11/22/2023 0 Comments Shannon entropy![]() ![]() Shannon's main result, the noisy-channel coding theorem showed that, in the limit of many channel uses, the rate of information that is asymptotically achievable is equal to the channel capacity, a quantity dependent merely on the statistics of the channel over which the messages are sent. In the case of communication of information over a noisy channel, this abstract concept was formalized in 1948 by Claude Shannon in a paper entitled A Mathematical Theory of Communication, in which information is thought of as a set of possible messages, and the goal is to send these messages over a noisy channel, and to have the receiver reconstruct the message with low probability of error, in spite of the channel noise. Abstractly, information can be thought of as the resolution of uncertainty. Information theory studies the transmission, processing, extraction, and utilization of information. The theory has also found applications in other areas, including statistical inference, cryptography, neurobiology, perception, linguistics, the evolution and function of molecular codes ( bioinformatics), thermal physics, molecular dynamics, quantum computing, black holes, information retrieval, intelligence gathering, plagiarism detection, pattern recognition, anomaly detection and even art creation. Its impact has been crucial to the success of the Voyager missions to deep space, the invention of the compact disc, the feasibility of mobile phones and the development of the Internet. for ZIP files), and channel coding/ error detection and correction (e.g. ![]() Important sub-fields of information theory include source coding, algorithmic complexity theory, algorithmic information theory and information-theoretic security.Īpplications of fundamental topics of information theory include source coding/ data compression (e.g. Some other important measures in information theory are mutual information, channel capacity, error exponents, and relative entropy. For example, identifying the outcome of a fair coin flip (with two equally likely outcomes) provides less information (lower entropy, less uncertainty) than specifying the outcome from a roll of a die (with six equally likely outcomes). Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. : vii The field, in applied mathematics, is at the intersection of probability theory, statistics, computer science, statistical mechanics, information engineering, and electrical engineering.Ī key measure in information theory is entropy. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. p 2) = I( p 1) + I( p 2): the information learned from independent events is the sum of the information learned from each event.Information theory is the mathematical study of the quantification, storage, and communication of information.I(1) = 0: events that always occur do not communicate information.I( p) is monotonically decreasing in p: an increase in the probability of an event decreases the information from an observed event, and vice versa. ![]() The amount of information acquired due to the observation of event i follows from Shannon's solution of the fundamental properties of information: To understand the meaning of −Σ p i log( p i), first define an information function I in terms of an event i with probability p i. This ratio is called metric entropy and is a measure of the randomness of the information. : 14–15Įntropy can be normalized by dividing it by information length. The entropy is zero: each toss of the coin delivers no new information as the outcome of each coin toss is always certain. 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. Entropy, then, can only decrease from the value associated with uniform probability. ![]() Uniform probability yields maximum uncertainty and therefore maximum entropy. H ( X ) := − ∑ x ∈ X p ( x ) log p ( x ) = E, ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |