![]() Thus, in information theory, information is a function, $I$, called self-information, that operates on probability values $p \in $:\ We discussed how surprise intuitively should correspond to probability in that an event with low probability elicits more surprise because it is unlikely to occur. In the first post of this series, we discussed how Shannon’s Information Theory defines the information content of an event as the degree of surprise that an agent experiences when the event occurs. In this second post, we will build on this foundation to discuss the concept of information entropy. In the first post, we discussed the concept of self-information. In this series of posts, I will attempt to describe my understanding of how, both philosophically and mathematically, information theory defines the polymorphic, and often amorphous, concept of information. ![]() The mathematical field of information theory attempts to mathematically describe the concept of “information”. 4 (3rd ed.), (Academic Press, New York), p.Information entropy (Foundations of information theory: Part 2) ![]() Meyers (ed.), Encyclopedia of Physical Science & Technology, Vol. (2001): " Cybernetics and Second Order Cybernetics", in: R.A. While GIT methods are under development, the probabilistic approach to information theory still dominates applications. Together with probability theory these are called Generalized Information Theory (GIT). These methods, involving concepts from fuzzy systems theory and possibility theory, lead to alternative information theories. We also note that there are other methods of weighting the state of a system which do not adhere to probability theory's additivity condition that the sum of the probabilities must be 1. Entropies, correlates to entropies, and correlates to such important results as Shannon's 10th Theorem and the Second Law of Thermodynamics have been sought in biology, ecology, psychology, sociology, and economics. H has been vigorously pursued as a measure for a number of higher-order relational concepts, including complexity and organization. If the observation completely determines the state of the system (H(after) = 0), then information I reduces to the initial entropy or uncertainty H.Īlthough Shannon came to disavow the use of the term "information" to describe this measure, because it is purely syntactic and ignores the meaning of the signal, his theory came to be known as Information Theory nonetheless. ![]() The information I we receive from an observation is equal to the degree to which uncertainty is reduced: through observation), then this will reduce our uncertainty about the system's state, by excluding-or reducing the probability of-a number of states. Indeed, if we get some information about the state of the system (e.g. This difference can also be interpreted in a different way, as information, and historically H was introduced by Shannon as a measure of the capacity for information transmission of a communication channel. We define constraint as that which reduces uncertainty, that is, the difference between maximal and actual uncertainty. In that case we have maximal certainty or complete information about what state the system is in. It is clear that H = 0, if and only if the probability of a certain state is 1 (and of all other states 0). Like variety, H expresses our uncertainty or ignorance about the system's state. Thus it is natural that in this case entropy H reduces to variety V. H reaches its maximum value if all states are equiprobable, that is, if we have no indication whatsoever to assume that one state is more probable than another state. Variety V can then be expressed as entropy H (as originally defined by Boltzmann for statistical mechanics): Assume that we do not know the precise state s of a system, but only the probability distribution P(s) that the system would be in state s. Variety and constraint, the basic concepts of cybernetics, can be measured in a more general form by introducing probabilities. Statistical entropy is a probabilistic measure of uncertainty or ignorance information is a measure of a reduction in that uncertaintyĮntropy (or uncertainty) and its complement, information, are perhaps the most fundamental quantitive measures in cybernetics, extending the more qualitative concepts of variety and constraint to the probabilistic domain. ![]()
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