Differential entropy (also referred to as continuous entropy) is a concept in information theory that began as an attempt by Claude Shannon to extend the idea of (Shannon) entropy, a measure of average (surprisal) of a random variable, to continuous probability distributions. Unfortunately, Shannon did not derive this formula, and rather just assumed it was the correct continuous analogue of discrete entropy, but it is not.[1]: 181–218 The actual continuous version of discrete entropy is the limiting density of discrete points (LDDP). Differential entropy (described here) is commonly encountered in the literature, but it is a limiting case of the LDDP, and one that loses its fundamental association with discrete entropy.
In terms of measure theory, the differential entropy of a probability measure is the negative relative entropy from that measure to the Lebesgue measure, where the latter is treated as if it were a probability measure, despite being unnormalized.
^Jaynes, E.T. (1963). "Information Theory And Statistical Mechanics" (PDF). Brandeis University Summer Institute Lectures in Theoretical Physics. 3 (sect. 4b).
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Differentialentropy (also referred to as continuous entropy) is a concept in information theory that began as an attempt by Claude Shannon to extend the...
In information theory, the conditional entropy quantifies the amount of information needed to describe the outcome of a random variable Y {\displaystyle...
in which case we say that the differentialentropy is not defined. As in the discrete case the joint differentialentropy of a set of random variables...
the discrete entropy. It is known since then that the differentialentropy may differ from the infinitesimal limit of the discrete entropy by an infinite...
_{x})-S(p_{x})\,} where S ( φ x ) {\displaystyle S(\varphi _{x})} is the differentialentropy of the Gaussian density with the same mean and variance as p x {\displaystyle...
properties; for example, differentialentropy may be negative. The differential analogies of entropy, joint entropy, conditional entropy, and mutual information...
d-dimensional entropy does not necessarily exist there. Finally, dimensional-rate bias generalizes the Shannon's entropy and differentialentropy, as one could...
Shannon for differentialentropy. It was formulated by Edwin Thompson Jaynes to address defects in the initial definition of differentialentropy. Shannon...
vector, it is distributed as a generalized chi-squared variable. The differentialentropy of the multivariate normal distribution is h ( f ) = − ∫ − ∞ ∞ ∫...
information theoretic quantities include Rényi entropy (a generalization of entropy), differentialentropy (a generalization of quantities of information...
V, Gilson MK (March 2010). "Thermodynamic and DifferentialEntropy under a Change of Variables". Entropy. 12 (3): 578–590. Bibcode:2010Entrp..12..578H...
{\displaystyle \operatorname {erf} (z)} is the error function. The differentialentropy is given by[citation needed] H = 1 + ln ( σ 2 ) + γ 2 {\displaystyle...
Clausius named the concept of "the differential of a quantity which depends on the configuration of the system", entropy (Entropie) after the Greek word...
entropy states that the probability distribution which best represents the current state of knowledge about a system is the one with largest entropy,...
mathematical theory of probability, the entropy rate or source information rate is a function assigning an entropy to a stochastic process. For a strongly...
learning, and time delay estimation it is useful to estimate the differentialentropy of a system or process, given some observations. The simplest and...
variable. The concept of mutual information is intimately linked to that of entropy of a random variable, a fundamental notion in information theory that quantifies...
I ( X ; Y ) {\displaystyle I(X;Y)} , writing it in terms of the differentialentropy: I ( X ; Y ) = h ( Y ) − h ( Y ∣ X ) = h ( Y ) − h ( X + Z ∣ X )...
distribution with λ = 1/μ has the largest differentialentropy. In other words, it is the maximum entropy probability distribution for a random variate...
\operatorname {Dir} ({\boldsymbol {\alpha }})} random variable, the differentialentropy of X (in nat units) is h ( X ) = E [ − ln f ( X ) ] = ln B...
Entropy is one of the few quantities in the physical sciences that require a particular direction for time, sometimes called an arrow of time. As one...
{\displaystyle H} is simply the entropy of a symbol) and the continuous-valued case (where H {\displaystyle H} is the differentialentropy instead). The definition...
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