:
377:
374:
371:
355:
352:
308:
299:
295:
291:
279:
274:
272:
267:
265:
260:
259:
257:
256:
251:
248:
246:
243:
241:
238:
236:
233:
232:
231:
230:
225:
222:
220:
217:
216:
215:
214:
209:
206:
204:
201:
199:
196:
194:
191:
189:
186:
184:
181:
179:
178:Joint entropy
176:
174:
171:
169:
166:
164:
161:
160:
159:
158:
154:
150:
149:
146:
143:
142:
139:
131:
128:
120:
117:February 2019
109:
106:
102:
99:
95:
92:
88:
85:
81:
78: –
77:
73:
72:Find sources:
66:
62:
56:
55:
50:This article
48:
44:
39:
38:
33:
19:
17105:Hutter Prize
17069:Quantization
17041:
16974:Compensation
16768:Quantization
16491:Compensation
16057:Shannon–Fano
15997:Entropy type
15932:
15927:Rosetta Code
15902:
15878:Online books
15868:
15834:
15819:
15793:
15784:
15773:, Springer,
15770:
15756:
15739:
15736:Thomas, J.A.
15707:
15706:
15685:
15678:
15666:. Retrieved
15641:
15637:
15627:
15615:. Retrieved
15588:
15578:
15569:
15560:
15550:20 September
15548:. Retrieved
15523:1509.05363v6
15513:
15503:
15491:. Retrieved
15485:
15472:
15460:
15431:
15425:
15413:. Retrieved
15401:
15388:
15376:. Retrieved
15364:
15351:
15308:
15304:
15294:
15284:
15254:. Retrieved
15250:the original
15239:
15227:. Retrieved
15202:
15198:
15188:
15155:
15151:
15141:
15122:
15118:
15108:
15099:
15093:
15076:
15043:
15039:
15016:. Retrieved
14996:
14992:
14979:
14961:
14956:
14930:. Retrieved
14921:. Santa Fe.
14914:
14907:
14898:
14889:
14870:
14830:
14810:
14803:
14787:. Springer.
14783:
14776:
14768:
14763:
14751:. Retrieved
14731:
14721:
14711:– via
14705:. Retrieved
14691:
14648:
14642:
14583:
14577:
14541:
14534:
14378:. We do see
14342:
14327:Typoglycemia
14140:
14126:
13999:
13993:
13933:
13927:
13924:
13911:
13908:
13798:
13438:
13316:
13312:
13308:
13302:
13298:
13294:
13180:
13175:
13171:
13112:
13108:
13104:
13099:
13095:
13089:
13081:
13071:
13066:
13059:
13053:
13047:
13044:
13037:
13033:
13026:
13022:
13016:
13009:
13005:
12949:
12796:
12790:
12784:
12776:
12772:
12765:
12758:
12754:
12747:
12740:
12560:
12549:
12545:
12542:
12437:
12433:
12426:
12414:
12406:
12377:
12365:
12361:
12319:
12315:
12308:
12298:
12287:
12281:
12275:
12271:
12265:
12255:
12245:
12242:
12081:
12075:
12069:
12062:
12058:
12054:
12050:
12046:
12042:
12036:
12026:
12020:
12011:
12009:
11930:
11927:
11825:
11819:
11813:
11805:
11801:
11795:
11789:
11787:
11765:
11760:
11758:
11598:
11321:
11021:
10893:
10792:
10788:
10762:
10759:
10752:
10748:
10742:
10735:
10732:
10724:
10710:
10704:
10701:
10543:
10539:
10533:
10511:
10463:
10017:
9814:
9638:
9632:
9626:
9585:
9580:
9577:
9448:entropy rate
9446:
9437:
9430:
9426:
9423:
9343:Markov model
9340:
9330:one-time pad
9327:
9322:
9285:
9273:
9264:
9240:
9232:
9221:
9214:
9210:
9205:
9204:its entropy
9202:
9191:
9181:entropy rate
9179:
9170:
9161:
9155:
9123:
9100:
9038:
9036:
9027:
9022:
9018:
9003:
8994:entropy rate
8984:. (See also
8966:
8932:
8923:
8913:
8903:
8897:
8892:
8888:
8875:
8869:
8846:
8787:
8770:
8766:
8757:
8745:
8743:
8732:
8650:to give the
8645:
8622:
8618:
8605:
8602:
8517:
8507:
8498:
8496:
8188:The entropy
8041:
8035:
7958:
7952:
7946:
7940:
7443:
7437:
7431:
7425:
7419:
7412:
7408:
7311:
7306:
7299:
7151:
7147:
7113:
7041:
6503:
6483:
6320:
6310:
6304:
6288:
6284:
6277:
6271:
6267:
6264:
6002:
5997:
5993:
5986:
5979:
5975:
5969:
5955:
5951:
5944:
5938:
5932:
5306:
5302:
5285:Continuity:
5276:
5271:
5267:
5260:
5255:
5245:
5241:
5237:
5232:
5228:
5225:
5211:
5207:
5205:
5199:
5192:
5185:
5179:
5121:
4893:
4232:
4157:
4061:
3954:
3947:
3939:
3931:
3919:
3907:
3896:
3890:
3886:equiprobable
3881:
3878:
3863:
3856:
3849:
3842:
3827:
3816:
3802:
3795:
3791:
3785:
3773:
3769:
3764:
3760:
3756:
3747:
3507:= 0.7, then
3503:
3497:
3493:
3487:
3481:
3478:
3216:
3209:
3183:
3176:
3161:
3123:
2918:
2635:
2538:
2536:
2371:
1910:
1831:
1761:
1754:
1744:
1740:
1729:
1718:
1710:
1696:
1596:
1558:
1521:
1422:
1261:Named after
1260:
1251:
1244:
1234:
1224:
1217:
1169:
1163:
1156:
1153:
1081:
913:
906:
715:of an event
712:
708:
701:
699:
694:
690:
687:
684:Introduction
606:
581:
571:
537:
293:
287:
203:Entropy rate
162:
138:
123:
114:
104:
97:
90:
83:
71:
59:Please help
54:verification
51:
17064:Prefix code
16917:Frame types
16738:Color space
16564:Convolution
16294:LZ77 + ANS
16205:Incremental
16178:Other types
16097:Levenshtein
15787:, Springer.
15732:Cover, T.M.
15668:16 December
15617:16 December
15415:31 December
15378:31 December
15256:27 November
15229:15 December
14322:Theil index
12382:along with
12300:Terence Tao
11601:log(Δ) → −∞
9292:uncertainty
9006:compression
8970:typical set
8924:application
5479:permutation
4707:introducing
3808:information
1231:information
17169:Categories
17121:Mark Adler
17079:Redundancy
16996:Daubechies
16979:Estimation
16912:Frame rate
16834:Daubechies
16794:Chain code
16753:Macroblock
16559:Companding
16496:Estimation
16416:Daubechies
16122:Lempel–Ziv
16082:Exp-Golomb
16010:Arithmetic
15816:Weaver, W.
15710:PlanetMath
15018:2 November
14526:References
14306:Randomness
14276:Perplexity
14134:to obtain
13566:Note that
13185:and hence
12441:, we have
11326:, we have
9011:redundancy
8978:Lempel–Ziv
8629:microstate
7038:Discussion
6841:Symmetry:
5966:Discussion
5296:Symmetry:
5291:continuous
5289:should be
5087:, so that
4230:, so that
3166:(i.e. the
2702:such that
2662:set family
1368:such that
1257:Definition
598:losslessly
87:newspapers
17098:Community
16922:Interlace
16308:Zstandard
16087:Fibonacci
16077:Universal
16035:Canonical
15916:"Entropy"
15909:EMS Press
15899:"Entropy"
15658:2168-2887
15493:18 August
15335:1466-8238
15219:0018-8646
15068:204177762
14942:cite book
14707:5 October
14491:→
14481::
14449:
14408:
14394:→
14357:
13876:≥
13865:−
13851:−
13762:−
13736:−
13722:−
13662:∑
13631:−
13617:−
13532:−
13523:
13504:−
13495:−
13483:
13467:−
13397:≤
13370:≤
13246:
13240:≤
13222:∈
13145:∈
13111:+1, ...,
12979:∈
12920:∈
12871:∑
12857:≤
12835:…
12783:{1, ...,
12717:∈
12698:…
12660:…
12630:−
12616:…
12482:∏
12478:≤
12470:−
12332:λ
12192:
12174:∫
12135:
12129:∫
12117:‖
11971:
11948:∞
11945:→
11900:Δ
11881:
11861:∞
11856:∞
11853:−
11849:∫
11722:
11702:∞
11697:∞
11694:−
11690:∫
11686:−
11675:Δ
11672:
11661:Δ
11641:→
11638:Δ
11558:
11538:∞
11533:∞
11530:−
11526:∫
11522:→
11490:
11484:Δ
11460:∞
11455:∞
11452:−
11442:∑
11407:∞
11402:∞
11399:−
11395:∫
11391:→
11384:Δ
11360:∞
11355:∞
11352:−
11342:∑
11302:Δ
11296:
11290:Δ
11266:∞
11261:∞
11258:−
11248:∑
11244:−
11216:
11210:Δ
11186:∞
11181:∞
11178:−
11168:∑
11164:−
11156:Δ
11124:Δ
11097:
11091:Δ
11067:∞
11062:∞
11059:−
11049:∑
11045:−
11037:Δ
11004:Δ
10980:∞
10975:∞
10972:−
10962:∑
10953:→
10950:Δ
10918:∞
10913:∞
10910:−
10906:∫
10858:Δ
10838:Δ
10831:∫
10824:Δ
10774:Δ
10720:Boltzmann
10716:H-theorem
10663:
10637:∫
10633:−
10612:
10606:−
10487:
10422:−
10381:∏
10374:
10334:−
10307:
10277:∑
10258:
10219:−
10192:
10159:∑
10140:
10100:
10048:∑
10044:−
10029:η
9988:
9948:
9896:∑
9892:−
9851:η
9764:
9724:∑
9695:∑
9675:∑
9671:−
9541:
9507:∑
9487:∑
9483:−
9396:
9380:∑
9377:−
9323:guesswork
9196:.) Other
9138:dominance
9104:broadcast
9078:Broadcast
8990:checksums
8826:
8711:ρ
8708:
8702:ρ
8675:−
8658:in 1927:
8641:Boltzmann
8574:
8558:∑
8545:−
8473:LogSumExp
8445:≤
8442:λ
8439:≤
8348:λ
8345:−
8312:λ
8309:≥
8290:λ
8287:−
8265:λ
8142:≤
8081:≤
8045:, we have
7907:≤
7383:
7370:≤
7351:…
7263:…
7207:…
7086:…
7003:−
6970:→
6926:…
6877:…
6810:…
6754:…
6540:≤
6433:μ
6430:
6424:⋅
6412:μ
6382:∩
6355:⋅
6346:∣
6265:Choosing
6222:…
6140:∑
6108:…
6047:…
5892:⏟
5867:…
5796:⏟
5779:…
5702:…
5669:≤
5650:…
5585:Maximum:
5440:…
5362:…
5060:∈
5034:≥
5022:
4968:∈
4951:for some
4930:
4909:
4832:−
4820:
4759:
4677:
4651:
4574:
4514:
4460:
4413:
4362:
4337:
4295:
4215:
4201:−
4114:
4093:
4036:
4030:−
4004:
3986:
3705:−
3699:⋅
3693:−
3684:−
3678:⋅
3672:−
3669:≈
3650:
3634:−
3622:
3606:−
3584:
3568:−
3556:
3540:−
3448:−
3442:⋅
3411:∑
3407:−
3383:
3342:∑
3338:−
3305:
3255:∑
3251:−
3171:surprisal
3110:μ
3087:Σ
3079:μ
3038:μ
3023:⊆
2998:μ
2891:μ
2878:∈
2871:∑
2853:μ
2803:∈
2762:∩
2753:μ
2721:
2717:∪
2710:μ
2674:⊆
2658:partition
2644:μ
2608:μ
2604:σ
2591:μ
2574:μ
2511:μ
2508:
2502:−
2485:μ
2481:σ
2450:surprisal
2432:Σ
2429:∈
2399:μ
2393:Σ
2109:
2063:×
2053:∈
2040:∑
2036:−
1882:
1861:→
1808:∈
1797:for some
1706:logarithm
1667:
1633:∈
1626:∑
1622:−
1573:
1490:
1484:−
1458:
1341:→
1200:
1036:
970:
957:−
899:logarithm
833:
709:surprisal
653:
647:−
525:logarithm
491:Σ
456:
429:∈
422:∑
418:−
366:→
356::
17084:Symmetry
17052:Timeline
17035:FM-index
16880:Bit rate
16873:Concepts
16721:Concepts
16584:Sampling
16537:Bit rate
16530:Concepts
16232:Sequitur
16067:Tunstall
16040:Modified
16030:Adaptive
15988:Lossless
15937:Archived
15919:Archived
15839:Archived
15755:(2003),
15738:(2006),
15662:Archived
15611:Archived
15544:Archived
15540:59361755
15406:Archived
15369:Archived
15343:85935463
15277:Archived
15223:Archived
15180:17870175
15009:Archived
14969:Archived
14932:4 August
14923:Archived
14747:Archived
14729:(2003).
14701:Archived
14680:Archived
14615:Archived
14313:(SampEn)
14161:See also
13077:, where
12398:Archived
12387:Archived
12371:per the
10738:bin size
9134:evenness
9130:richness
9050:exabytes
8943:Landauer
8794:equation
8643:(1872).
6283:= ... =
5992:+ ... +
5477:for any
5212:messages
5196:nats or
5155:for the
5144:for the
5130:for the
5049:for all
4816:′
4775:′
4755:′
4673:″
4647:′
4570:″
4510:′
4456:′
4409:′
3835:I(1) = 0
3491:, where
3181:, where
3168:expected
3158:Entropy
2656:-almost
2539:expected
691:will not
547:hartleys
533:shannons
17042:Entropy
16991:Wavelet
16970:Motion
16829:Wavelet
16809:Fractal
16804:Deflate
16787:Methods
16574:Latency
16487:Motion
16411:Wavelet
16328:LHA/LZH
16278:Deflate
16227:Re-Pair
16222:Grammar
16052:Shannon
16025:Huffman
15981:methods
15934:Entropy
15911:, 2001
15818:(1949)
15313:Bibcode
15286:Science
15160:Bibcode
15119:Entropy
15060:1426210
14896:at the
14894:Entropy
14713:YouTube
13439:where
13320:. Then
13297:0 <
12771:, ...,
12753:, ...,
12558:in the
12554:is the
12259:is the
12249:is the
10714:in the
9256:program
9235:) = log
9067:Storage
9040:Science
8974:Huffman
8882:is the
8746:changes
8612:is the
8525:is the
8514:entropy
8499:entropy
8482:Aspects
8221:concave
6300:(1) = 0
5950:, ...,
5266:, ...,
5208:meaning
5175:(2) = 1
3942:) + log
3934:) = log
3196:6 bits.
3150:Example
1714:are 2,
1704:of the
1700:is the
1552:is the
1544:is the
1273:) of a
1241:Example
897:is the
609:entropy
566:
553:of the
294:entropy
163:Entropy
101:scholar
17153:codecs
17114:People
17017:Theory
16984:Vector
16501:Vector
16318:Brotli
16268:Hybrid
16167:Snappy
16020:Golomb
15866:about
15849:
15826:
15801:
15777:
15763:
15746:
15693:
15656:
15603:
15538:
15448:
15341:
15333:
15217:
15178:
15085:
15066:
15058:
14877:
14818:
14791:
14753:9 June
14739:
14549:
14186:(ApEn)
14084:given
13093:. Let
13085:|
13079:|
13075:|
12950:where
12543:where
11820:Δ
11761:is not
11599:Note;
9767:
9758:
9630:given
9578:where
9516:
9424:where
9258:for a
9144:, the
9136:, and
8916:Jaynes
8847:where
8631:. The
8616:, and
8603:where
8243:, i.e.
7738:yields
7618:where
5984:where
5275:) = Η(
4860:
4843:
4801:
4784:
4694:
4625:
4608:
4488:
4447:
4329:
4252:Proof
3862:) + I(
3855:) = I(
3724:0.8816
2448:. The
2444:be an
2418:. Let
2182:where
1748:, and
1694:where
1548:, and
1267:Η
877:where
625:axioms
523:, the
483:where
292:, the
103:
96:
89:
82:
74:
16944:parts
16942:Codec
16907:Frame
16865:Video
16849:SPIHT
16758:Pixel
16713:Image
16667:ACELP
16638:ADPCM
16628:μ-law
16623:A-law
16616:parts
16614:Codec
16522:Audio
16461:ACELP
16449:ADPCM
16426:SPIHT
16367:Lossy
16351:bzip2
16342:LZHAM
16298:LZFSE
16200:Delta
16092:Gamma
16072:Unary
16047:Range
15536:S2CID
15518:arXiv
15409:(PDF)
15398:(PDF)
15372:(PDF)
15361:(PDF)
15339:S2CID
15176:S2CID
15064:S2CID
15056:JSTOR
15012:(PDF)
14989:(PDF)
14926:(PDF)
14919:(PDF)
14334:Notes
13301:<
12745:: if
12079:with
11605:Δ → 0
11324:Δ → 0
9586:state
9584:is a
9184:of a
9092:0.281
9084:1900
9062:2007
9009:less
9004:If a
8739:trace
8521:of a
6506:Aczél
5236:= Pr(
5198:0.301
5191:0.693
4894:This
3708:1.737
3687:0.515
2939:be a
2660:is a
2446:event
2414:be a
1843:limit
1829:0 log
1522:Here
1172:= 1/2
531:(or "
296:of a
108:JSTOR
94:books
16956:DPCM
16763:PSNR
16694:MDCT
16687:WLPC
16672:CELP
16633:DPCM
16481:WLPC
16466:CELP
16444:DPCM
16394:MDCT
16338:LZMA
16239:LDCT
16217:DPCM
16162:LZWL
16152:LZSS
16147:LZRW
16137:LZJB
15847:ISBN
15824:ISBN
15799:ISBN
15775:ISBN
15761:ISBN
15744:ISBN
15691:ISBN
15670:2021
15654:ISSN
15619:2021
15601:ISBN
15552:2023
15495:2014
15446:ISBN
15417:2013
15380:2013
15331:ISSN
15258:2008
15231:2021
15215:ISSN
15083:ISBN
15020:2022
14948:link
14934:2017
14875:ISBN
14816:ISBN
14789:ISBN
14755:2014
14737:ISBN
14709:2021
14677:here
14612:here
14547:ISBN
14287:for
13306:let
13107:−1,
12049:) =
9451:is:
9227:−log
9206:rate
9178:the
9169:the
9160:the
9073:295
9059:1986
9023:less
9019:more
8961:and
8896:= 1/
8428:and
8039:and
7944:and
7423:and
6490:dual
5961:box.
5814:<
5251:and
5206:The
5153:bans
5142:nats
5128:bits
5098:<
4257:Let
4187:for
4172:>
4140:<
4129:for
4082:are
3768:log(
3727:<
3485:and
3204:and
3179:= 1)
2919:Let
2745:and
2537:The
2265:and
1979:and
1935:and
1757:= 10
1752:for
1750:bans
1738:for
1736:nats
1727:for
1725:bits
1702:base
1180:trit
1162:1 −
700:The
695:will
619:and
529:bits
80:news
17001:DWT
16951:DCT
16895:VBR
16890:CBR
16885:ABR
16844:EZW
16839:DWT
16824:RLE
16814:KLT
16799:DCT
16682:LSP
16677:LAR
16662:LPC
16655:FFT
16552:VBR
16547:CBR
16542:ABR
16476:LSP
16471:LAR
16456:LPC
16421:DWT
16406:FFT
16401:DST
16389:DCT
16288:LZS
16283:LZX
16259:RLE
16254:PPM
16249:PAQ
16244:MTF
16212:DMC
16190:CTW
16185:BWT
16157:LZW
16142:LZO
16132:LZ4
16127:842
15925:at
15646:doi
15593:doi
15528:doi
15436:doi
15321:doi
15207:doi
15168:doi
15156:106
15127:doi
15048:doi
15001:doi
14966:url
14901:Lab
14673:PDF
14661:hdl
14653:doi
14608:PDF
14596:hdl
14588:doi
14446:log
14405:log
14387:lim
14354:log
14259:in
14148:or
13914:+ 1
13514:log
13474:log
13243:log
13042:).
13020:in
12189:log
12132:log
12030:is
11968:log
11878:log
11790:not
11774:).
11768:→ ∞
11719:log
11669:log
11634:lim
11603:as
11555:log
11487:log
11322:As
11293:log
11213:log
11094:log
10946:lim
10718:of
10660:log
10609:log
10478:log
10365:log
10298:log
10249:log
10183:log
10131:log
10091:log
9979:log
9939:log
9761:log
9538:log
9393:log
9307:127
9286:In
9268:log
9231:(1/
9095:65
9081:432
9070:2.6
9028:all
8980:or
8937:).
8781:or
8508:In
8471:is
8219:is
7938:If
7876:so
7583:If
7374:log
7304:log
6963:lim
6463:log
6292:= 1
5538:of
5171:log
5160:log
5135:log
4927:log
4212:log
4111:log
4033:log
4001:log
3926:log
3914:log
3902:log
3825:in
3821:is
3759:−Σ
3696:0.3
3675:0.7
3656:0.3
3641:log
3637:0.3
3628:0.7
3613:log
3609:0.7
3575:log
3547:log
3374:log
3296:log
3192:log
3186:= 1
3175:Pr(
3124:all
3016:sup
2983:is
2943:on
2838:is
2561:is
2472:is
2452:of
2106:log
1879:log
1854:lim
1835:(0)
1732:= 2
1658:log
1556:of
1487:log
1271:eta
1227:= 1
1222:or
1220:= 0
1191:log
1176:bit
1151:).
1027:log
961:log
934:by
903:log
885:log
830:log
711:or
650:log
611:in
543:nat
511:log
453:log
288:In
63:by
17171::
16819:LP
16650:FT
16643:DM
16195:CM
15907:,
15901:,
15814:,
15734:,
15660:.
15652:.
15640:.
15636:.
15609:.
15599:.
15542:.
15534:.
15526:.
15516:.
15512:.
15484:.
15444:.
15404:.
15400:.
15367:.
15363:.
15337:.
15329:.
15319:.
15309:12
15307:.
15303:.
15266:^
15221:.
15213:.
15201:.
15197:.
15174:.
15166:.
15154:.
15150:.
15123:17
15121:.
15117:.
15062:.
15054:.
15042:.
15028:^
15007:.
14997:25
14995:.
14991:.
14944:}}
14940:{{
14843:^
14745:.
14699:.
14659:.
14649:27
14647:.
14641:.
14626:^
14594:.
14584:27
14582:.
14576:.
14561:^
13986:.
13784:1.
13311:=
13065:Η(
12436:⊆
12419:.
12375:.
12366:H'
12324:=
12306:.
12291:.
12274:=
12063:dx
12047:dx
11812:1/
11042::=
10730:.
10722:.
9220:1/
9132:,
9110:.
9013:.
8976:,
8823:ln
8796::
8769:/
8741:.
8705:ln
8571:ln
8505:.
8460:.
7155::
7140:.
7111:.
6427:ln
6318:.
6275:,
6270:=
6006:,
6001:=
5281:.
5240:=
5162:10
5151:,
5149:ln
5140:,
5113:.
3932:mn
3897:mn
3841:I(
3815:I(
3810::
3730:1.
3496:≠
3461:1.
3214:.
3160:Η(
3146:.
2636:A
2505:ln
2364:.
2221::=
1897:0.
1845::
1759:.
1743:=
1734:,
1562:.
1420::
1388::=
911:.
604:.
415::=
17155:)
17151:(
15971:e
15964:t
15957:v
15853:.
15807:.
15716:.
15699:.
15672:.
15648::
15642:4
15621:.
15595::
15554:.
15530::
15520::
15497:.
15454:.
15438::
15419:.
15382:.
15345:.
15323::
15315::
15260:.
15233:.
15209::
15203:5
15182:.
15170::
15162::
15135:.
15129::
15070:.
15050::
15044:6
15022:.
15003::
14950:)
14936:.
14899:n
14883:.
14824:.
14797:.
14757:.
14715:.
14686:)
14671:(
14669:.
14663::
14655::
14621:)
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14598::
14590::
14555:.
14506:]
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14494:(
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14423:0
14420:=
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14414:x
14411:(
14402:x
14397:0
14391:x
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14360:(
14112:X
14092:X
14072:Y
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14029:X
14026:,
14023:Y
14020:(
14017:G
14014:I
13972:)
13969:n
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13464:=
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13419:)
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12958:(
12935:]
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12913:)
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12838:,
12832:,
12827:1
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12819:(
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12613:,
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12600:(
12597:{
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12588:A
12585:(
12580:i
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12561:i
12550:i
12546:P
12527:|
12523:)
12520:A
12517:(
12512:i
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12503:|
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12186:)
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12171:=
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11910:x
11907:d
11903:)
11896:)
11893:x
11890:(
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11884:(
11875:)
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11826:x
11814:x
11808:)
11806:x
11804:(
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11766:n
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11725:f
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11710:(
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11630:=
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11577:x
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11570:)
11567:x
11564:(
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11552:)
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11546:(
11543:f
11515:)
11512:)
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11499:(
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11468:(
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11421:)
11418:x
11415:(
11412:f
11381:)
11376:i
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11368:(
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11349:=
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11305:)
11299:(
11287:)
11282:i
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11255:=
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11241:)
11238:)
11233:i
11229:x
11225:(
11222:f
11219:(
11207:)
11202:i
11198:x
11194:(
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11175:=
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11128:)
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11116:i
11112:x
11108:(
11105:f
11101:(
11088:)
11083:i
11079:x
11075:(
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11056:=
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11032:H
11007:,
11001:)
10996:i
10992:x
10988:(
10985:f
10969:=
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10942:=
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10932:)
10929:x
10926:(
10923:f
10894:f
10879:x
10876:d
10872:)
10869:x
10866:(
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10849:+
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10827:=
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10812:x
10808:(
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10793:i
10789:x
10763:f
10753:n
10749:p
10743:n
10711:Η
10705:h
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10684:x
10680:d
10675:)
10672:x
10669:(
10666:f
10657:)
10654:x
10651:(
10648:f
10642:X
10630:=
10627:]
10624:)
10621:X
10618:(
10615:f
10603:[
10599:E
10595:=
10592:)
10589:X
10586:(
10582:H
10557:X
10546:)
10544:x
10542:(
10540:f
10512:b
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10493:n
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10441:)
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10432:x
10428:(
10425:p
10418:)
10412:i
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10404:(
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10353:)
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10340:(
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10330:)
10324:i
10320:x
10316:(
10313:p
10310:(
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10284:=
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10267:)
10264:n
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10253:b
10243:)
10238:)
10233:i
10229:x
10225:(
10222:p
10215:)
10209:i
10205:x
10201:(
10198:p
10195:(
10187:b
10174:n
10169:1
10166:=
10163:i
10155:=
10149:)
10146:n
10143:(
10135:b
10125:)
10122:)
10117:i
10113:x
10109:(
10106:p
10103:(
10095:b
10087:)
10082:i
10078:x
10074:(
10071:p
10063:n
10058:1
10055:=
10052:i
10041:=
10038:)
10035:X
10032:(
10003:.
9997:)
9994:n
9991:(
9983:b
9973:)
9970:)
9965:i
9961:x
9957:(
9954:p
9951:(
9943:b
9935:)
9930:i
9926:x
9922:(
9919:p
9911:n
9906:1
9903:=
9900:i
9889:=
9882:x
9879:a
9876:m
9872:H
9868:H
9863:=
9860:)
9857:X
9854:(
9825:X
9795:.
9792:)
9789:k
9786:(
9781:j
9778:,
9775:i
9771:p
9755:)
9752:k
9749:(
9744:j
9741:,
9738:i
9734:p
9728:k
9720:)
9717:j
9714:(
9709:i
9705:p
9699:j
9689:i
9685:p
9679:i
9668:=
9665:)
9660:S
9655:(
9651:H
9633:i
9627:j
9612:)
9609:j
9606:(
9601:i
9597:p
9581:i
9563:,
9560:)
9557:j
9554:(
9549:i
9545:p
9535:)
9532:j
9529:(
9524:i
9520:p
9511:j
9501:i
9497:p
9491:i
9480:=
9477:)
9472:S
9467:(
9463:H
9438:i
9431:i
9427:p
9409:,
9404:i
9400:p
9388:i
9384:p
9374:=
9371:)
9366:S
9361:(
9357:H
9303:2
9276:)
9274:n
9272:(
9270:2
9243:)
9241:N
9239:(
9237:2
9233:N
9229:2
9222:N
9215:N
9142:D
8907:B
8904:k
8898:W
8893:i
8889:p
8879:B
8876:k
8870:W
8855:S
8832:,
8829:W
8818:B
8814:k
8810:=
8807:S
8774:B
8771:k
8767:S
8761:B
8758:k
8718:,
8714:)
8699:(
8694:r
8691:T
8683:B
8679:k
8672:=
8669:S
8623:i
8619:p
8609:B
8606:k
8588:,
8582:i
8578:p
8566:i
8562:p
8553:B
8549:k
8542:=
8539:S
8518:S
8475:.
8448:1
8436:0
8414:2
8410:p
8406:,
8401:1
8397:p
8372:)
8367:2
8363:p
8359:(
8355:H
8351:)
8342:1
8339:(
8336:+
8333:)
8328:1
8324:p
8320:(
8316:H
8306:)
8301:2
8297:p
8293:)
8284:1
8281:(
8278:+
8273:1
8269:p
8262:(
8258:H
8231:p
8207:)
8204:p
8201:(
8197:H
8173:)
8170:Y
8167:(
8163:H
8159:+
8156:)
8153:X
8150:(
8146:H
8139:)
8136:Y
8133:,
8130:X
8127:(
8123:H
8107:.
8095:)
8092:X
8089:(
8085:H
8078:)
8075:Y
8071:|
8067:X
8064:(
8060:H
8042:Y
8036:X
8015:.
8012:)
8009:X
8006:(
8002:H
7998:=
7995:)
7992:Y
7988:|
7984:X
7981:(
7977:H
7959:X
7953:Y
7947:Y
7941:X
7921:)
7918:X
7915:(
7911:H
7904:)
7901:)
7898:X
7895:(
7892:f
7889:(
7885:H
7860:,
7857:)
7854:)
7851:X
7848:(
7845:f
7841:|
7837:X
7834:(
7830:H
7826:+
7823:)
7820:)
7817:X
7814:(
7811:f
7808:(
7804:H
7800:=
7797:)
7794:X
7790:|
7786:)
7783:X
7780:(
7777:f
7774:(
7770:H
7766:+
7763:)
7760:X
7757:(
7753:H
7726:)
7723:)
7720:X
7717:(
7714:f
7711:,
7708:X
7705:(
7701:H
7680:0
7677:=
7674:)
7671:X
7667:|
7663:)
7660:X
7657:(
7654:f
7651:(
7647:H
7626:f
7606:)
7603:X
7600:(
7597:f
7594:=
7591:Y
7565:.
7562:)
7559:X
7556:(
7552:H
7548:+
7545:)
7542:X
7538:|
7534:Y
7531:(
7527:H
7523:=
7520:)
7517:Y
7514:(
7510:H
7506:+
7503:)
7500:Y
7496:|
7492:X
7489:(
7485:H
7481:=
7478:)
7475:Y
7472:,
7469:X
7466:(
7462:H
7444:Y
7438:X
7432:Y
7426:Y
7420:X
7415:)
7413:Y
7411:,
7409:X
7407:(
7398:.
7386:n
7378:b
7367:)
7362:n
7358:p
7354:,
7348:,
7343:1
7339:p
7335:(
7331:H
7314:)
7312:n
7310:(
7307:b
7300:n
7291:.
7279:)
7274:n
7270:p
7266:,
7260:,
7255:1
7251:p
7247:(
7242:n
7237:H
7232:=
7229:)
7226:0
7223:,
7218:n
7214:p
7210:,
7204:,
7199:1
7195:p
7191:(
7186:1
7183:+
7180:n
7175:H
7152:X
7123:H
7097:n
7093:p
7089:,
7083:,
7078:1
7074:p
7051:H
7033:.
7021:0
7018:=
7015:)
7012:q
7009:,
7006:q
7000:1
6997:(
6992:2
6987:H
6978:+
6974:0
6967:q
6951:.
6937:n
6933:p
6929:,
6923:,
6918:1
6914:p
6893:)
6888:n
6884:p
6880:,
6874:,
6869:1
6865:p
6861:(
6856:n
6851:H
6826:)
6821:n
6817:p
6813:,
6807:,
6802:1
6798:p
6794:(
6789:n
6784:H
6779:=
6776:)
6773:0
6770:,
6765:n
6761:p
6757:,
6751:,
6746:1
6742:p
6738:(
6733:1
6730:+
6727:n
6722:H
6697:Y
6694:,
6691:X
6671:)
6668:Y
6665:(
6661:H
6657:+
6654:)
6651:X
6648:(
6644:H
6640:=
6637:)
6634:Y
6631:,
6628:X
6625:(
6621:H
6609:.
6597:Y
6594:,
6591:X
6571:)
6568:Y
6565:(
6561:H
6557:+
6554:)
6551:X
6548:(
6544:H
6537:)
6534:Y
6531:,
6528:X
6525:(
6521:H
6467:2
6442:)
6439:A
6436:(
6421:)
6418:A
6415:(
6388:)
6385:B
6379:A
6376:(
6373:P
6370:=
6367:)
6364:B
6361:(
6358:P
6352:)
6349:B
6343:A
6340:(
6337:P
6311:n
6305:n
6298:1
6296:Η
6289:n
6285:b
6281:1
6278:b
6272:n
6268:k
6250:.
6246:)
6238:i
6234:b
6230:1
6225:,
6219:,
6212:i
6208:b
6204:1
6198:(
6190:i
6186:b
6180:H
6172:n
6167:i
6163:b
6155:k
6150:1
6147:=
6144:i
6136:+
6132:)
6126:n
6121:k
6117:b
6111:,
6105:,
6100:n
6095:1
6091:b
6084:(
6078:k
6073:H
6068:=
6064:)
6058:n
6055:1
6050:,
6044:,
6039:n
6036:1
6030:(
6024:n
6019:H
6003:n
5998:k
5994:b
5990:1
5987:b
5980:i
5976:b
5956:k
5952:b
5948:1
5945:b
5939:k
5933:n
5916:.
5911:)
5904:1
5901:+
5898:n
5885:1
5882:+
5879:n
5875:1
5870:,
5864:,
5858:1
5855:+
5852:n
5848:1
5837:(
5830:1
5827:+
5824:n
5819:H
5809:)
5802:n
5790:n
5787:1
5782:,
5776:,
5771:n
5768:1
5757:(
5750:n
5745:H
5732:.
5719:)
5713:n
5710:1
5705:,
5699:,
5694:n
5691:1
5685:(
5679:n
5674:H
5666:)
5661:n
5657:p
5653:,
5647:,
5642:1
5638:p
5634:(
5629:n
5624:H
5600:n
5595:H
5582:.
5570:}
5567:n
5564:,
5561:.
5558:.
5555:.
5552:,
5549:1
5546:{
5526:}
5521:n
5517:i
5513:,
5510:.
5507:.
5504:.
5501:,
5496:1
5492:i
5488:{
5464:)
5456:n
5452:i
5447:p
5443:,
5437:,
5430:2
5426:i
5421:p
5417:,
5410:1
5406:i
5401:p
5396:(
5390:n
5385:H
5380:=
5376:)
5370:n
5366:p
5359:,
5354:2
5350:p
5346:,
5341:1
5337:p
5332:(
5326:n
5321:H
5307:i
5303:x
5298:H
5287:H
5279:)
5277:X
5272:n
5268:p
5264:1
5261:p
5259:(
5256:n
5253:Η
5249:)
5246:i
5242:x
5238:X
5233:i
5229:p
5200:n
5193:n
5186:n
5180:n
5173:2
5137:2
5126:(
5101:0
5095:k
5075:]
5072:1
5069:,
5066:0
5063:[
5057:p
5037:0
5031:)
5028:p
5025:(
5019:I
4999:0
4996:=
4993:c
4972:R
4965:c
4962:,
4959:k
4939:c
4936:+
4933:u
4924:k
4921:=
4918:)
4915:u
4912:(
4906:I
4875:k
4866:,
4863:u
4848:0
4840:=
4835:k
4829:)
4826:u
4823:(
4813:I
4809:u
4789:0
4781:=
4772:)
4768:)
4765:u
4762:(
4752:I
4748:u
4745:(
4733:2
4729:p
4723:1
4719:p
4715:=
4712:u
4699:0
4691:=
4686:)
4683:u
4680:(
4670:I
4666:u
4663:+
4660:)
4657:u
4654:(
4644:I
4633:2
4629:p
4613:0
4605:=
4600:)
4595:2
4591:p
4585:1
4581:p
4577:(
4567:I
4561:2
4557:p
4551:1
4547:p
4543:+
4540:)
4535:2
4531:p
4525:1
4521:p
4517:(
4507:I
4496:1
4492:p
4476:)
4471:1
4467:p
4463:(
4453:I
4444:=
4439:)
4434:2
4430:p
4424:1
4420:p
4416:(
4406:I
4400:2
4396:p
4378:)
4373:2
4369:p
4365:(
4359:I
4356:+
4353:)
4348:1
4344:p
4340:(
4334:I
4326:=
4321:)
4316:2
4312:p
4306:1
4302:p
4298:(
4292:I
4265:I
4233:x
4218:x
4208:/
4204:1
4198:=
4195:k
4175:1
4169:x
4158:k
4143:0
4137:k
4117:u
4108:k
4105:=
4102:)
4099:u
4096:(
4090:I
4070:I
4048:.
4045:)
4042:p
4039:(
4027:=
4023:)
4017:p
4014:1
4008:(
3998:=
3995:)
3992:p
3989:(
3983:I
3963:I
3950:)
3948:n
3946:(
3944:2
3940:m
3938:(
3936:2
3930:(
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3922:)
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3916:2
3910:)
3908:n
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3904:2
3891:m
3882:n
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3857:p
3853:2
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3848:·
3846:1
3843:p
3828:p
3819:)
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3803:i
3796:i
3792:p
3786:i
3781:I
3777:)
3774:i
3770:p
3765:i
3761:p
3721:=
3711:)
3702:(
3690:)
3681:(
3659:)
3653:(
3645:2
3631:)
3625:(
3617:2
3603:=
3593:)
3590:q
3587:(
3579:2
3571:q
3565:)
3562:p
3559:(
3551:2
3543:p
3537:=
3530:)
3527:X
3524:(
3520:H
3504:p
3498:q
3494:p
3488:q
3482:p
3458:=
3454:)
3451:1
3445:(
3437:2
3434:1
3426:2
3421:1
3418:=
3415:i
3404:=
3391:2
3388:1
3378:2
3368:2
3365:1
3357:2
3352:1
3349:=
3346:i
3335:=
3324:)
3319:i
3315:x
3311:(
3308:p
3300:b
3292:)
3287:i
3283:x
3279:(
3276:p
3270:n
3265:1
3262:=
3259:i
3248:=
3241:)
3238:X
3235:(
3231:H
3194:2
3184:X
3177:X
3164:)
3162:X
3134:X
3090:)
3084:(
3074:H
3052:.
3049:)
3046:P
3043:(
3033:H
3026:M
3020:P
3012:=
3009:)
3006:M
3003:(
2993:H
2971:M
2951:X
2927:M
2905:.
2902:)
2899:A
2896:(
2887:h
2881:P
2875:A
2867:=
2864:)
2861:P
2858:(
2848:H
2826:P
2806:P
2800:B
2797:,
2794:A
2774:0
2771:=
2768:)
2765:B
2759:A
2756:(
2733:1
2730:=
2727:)
2724:P
2713:(
2690:)
2687:X
2684:(
2679:P
2671:P
2622:.
2619:)
2616:A
2613:(
2600:)
2597:A
2594:(
2588:=
2585:)
2582:A
2579:(
2570:h
2549:A
2523:.
2520:)
2517:A
2514:(
2499:=
2496:)
2493:A
2490:(
2460:A
2426:A
2402:)
2396:,
2390:,
2387:X
2384:(
2352:Y
2332:X
2312:]
2309:y
2306:=
2303:Y
2300:[
2296:P
2292:=
2289:)
2286:y
2283:(
2278:Y
2274:p
2253:]
2250:y
2247:=
2244:Y
2241:,
2238:x
2235:=
2232:X
2229:[
2225:P
2218:)
2215:y
2212:,
2209:x
2206:(
2201:Y
2198:,
2195:X
2191:p
2170:,
2164:)
2161:y
2158:(
2153:Y
2149:p
2143:)
2140:y
2137:,
2134:x
2131:(
2126:Y
2123:,
2120:X
2116:p
2103:)
2100:y
2097:,
2094:x
2091:(
2086:Y
2083:,
2080:X
2076:p
2068:Y
2058:X
2050:y
2047:,
2044:x
2033:=
2030:)
2027:Y
2023:|
2019:X
2016:(
2012:H
1989:Y
1965:X
1943:Y
1923:X
1894:=
1891:)
1888:p
1885:(
1876:p
1869:+
1865:0
1858:p
1839:0
1832:b
1813:X
1805:x
1785:0
1782:=
1779:)
1776:x
1773:(
1770:p
1755:b
1745:e
1741:b
1730:b
1719:e
1711:b
1697:b
1682:,
1679:)
1676:x
1673:(
1670:p
1662:b
1654:)
1651:x
1648:(
1645:p
1638:X
1630:x
1619:=
1616:)
1613:X
1610:(
1606:H
1582:)
1579:X
1576:(
1570:I
1559:X
1550:I
1531:E
1508:.
1505:]
1502:)
1499:X
1496:(
1493:p
1481:[
1477:E
1473:=
1470:]
1467:)
1464:X
1461:(
1455:I
1452:[
1448:E
1444:=
1441:)
1438:X
1435:(
1431:H
1408:]
1405:x
1402:=
1399:X
1396:[
1392:P
1385:)
1382:x
1379:(
1376:p
1356:]
1353:1
1350:,
1347:0
1344:[
1336:X
1331::
1328:p
1306:X
1284:X
1235:p
1225:p
1218:p
1203:3
1195:2
1170:p
1164:p
1157:p
1139:2
1135:/
1131:1
1128:=
1125:p
1105:6
1101:/
1097:1
1094:=
1091:p
1083:(
1068:.
1064:)
1058:)
1055:E
1052:(
1049:p
1045:1
1040:(
1031:2
1023:=
1020:)
1017:E
1014:(
1011:I
991:,
988:)
985:)
982:E
979:(
976:p
973:(
965:2
954:=
951:)
948:E
945:(
942:I
922:E
865:,
861:)
855:)
852:E
849:(
846:p
842:1
837:(
810:)
807:E
804:(
801:p
781:)
778:E
775:(
772:p
752:)
749:E
746:(
743:p
723:E
705:,
668:]
665:)
662:X
659:(
656:p
644:[
640:E
538:e
471:,
468:)
465:x
462:(
459:p
450:)
447:x
444:(
441:p
434:X
426:x
412:)
409:X
406:(
402:H
381:]
378:1
375:,
372:0
369:[
361:X
353:p
331:X
309:X
277:e
270:t
263:v
130:)
124:(
119:)
115:(
105:·
98:·
91:·
84:·
57:.
34:.
20:)
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