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Information theory is the mathematical study of quantifying, storing, and communicating a mathematically defined type of information. Conceived by Claude Shannon in the 1940s, it treats information as a way to measure uncertainty: before a message is known, uncertainty is high, and after the message is observed, uncertainty decreases. A central example is a fair coin flip, where the initial uncertainty corresponds to 1 bit of information, and becomes 0 bits once the outcome is revealed. In Shannonās probabilistic framework, information is modeled as a set of possible messages sent over a (possibly noisy) communication channel, with the goal of enabling reliable reconstruction at the receiver. This leads to foundational results such as the noisy-channel coding theorem, which links the maximum achievable information rate to the channel capacity determined by the channelās statistical behavior. The scope of information theory therefore includes core measures of information (like entropy, mutual information, and related quantities), and the design of coding and communication methods for efficient data compression and error correction, with broad applications across mathematics, statistics, computer science, and engineering as well as fields such as cryptography, signal processing, and biology.
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In information theory, uncertainty represents the lack of knowledge about the outcome of a random variable. Information is understood as the reduction or resolution of this uncertainty when an outcome becomes known. For example, before a fair coin is flipped, the result is uncertain and has an entropy of 1 bit; after the result is observed, the uncertainty and remaining entropy are zero. Shannon entropy provides a mathematical measure of uncertainty based on the probabilities of possible outcomes. It is calculated as the expected value of self-information, with less probable events carrying more information when they occur. Entropy is highest when all possible outcomes are equally likely and lower when one outcome is more predictable. This model supports applications such as data compression, reliable communication, statistical inference, and error correction.
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