"The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s."
It studies the quantification, transmission, and processing of information, including topics such as data compression, error correction, and encryption.
Basics of probability: Probability is the foundation of Information Theory. One must have a clear understanding of basic probability concepts such as random variables, probability distributions, conditional probabilities, joint probabilities, etc.
Entropy: Entropy is a measure of uncertainty or unpredictability in a system. It is a central concept in information theory, and understanding it is key to understanding the field as a whole.
Information: Information theory deals with measuring the amount of information in a message, where information is defined as the amount of uncertainty that is removed by receiving the message.
Coding Theory: Coding theory deals with the design of efficient codes for transmitting data over noisy channels. Coding theory is a vital part of information theory because it provides the methods for efficient transmission of information.
Compression: Compression is the process of reducing the amount of data required to represent a message while preserving the original message's meaning. Compression is an essential part of information theory because it allows us to send more data over a limited channel.
Cryptography: Cryptography is the science of secure communication. It involves techniques for hiding the content of a message from outsiders, ensuring the integrity and confidentiality of the message.
Shannon's Information Theory: Shannon's information theory is the foundation of modern information theory. It includes concepts such as entropy, information, noise, channel capacity, and coding theory.
Channel Capacity: Channel capacity is the maximum rate at which information can be transmitted over a communication channel with a given amount of noise. Understanding channel capacity is critical for designing efficient communication systems.
Mutual Information: Mutual information is the amount of information that two random variables share. It is a crucial concept in information theory, especially in understanding the joint information of two sources.
Error-Correcting Codes: Error-correcting codes are codes that can detect and correct errors that occur during transmission. They are used to improve the reliability of communication systems.
Source Coding: Source coding is the process of encoding data to reduce redundancy and compression. It is used to reduce the amount of data needed to represent a message.
Channel Coding: Channel coding is the process of adding redundancy to transmitted data to allow it to be recovered in the presence of noise. It is used to improve the reliability of communication systems.
Rate-Distortion Theory: Rate-distortion theory deals with the problem of representing data at a given rate while minimizing the distortion of the signal. It is used in image and video compression.
Kolmogorov Complexity: Kolmogorov complexity is the measure of the amount of information in a sequence that cannot be compressed. It is used to study the complexity of algorithms and the limits of computation.
Information Theory in Biology: Information theory has applications in many branches of science, including biology. It is used to study the coding of genetic information and the processing of sensory information in animals.
Shannon Information Theory: Developed by Claude Shannon, this theory studies the fundamental limits of information transmission and storage in communication systems. It is mainly concerned with measuring the amount of information contained in a message or signal.
Kolmogorov Complexity Theory: This theory studies the amount of information necessary to describe an object or a sequence of symbols. It proposes that the complexity of an object can be measured by the length of the shortest possible description of it.
Algorithmic Information Theory: This theory studies the amount of information contained in an object or sequence of symbols, as well as the minimum amount of information necessary to describe it. It is concerned with the concept of algorithmic randomness and complexity.
Computational Complexity Theory: This theory studies the complexity of algorithms and problems in terms of time, space, and other resources required for their computation. It is concerned with the classification of computational problems into classes based on the resources they require.
Coding Theory: This theory studies how to efficiently encode information for transmission or storage, and how to decode it when it is received or retrieved. It involves the design of error-correcting codes for reliable communication and storage of information.
Information Retrieval Theory: This theory studies the retrieval of information from large collections of data, such as text, images, and multimedia. It is concerned with techniques for indexing, searching, and retrieving relevant information from such collections.
Information Visualization Theory: This theory studies the design and evaluation of visual representations of information for effective communication and understanding of complex data. It is concerned with the development of techniques for visualizing data in a meaningful and informative way.
Quantum Information Theory: This theory studies the transmission, storage, and processing of information using quantum systems. It is concerned with the development of quantum algorithms and protocols for secure communication and computation.
Game Theory: This theory studies the strategic interactions between individuals or groups in situations involving conflicting interests. It is concerned with the analysis of decision-making strategies and the prediction of outcomes in such situations.
Information Ethics: This theory studies the ethical and moral implications of the collection, storage, processing, and dissemination of information in society. It is concerned with issues such as privacy, freedom of expression, intellectual property, and social justice.
"The field, in applied mathematics, is at the intersection of probability theory, statistics, computer science, statistical mechanics, information engineering, and electrical engineering."
"Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process."
"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)."
"Some other important measures in information theory are mutual information, channel capacity, error exponents, and relative entropy."
"Important sub-fields of information theory include source coding, algorithmic complexity theory, algorithmic information theory, and information-theoretic security."
"Applications of fundamental topics of information theory include source coding/data compression (e.g. for ZIP files), and channel coding/error detection and correction (e.g. for DSL)."
"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."
"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."