Information Theory Fundamentals and Applications

Information theory foundations

Information theory is the mathematical study of how information is quantified, stored, compressed, transmitted, and recovered in the presence of uncertainty and noise. Claude Shannon established the field formally in the 1940s, building on earlier work by Harry Nyquist and Ralph Hartley. The central idea is that information can be understood as the reduction of uncertainty: an unlikely event conveys more information than a likely one. Information is commonly measured in bits using base-2 logarithms; for example, learning the result of a fair coin toss provides one bit of information. Entropy is the foundational measure of uncertainty in a random variable and represents the average amount of information produced by its possible outcomes. Related concepts include joint entropy, conditional entropy, mutual information, and Kullback–Leibler divergence, which describe combined uncertainty, remaining uncertainty after observation, shared information, and the difference between probability distributions. Shannon's source-coding and noisy-channel coding theorems establish fundamental limits for data compression and reliable communication, while channel capacity specifies the maximum achievable information rate over a noisy channel. These foundations support applications such as compression, error correction, cryptography, statistics, signal processing, artificial intelligence, and communications engineering.

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Probability and statistical information

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