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Information theory quantifies uncertainty and information using probabilistic models, with entropy as a key measure.
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 Shannon’s 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 channel’s 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.
Information theory quantifies uncertainty and information using probabilistic models, with entropy as a key measure.
Shannon’s communication framework models sending messages over noisy channels and characterizes achievable rates via channel capacity (noisy-channel coding theorem).
The field covers both fundamental information measures and practical coding methods for source coding (compression) and channel coding (error correction), with wide interdisciplinary applications.
A measure of the uncertainty of a random variable, representing the expected information content of outcomes.
The information associated with a specific event, defined as −log(p) for an event with probability p.
The entropy of a pair (or vector) of random variables, capturing uncertainty about their combined outcomes.
The average remaining uncertainty about one random variable given knowledge of another.
A measure of how much observing one random variable reduces uncertainty about another, quantifying shared information.
The maximum reliable communication rate over a noisy channel, determined by the channel’s statistics.
A result stating that, in the limit of many channel uses, the highest achievable information rate equals the channel capacity.
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