Informationtheoretic

Information-theoretic ideas arise from information theory, a mathematical framework for quantifying information, its transmission, processing, and storage. Developed largely by Claude Shannon in the 1940s, information theory seeks the fundamental limits governing data compression and reliable communication, independent of particular devices or protocols. The field introduces measures such as entropy, which quantifies average uncertainty in a source; mutual information, which captures the amount of information shared between variables; and divergences that compare probability distributions.

Core results include the source coding theorem, which states that a stationary source can be losslessly compressed to its entropy rate, and the channel coding theorem, which identifies the maximum reliable communication rate, or channel capacity, of a given communication channel. Rate-distortion theory generalizes compression by trading rate for acceptable distortion. Practical coding schemes—error-correcting codes and source coders—aim to approach these limits in real systems. Information theory also informs security through concepts of secrecy and one-time pad perfect secrecy as an information-theoretic benchmark.

Applications span telecommunications, data compression, multimedia transmission, and cryptography. In statistics and machine learning, information-theoretic quantities such as mutual information are used for feature selection and measuring dependencies. The theory emphasizes limits and idealized models; real-world systems face finite block lengths, modeling errors, and practical constraints that can prevent achieving capacity or entropy bounds. Critics note that information theory treats semantic content as orthogonal to information content, focusing on syntactic information instead.