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Principle of Maximum Entropy: Simple Form

In the last chapter, we discussed one technique of estimating the input probabilities of a process given that the output event is known. This technique, which relies on the use of Bayes’ Theorem, only works if the process is lossless (in which case the input can be identified with certainty) or the a priori input probability distribution is assumed (in which case the technique refines the initial probability distribution to take account of the known output). The Principle of Maximum Entropy is a technique that can be used to estimate input probabilities more generally. The result is a probability distribution that is consistent with known constraints expressed in terms of averages, or expected values, of one or more quantities, but is otherwise as unbiased as possible. This principle is described first for the simple case of one constraint and three input events, in which case the technique can be carried out analytically. Then it is described more generally in Chapter 10. This principle has applications in many domains, but was originally motivated by statistical physics, which attempts to relate macroscopic, measurable properties of physical systems to a description at the atomic or molecular level. It can be used to approach physical systems from the point of view of information theory, because the probability distributions can be derived by avoiding the assumption that the observer has more information than is actually available. Information theory, particularly the definition of information in terms of probability distributions, provides a quantitative measure of ignorance (or uncertainty, or entropy) that can be maximized mathematically to find the probability distribution that is maximally unbiased. This approach to statistical physics was pioneered by Edwin T. Jaynes (1922 ­ 1998), a professor at Washington University in St. Louis, and previously Stanford University. The seminal publication was