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The Shannon entropy is restricted to random variables taking discrete values. The corresponding formula for a continuous random variable with probability density function f(x) with finite or infinite support on the real line is defined by analogy, using the above form of the entropy as an expectation: [11]: 224
If is a continuous random variable with probability density (), then the differential entropy of is defined as [1] [2] [3] = .If is a discrete random variable with distribution given by
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 ...
One must take care in trying to apply properties of discrete entropy to differential entropy, since probability density functions can be greater than 1. For example, the uniform distribution U ( 0 , 1 / 2 ) {\displaystyle {\mathcal {U}}(0,1/2)} has negative differential entropy; i.e., it is better ordered than U ( 0 , 1 ) {\displaystyle ...
where is the probability of the message taken from the message space M, and b is the base of the logarithm used. Common values of b are 2, Euler's number e, and 10, and the unit of entropy is shannon (or bit) for b = 2, nat for b = e, and hartley for b = 10. [1]
It can be shown that if a system is described by a probability density in phase space, then Liouville's theorem implies that the joint information (negative of the joint entropy) of the distribution remains constant in time. The joint information is equal to the mutual information plus the sum of all the marginal information (negative of the ...
4.3 Extremal principle of entropy to fix the free parameter ... Its probability density function at the neighborhood of 0 has been characterized [35] ...
In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the ...