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Many of the concepts in information theory have separate definitions and formulas for continuous and discrete cases. For example, entropy is usually defined for discrete random variables, whereas for continuous random variables the related concept of differential entropy, written (), is used (see Cover and Thomas, 2006, chapter 8).
One early commercial application of information theory was in the field of seismic oil exploration. Work in this field made it possible to strip off and separate the unwanted noise from the desired seismic signal. Information theory and digital signal processing offer a major improvement of resolution and image clarity over previous analog methods.
Information flow (information theory) Information fluctuation complexity; Information–action ratio; Information projection; Information source (mathematics) Information theory and measure theory; Integrated information theory; Interaction information; Interactions of actors theory; Interference channel
Given a discrete-time stationary ergodic stochastic process on the probability space (,,), the asymptotic equipartition property is an assertion that, almost surely, (,, …,) where () or simply denotes the entropy rate of , which must exist for all discrete-time stationary processes including the ergodic ones.
Capacity of the two-way channel: The capacity of the two-way channel (a channel in which information is sent in both directions simultaneously) is unknown. [ 5 ] [ 6 ] Capacity of Aloha : The ALOHAnet used a very simple access scheme for which the capacity is still unknown, though it is known in a few special cases.
In the view of Jaynes (1957), [20] thermodynamic entropy, as explained by statistical mechanics, should be seen as an application of Shannon's information theory: the thermodynamic entropy is interpreted as being proportional to the amount of further Shannon information needed to define the detailed microscopic state of the system, that remains ...
During his 48-year career as a professor of Electrical Engineering and Statistics at Stanford University, he graduated 64 PhD students, authored over 120 journal papers in learning, information theory, statistical complexity, pattern recognition, and portfolio theory; and he partnered with Joy A. Thomas to coauthor the book Elements of ...
For a given probability space, the measurement of rarer events are intuitively more "surprising", and yield more information content, than more common values. Thus, self-information is a strictly decreasing monotonic function of the probability, or sometimes called an "antitonic" function.