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- Entropy_(information_theory) abstract "In information theory, entropy (more specifically, Shannon entropy) is the expected value (average) of the information contained in each message received. 'Messages' don't have to be text; in this context a 'message' is simply any flow of information. The entropy of the message is its amount of uncertainty; it increases when the message is closer to random, and decreases when it is less random. The idea here is that the less likely an event is, the more information it provides when it occurs. This seems backwards at first: it seems like messages which have more structure would contain more information, but this is not true. For example, the message 'aaaaaaaaaa' (which appears to be very structured and not random at all [although in fact it could result from a random process]) contains much less information than the message 'alphabet' (which is somewhat structured, but more random) or even the message 'axraefy6h' (which is very random). In information theory, 'information' doesn't necessarily mean useful information; it simply describes the amount of randomness of the message, so in the example above the first message has the least information and the last message has the most information, even though in everyday terms we would say that the middle message, 'alphabet', contains more information than a stream of random letters. Therefore, we would say in information theory that the first message has low entropy, the second has higher entropy, and the third has the highest entropy.In a more technical sense, there are reasons (explained below) to define information as the negative of the logarithm of the probability distribution. The probability distribution of the events, coupled with the information amount of every event, forms a random variable whose average (also termed expected value) is the average amount of information, a.k.a. entropy, generated by this distribution. Units of entropy are the shannon, nat, or hartley, depending on the base of the logarithm used to define it, though the shannon is commonly referred to as a bit.The logarithm of the probability distribution is useful as a measure of entropy because it is additive for independent sources. For instance, the entropy of a coin toss is 1 shannon, whereas of m tosses it is m shannons. Generally, you need log2(n) bits to represent a variable that can take one of n values if n is a power of 2. If these values are equiprobable, the entropy (in shannons) is equal to the number of bits. Equality between number of bits and shannons holds only while all outcomes are equally probable. If one of the events is more probable than others, observation of that event is less informative. Conversely, observing rarer events compensate by providing more information when observed. Since observation of less probable events occurs more rarely, the net effect is that the entropy (thought of as the average information) received from non-uniformly distributed data is less than log2(n). Entropy is zero when one outcome is certain. Shannon entropy quantifies all these considerations exactly when a probability distribution of the source is known. The meaning of the events observed (a.k.a. the meaning of messages) do 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 probability distribution, not the meaning of the events themselves.Generally, entropy refers to disorder or uncertainty. Shannon entropy was introduced by Claude E. Shannon in his 1948 paper "A Mathematical Theory of Communication". Shannon entropy provides an absolute limit on the best possible average length of lossless encoding or compression of an information source. Rényi entropy generalizes Shannon entropy.".
- Entropy_(information_theory) thumbnail Entropy_flip_2_coins.jpg?width=300.
- Entropy_(information_theory) wikiPageExternalLink books?id=_77lvx7y8joC.
- Entropy_(information_theory) wikiPageExternalLink 1404.1998.
- Entropy_(information_theory) wikiPageExternalLink InfoTheoryBookMain.html.
- Entropy_(information_theory) wikiPageExternalLink ENTROPY.
- Entropy_(information_theory) wikiPageExternalLink ENTRINFO.html.
- Entropy_(information_theory) wikiPageExternalLink infogain.html.
- Entropy_(information_theory) wikiPageExternalLink entropy.
- Entropy_(information_theory) wikiPageExternalLink 6.html.
- Entropy_(information_theory) wikiPageExternalLink www.shannonentropy.netmark.pl.
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- Entropy_(information_theory) wikiPageWikiLink A_Mathematical_Theory_of_Communication.
- Entropy_(information_theory) wikiPageWikiLink Absolute_continuity.
- Entropy_(information_theory) wikiPageWikiLink Absolutely_continuous.
- Entropy_(information_theory) wikiPageWikiLink Arithmetic_coding.
- Entropy_(information_theory) wikiPageWikiLink Base_(exponentiation).
- Entropy_(information_theory) wikiPageWikiLink Bernoulli_process.
- Entropy_(information_theory) wikiPageWikiLink Bin_size.
- Entropy_(information_theory) wikiPageWikiLink Bit.
- Entropy_(information_theory) wikiPageWikiLink Boltzmann.
- Entropy_(information_theory) wikiPageWikiLink Boltzmann_constant.
- Entropy_(information_theory) wikiPageWikiLink Boltzmanns_constant.
- Entropy_(information_theory) wikiPageWikiLink Broadcast.
- Entropy_(information_theory) wikiPageWikiLink Broadcasting.
- Entropy_(information_theory) wikiPageWikiLink Category:Entropy_and_information.
- Entropy_(information_theory) wikiPageWikiLink Category:Information_theory.
- Entropy_(information_theory) wikiPageWikiLink Category:Randomness.
- Entropy_(information_theory) wikiPageWikiLink Category:Statistical_theory.
- Entropy_(information_theory) wikiPageWikiLink Characterization_(mathematics).
- Entropy_(information_theory) wikiPageWikiLink Checksum.
- Entropy_(information_theory) wikiPageWikiLink Claude_E._Shannon.
- Entropy_(information_theory) wikiPageWikiLink Claude_Shannon.
- Entropy_(information_theory) wikiPageWikiLink Combinatorics.
- Entropy_(information_theory) wikiPageWikiLink Computer_program.
- Entropy_(information_theory) wikiPageWikiLink Conditional_entropy.
- Entropy_(information_theory) wikiPageWikiLink Continuous_function.
- Entropy_(information_theory) wikiPageWikiLink Counting_measure.
- Entropy_(information_theory) wikiPageWikiLink Cross_entropy.
- Entropy_(information_theory) wikiPageWikiLink Cryptanalysis.
- Entropy_(information_theory) wikiPageWikiLink Data_compression.
- Entropy_(information_theory) wikiPageWikiLink Density_matrix.
- Entropy_(information_theory) wikiPageWikiLink Differential_entropy.
- Entropy_(information_theory) wikiPageWikiLink Discrete_probability_distribution.
- Entropy_(information_theory) wikiPageWikiLink Discrete_random_variable.
- Entropy_(information_theory) wikiPageWikiLink Diversity_index.
- Entropy_(information_theory) wikiPageWikiLink Dynamical_system.
- Entropy_(information_theory) wikiPageWikiLink E_(mathematical_constant).
- Entropy_(information_theory) wikiPageWikiLink Edwin_Thompson_Jaynes.
- Entropy_(information_theory) wikiPageWikiLink Entropy.
- Entropy_(information_theory) wikiPageWikiLink Entropy_(arrow_of_time).
- Entropy_(information_theory) wikiPageWikiLink Entropy_(information_theory).
- Entropy_(information_theory) wikiPageWikiLink Entropy_(statistical_thermodynamics).
- Entropy_(information_theory) wikiPageWikiLink Entropy_encoding.
- Entropy_(information_theory) wikiPageWikiLink Entropy_estimation.
- Entropy_(information_theory) wikiPageWikiLink Entropy_power_inequality.
- Entropy_(information_theory) wikiPageWikiLink Entropy_rate.
- Entropy_(information_theory) wikiPageWikiLink Eta.
- Entropy_(information_theory) wikiPageWikiLink Expected_value.
- Entropy_(information_theory) wikiPageWikiLink FLAC.
- Entropy_(information_theory) wikiPageWikiLink Fisher_information.
- Entropy_(information_theory) wikiPageWikiLink Gibbs_entropy.
- Entropy_(information_theory) wikiPageWikiLink H-theorem.
- Entropy_(information_theory) wikiPageWikiLink Hamming_distance.
- Entropy_(information_theory) wikiPageWikiLink Hartley_(unit).
- Entropy_(information_theory) wikiPageWikiLink Histogram.
- Entropy_(information_theory) wikiPageWikiLink History_of_entropy.
- Entropy_(information_theory) wikiPageWikiLink History_of_information_theory.
- Entropy_(information_theory) wikiPageWikiLink Huffman_coding.
- Entropy_(information_theory) wikiPageWikiLink ISBN.
- Entropy_(information_theory) wikiPageWikiLink Independence_(probability_theory).
- Entropy_(information_theory) wikiPageWikiLink Information_geometry.
- Entropy_(information_theory) wikiPageWikiLink Information_theory.
- Entropy_(information_theory) wikiPageWikiLink International_Standard_Book_Number.
- Entropy_(information_theory) wikiPageWikiLink J._Willard_Gibbs.
- Entropy_(information_theory) wikiPageWikiLink Jensen_inequality.
- Entropy_(information_theory) wikiPageWikiLink Jensens_inequality.
- Entropy_(information_theory) wikiPageWikiLink John_von_Neumann.
- Entropy_(information_theory) wikiPageWikiLink Joint_entropy.
- Entropy_(information_theory) wikiPageWikiLink Josiah_Willard_Gibbs.
- Entropy_(information_theory) wikiPageWikiLink Kolmogorov-Sinai_entropy.
- Entropy_(information_theory) wikiPageWikiLink Kolmogorov_complexity.
- Entropy_(information_theory) wikiPageWikiLink Kullback–Leibler_divergence.
- Entropy_(information_theory) wikiPageWikiLink LZW.
- Entropy_(information_theory) wikiPageWikiLink Landauers_principle.
- Entropy_(information_theory) wikiPageWikiLink Lebesgue_measure.
- Entropy_(information_theory) wikiPageWikiLink Lempel–Ziv–Welch.
- Entropy_(information_theory) wikiPageWikiLink Level_of_measurement.
- Entropy_(information_theory) wikiPageWikiLink Levenshtein_distance.
- Entropy_(information_theory) wikiPageWikiLink Limit_of_a_function.
- Entropy_(information_theory) wikiPageWikiLink Limiting_density_of_discrete_points.
- Entropy_(information_theory) wikiPageWikiLink Logarithm.
- Entropy_(information_theory) wikiPageWikiLink Loomis-Whitney_inequality.
- Entropy_(information_theory) wikiPageWikiLink Loomis–Whitney_inequality.
- Entropy_(information_theory) wikiPageWikiLink Lossless.
- Entropy_(information_theory) wikiPageWikiLink Lossless_compression.