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Principle of maximum entropy
The principle of maximum entropy states that the probability distribution which best represents the current state of knowledge about a system is the one with largest entropy, in the context of precisely stated prior data (such as a proposition that expresses testable information).
Another way of stating this: Take precisely stated prior data or testable information about a probability distribution function. Consider the set of all trial probability distributions that would encode the prior data. According to this principle, the distribution with maximal information entropy is the best choice.
The principle was first expounded by E. T. Jaynes in two papers in 1957, where he emphasized a natural correspondence between statistical mechanics and information theory. In particular, Jaynes argued that the Gibbsian method of statistical mechanics is sound by also arguing that the entropy of statistical mechanics and the information entropy of information theory are the same concept. Consequently, statistical mechanics should be considered a particular application of a general tool of logical inference and information theory.
In most practical cases, the stated prior data or testable information is given by a set of conserved quantities (average values of some moment functions), associated with the probability distribution in question. This is the way the maximum entropy principle is most often used in statistical thermodynamics. Another possibility is to prescribe some symmetries of the probability distribution. The equivalence between conserved quantities and corresponding symmetry groups implies a similar equivalence for these two ways of specifying the testable information in the maximum entropy method.
The maximum entropy principle is also needed to guarantee the uniqueness and consistency of probability assignments obtained by different methods, statistical mechanics and logical inference in particular.
The maximum entropy principle makes explicit our freedom in using different forms of prior data. As a special case, a uniform prior probability density (Laplace's principle of indifference, sometimes called the principle of insufficient reason), may be adopted. Thus, the maximum entropy principle is not merely an alternative way to view the usual methods of inference of classical statistics, but represents a significant conceptual generalization of those methods.
However these statements do not imply that thermodynamical systems need not be shown to be ergodic to justify treatment as a statistical ensemble.
In ordinary language, the principle of maximum entropy can be said to express a claim of epistemic modesty, or of maximum ignorance. The selected distribution is the one that makes the least claim to being informed beyond the stated prior data, that is to say the one that admits the most ignorance beyond the stated prior data.
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Principle of maximum entropy AI simulator
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Principle of maximum entropy
The principle of maximum entropy states that the probability distribution which best represents the current state of knowledge about a system is the one with largest entropy, in the context of precisely stated prior data (such as a proposition that expresses testable information).
Another way of stating this: Take precisely stated prior data or testable information about a probability distribution function. Consider the set of all trial probability distributions that would encode the prior data. According to this principle, the distribution with maximal information entropy is the best choice.
The principle was first expounded by E. T. Jaynes in two papers in 1957, where he emphasized a natural correspondence between statistical mechanics and information theory. In particular, Jaynes argued that the Gibbsian method of statistical mechanics is sound by also arguing that the entropy of statistical mechanics and the information entropy of information theory are the same concept. Consequently, statistical mechanics should be considered a particular application of a general tool of logical inference and information theory.
In most practical cases, the stated prior data or testable information is given by a set of conserved quantities (average values of some moment functions), associated with the probability distribution in question. This is the way the maximum entropy principle is most often used in statistical thermodynamics. Another possibility is to prescribe some symmetries of the probability distribution. The equivalence between conserved quantities and corresponding symmetry groups implies a similar equivalence for these two ways of specifying the testable information in the maximum entropy method.
The maximum entropy principle is also needed to guarantee the uniqueness and consistency of probability assignments obtained by different methods, statistical mechanics and logical inference in particular.
The maximum entropy principle makes explicit our freedom in using different forms of prior data. As a special case, a uniform prior probability density (Laplace's principle of indifference, sometimes called the principle of insufficient reason), may be adopted. Thus, the maximum entropy principle is not merely an alternative way to view the usual methods of inference of classical statistics, but represents a significant conceptual generalization of those methods.
However these statements do not imply that thermodynamical systems need not be shown to be ergodic to justify treatment as a statistical ensemble.
In ordinary language, the principle of maximum entropy can be said to express a claim of epistemic modesty, or of maximum ignorance. The selected distribution is the one that makes the least claim to being informed beyond the stated prior data, that is to say the one that admits the most ignorance beyond the stated prior data.