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{{Short description|Concept in information theory}}

{{Short description|Concept in information theory}}

Differential entropy (also referred to as continuous entropy) is a concept in information theory that began as an attempt by 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.

Differential entropy (also referred to as continuous entropy) is a concept in information theory that began as an attempt by 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.

微分熵(也称为连续熵)是信息论中的一个概念，最初由香农尝试将(香农)熵的概念扩展到连续的概率分布，香农熵是衡量一个随机变量的平均惊人程度的指标。不幸的是，香农没有推导出这个公式，而只是假设它是离散熵的正确连续模拟，但它不是。

微分熵(也称为连续熵)是信息论中的一个概念，最初由香农尝试将(香农)熵的概念扩展到连续的概率分布，香农熵是衡量一个随机变量的平均惊人程度的指标。不幸的是，香农没有推导出这个公式，而只是假设它是离散熵的正确连续模拟，但事实上它不是。

==Definition==

==Definition==

 

Let <math>X</math> be a random variable with a [[probability density function]] <math>f</math> whose [[support (mathematics)|support]] is a set <math>\mathcal X</math>. The ''differential entropy'' <math>h(X)</math> or <math>h(f)</math> is defined as<ref name="cover_thomas">{{cite book|first1=Thomas M.|first2=Joy A.|last1=Cover|last2=Thomas|isbn=0-471-06259-6|title=Elements of Information Theory|year=1991|publisher=Wiley|location=New York|url=https://archive.org/details/elementsofinform0000cove|url-access=registration}}</ref>{{rp|243}}

Let <math>X</math> be a random variable with a [[probability density function]] <math>f</math> whose [[support (mathematics)|support]] is a set <math>\mathcal X</math>. The ''differential entropy'' <math>h(X)</math> or <math>h(f)</math> is defined as<ref name="cover_thomas">{{cite book|first1=Thomas M.|first2=Joy A.|last1=Cover|last2=Thomas|isbn=0-471-06259-6|title=Elements of Information Theory|year=1991|publisher=Wiley|location=New York|url=https://archive.org/details/elementsofinform0000cove|url-access=registration}}</ref>{{rp|243}}