# 微分熵

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.[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.

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.

$\displaystyle{ h(X_1, \ldots, X_n) = \sum_{i=1}^{n} h(X_i|X_1, \ldots, X_{i-1}) \leq \sum_{i=1}^{n} h(X_i) }$.

## Definition定义

Let $\displaystyle{ X }$ be a random variable with a probability density function $\displaystyle{ f }$ whose support is a set $\displaystyle{ \mathcal X }$. The differential entropy $\displaystyle{ h(X) }$ or $\displaystyle{ h(f) }$ is defined as[2] $\displaystyle{ h(X+c) = h(X) }$

 --CecileLi(讨论)  【审校】此处缺无格式的英文及翻译 补充：设随机变量X，其概率密度函数F的的定义域是X的集合

$\displaystyle{ h(X) = -\int_\mathcal{X} f(x)\log f(x)\,dx }$

For probability distributions which don't have an explicit density function expression, but have an explicit quantile function expression, $\displaystyle{ Q(p) }$, then $\displaystyle{ h(Q) }$ can be defined in terms of the derivative of $\displaystyle{ Q(p) }$ i.e. the quantile density function $\displaystyle{ Q'(p) }$ as [3]:54–59

--CecileLi(讨论)  【审校】此处缺无格式的英文及翻译 补充：For probability distributions which don't have an explicit density function expression, but have an explicit quantile function expression, , then  can be defined in terms of the derivative of  i.e. the quantile density function as


$\displaystyle{ h(Q) = \int_0^1 \log Q'(p)\,dp }$.

A modification of differential entropy that addresses these drawbacks is the relative information entropy, also known as the Kullback–Leibler divergence, which includes an invariant measure factor (see limiting density of discrete points).

As with its discrete analog, the units of differential entropy depend on the base of the logarithm, which is usually 2 (i.e., the units are bits). See logarithmic units for logarithms taken in different bases. Related concepts such as joint, conditional differential entropy, and relative entropy are defined in a similar fashion. Unlike the discrete analog, the differential entropy has an offset that depends on the units used to measure $\displaystyle{ X }$.[4]:183–184 For example, the differential entropy of a quantity measured in millimeters will be 模板:Not a typo more than the same quantity measured in meters; a dimensionless quantity will have differential entropy of 模板:Not a typo more than the same quantity divided by 1000.

--CecileLi(讨论) 【审校】补充翻译：与离散模型一样，微分熵的单位取决于对数的底数，通常是2（单位：比特；请参阅对数单位，了解不同基数的对数。）相对熵的定义与联合熵、条件差分熵等概念相对熵的概念存在类似之处。与离散模型不同，差分熵的偏移量取决于测量单位。例如，以毫米为单位测量的量的差分熵将大于以米为单位测量的相同量；无量纲量的差分熵将大于相同量除以1000。

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 $\displaystyle{ \mathcal{U}(0,1/2) }$ has negative differential entropy

--CecileLi(讨论) 【审校】补充翻译：在尝试将离散熵的性质应用于微分熵时必须小心，因为概率密度函数可以大于1。例如，均匀分布具有“负”微分熵

$\displaystyle{ \int_0^\frac{1}{2} -2\log(2)\,dx=-\log(2)\, }$.

Thus, differential entropy does not share all properties of discrete entropy. --CecileLi(讨论) 【审校】补充翻译：因此，微分熵并不具有离散熵的所有性质。

Note that the continuous mutual information $\displaystyle{ I(X;Y) }$ has the distinction of retaining its fundamental significance as a measure of discrete information since it is actually the limit of the discrete mutual information of partitions of $\displaystyle{ X }$ and $\displaystyle{ Y }$ as these partitions become finer and finer. Thus it is invariant under non-linear homeomorphisms (continuous and uniquely invertible maps), [5] including linear [6] transformations of $\displaystyle{ X }$ and $\displaystyle{ Y }$, and still represents the amount of discrete information that can be transmitted over a channel that admits a continuous space of values.

--CecileLi(讨论) 【审校】补充翻译：注意，连续相互变量I（X；Y）具有保留其作为离散信息度量的基本意义的区别，因为它实际上是X和Y的“分区”的离散互信息的极限，因为这些分区变得越来越细。因此，它在非线性同胚（连续且唯一可逆的映射）下是不变的，并且仍然表示可在允许连续值空间的信道上传输的离散信息量。

For the direct analogue of discrete entropy extended to the continuous space, see limiting density of discrete points.

--CecileLi(讨论) 【审校】补充翻译：对于扩展到连续空间的离散熵的直接模拟，参见离散点的极限密度

## Properties of differential entropy

• For probability densities $\displaystyle{ f }$ and $\displaystyle{ g }$, the Kullback–Leibler divergence $\displaystyle{ D_{KL}(f || g) }$ is greater than or equal to 0 with equality only if $\displaystyle{ f=g }$ almost everywhere. Similarly, for two random variables $\displaystyle{ X }$ and $\displaystyle{ Y }$, $\displaystyle{ I(X;Y) \ge 0 }$ and $\displaystyle{ h(X|Y) \le h(X) }$ with equality if and only if $\displaystyle{ X }$ and $\displaystyle{ Y }$ are independent.

--CecileLi(讨论) 【审校】补充翻译：*对于概率密度f和g，Kullback–Leibler散度D{KL}（f | | g）[/itex]只有在f=g几乎处处时才大于或等于0且相等。类似地，对于两个随机变量X和Y，I（X；Y）\ge 和h（X | Y）\le h（X），等式：当且仅当>X和Y是独立性

• The chain rule for differential entropy holds as in the discrete case[2]:253

--CecileLi(讨论) 【审校】补充翻译：微分熵的链式法则在离散情况下成立

$\displaystyle{ h(X_1, \ldots, X_n) = \sum_{i=1}^{n} h(X_i|X_1, \ldots, X_{i-1}) \leq \sum_{i=1}^{n} h(X_i) }$.
• Differential entropy is translation invariant, i.e. for a constant $\displaystyle{ c }$.[2]:253

--CecileLi(讨论) 【审校】补充翻译:微分熵是平移不变的，即对于常数c存在

$\displaystyle{ h(X+c) = h(X) }$
• Differential entropy is in general not invariant under arbitrary invertible maps.

--CecileLi(讨论) 【审校】补充翻译：在任意可逆映射下，微分熵一般是不不变的。

In particular, for a constant $\displaystyle{ a }$

--CecileLi(讨论) 【审校】补充翻译：特别地，对于一个常数a存在

$\displaystyle{ h(aX) = h(X)+ \log |a| }$
For a vector valued random variable $\displaystyle{ \mathbf{X} }$ and an invertible (square) matrix $\displaystyle{ \mathbf{A} }$

--CecileLi(讨论) 【审校】补充翻译：对于向量值随机变量X和可逆（平方）矩阵存在

$\displaystyle{ h(\mathbf{A}\mathbf{X})=h(\mathbf{X})+\log \left( |\det \mathbf{A}| \right) }$[2]:253
• In general, for a transformation from a random vector to another random vector with same dimension $\displaystyle{ \mathbf{Y}=m \left(\mathbf{X}\right) }$, the corresponding entropies are related via

--CecileLi(讨论) 【审校】补充翻译：一般地，对于从一个随机向量到另一个具有相同维数（X,Y）的随机向量的变换，相应的熵通过

$\displaystyle{ h(\mathbf{Y}) \leq h(\mathbf{X}) + \int f(x) \log \left\vert \frac{\partial m}{\partial x} \right\vert dx }$
where $\displaystyle{ \left\vert \frac{\partial m}{\partial x} \right\vert }$ is the Jacobian of the transformation $\displaystyle{ m }$.[7] The above inequality becomes an equality if the transform is a bijection. Furthermore, when $\displaystyle{ m }$ is a rigid rotation, translation, or combination thereof, the Jacobian determinant is always 1, and $\displaystyle{ h(Y)=h(X) }$.

--CecileLi(讨论) 【审校】补充翻译：其中(m,x)是变换m的 Jacobian。如果变换是双射，则上述不等式变为等式。此外，当m是刚性旋转、平移或其组合时，雅可比行列式总是1，并且h（Y）=h（X）

• If a random vector $\displaystyle{ X \in \mathbb{R}^n }$ has mean zero and covariance matrix $\displaystyle{ K }$, $\displaystyle{ h(\mathbf{X}) \leq \frac{1}{2} \log(\det{2 \pi e K}) = \frac{1}{2} \log[(2\pi e)^n \det{K}] }$ with equality if and only if $\displaystyle{ X }$ is jointly gaussian (see below).[2]:254

--CecileLi(讨论) 【审校】补充翻译：如果一个随机向量X具有均值零和协方差矩阵$\displaystyle{ K }$$\displaystyle{ h（\mathbf{X}）\leq\frac{1}{2}\log（\det{2\pi e K}）=\frac{1}{2}\log[（2\pi e）^n\det{K}] }$等式当且仅当X为多元正态分布/联合正态性/联合高斯（见下文[[#正态分布中的最大化]）。

• It is not invariant under change of variables, and is therefore most useful with dimensionless variables.

• It can be negative.

A modification of differential entropy that addresses these drawbacks is the relative information entropy, also known as the Kullback–Leibler divergence, which includes an invariant measure factor (see limiting density of discrete points).

--CecileLi(讨论) 【审校】补充翻译： 解决这些缺点的微分熵的一种改进是“相对信息熵”，也称为Kullback–Leibler散度，它包括一个“不变测度”因子（参见：离散点的极限密度）。

## Maximization in the normal distribution

### Proof

Let $\displaystyle{ g(x) }$ be a Gaussian PDF with mean μ and variance $\displaystyle{ \sigma^2 }$ and $\displaystyle{ f(x) }$ an arbitrary PDF with the same variance. Since differential entropy is translation invariant we can assume that $\displaystyle{ f(x) }$ has the same mean of $\displaystyle{ \mu }$ as $\displaystyle{ g(x) }$.

--CecileLi(讨论) 【审校】补充翻译：设$\displaystyle{ g（x） }$是一个高斯 PDF，具有均值μ和方差$\displaystyle{ \sigma^2 }$$\displaystyle{ f（x） }$具有相同方差的任意 PDF。由于微分熵是平移不变性的，我们可以假设$\displaystyle{ f（x） }$$\displaystyle{ g（x） }$具有相同的$\displaystyle{ \mu }$平均值。

Consider the Kullback–Leibler divergence between the two distributions

$\displaystyle{ 0 \leq D_{KL}(f || g) = \int_{-\infty}^\infty f(x) \log \left( \frac{f(x)}{g(x)} \right) dx = -h(f) - \int_{-\infty}^\infty f(x)\log(g(x)) dx. }$

Now note that

\displaystyle{ \begin{align} \int_{-\infty}^\infty f(x)\log(g(x)) dx &= \int_{-\infty}^\infty f(x)\log\left( \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\right) dx \\ &= \int_{-\infty}^\infty f(x) \log\frac{1}{\sqrt{2\pi\sigma^2}} dx + \log(e)\int_{-\infty}^\infty f(x)\left( -\frac{(x-\mu)^2}{2\sigma^2}\right) dx \\ &= -\tfrac{1}{2}\log(2\pi\sigma^2) - \log(e)\frac{\sigma^2}{2\sigma^2} \\ &= -\tfrac{1}{2}\left(\log(2\pi\sigma^2) + \log(e)\right) \\ &= -\tfrac{1}{2}\log(2\pi e \sigma^2) \\ &= -h(g) \end{align} }

--CecileLi(讨论) 【审校】补充翻译：考虑两个分布之间的Kullback–Leibler散度

$\displaystyle{ 0\leq D{KL}（f{g）=\int{-\infty}^\infty f（x）\log\left（\frac{f（x）}{g（x）}\right）dx=-h（f）-\int{-\infty}^\infty f（x）\log（g（x））dx。 }$

\displaystyle{ \begin{align} \int_{-\infty}^\infty f(x)\log(g(x)) dx &= \int_{-\infty}^\infty f(x)\log\left( \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\right) dx \\ &= \int_{-\infty}^\infty f(x) \log\frac{1}{\sqrt{2\pi\sigma^2}} dx + \log(e)\int_{-\infty}^\infty f(x)\left( -\frac{(x-\mu)^2}{2\sigma^2}\right) dx \\ &= -\tfrac{1}{2}\log(2\pi\sigma^2) - \log(e)\frac{\sigma^2}{2\sigma^2} \\ &= -\tfrac{1}{2}\left(\log(2\pi\sigma^2) + \log(e)\right) \\ &= -\tfrac{1}{2}\log(2\pi e \sigma^2) \\ &= -h(g) \end{align} }

because the result does not depend on $\displaystyle{ f(x) }$ other than through the variance. Combining the two results yields

--CecileLi(讨论) 【审校】补充翻译：因为结果不依赖于$\displaystyle{ f（x） }$而不是通过方差。将这两个结果结合起来就得到了

$\displaystyle{ h(g) - h(f) \geq 0 \! }$

with equality when $\displaystyle{ f(x)=g(x) }$ following from the properties of Kullback–Leibler divergence.

--CecileLi(讨论) 【审校】补充翻译：当f（x）=g（x）[/itex]遵循Kullback-Leibler散度的性质时相等。

### Alternative proof

This result may also be demonstrated using the variational calculus. A Lagrangian function with two Lagrangian multipliers may be defined as:

$\displaystyle{ L=\int_{-\infty}^\infty g(x)\ln(g(x))\,dx-\lambda_0\left(1-\int_{-\infty}^\infty g(x)\,dx\right)-\lambda\left(\sigma^2-\int_{-\infty}^\infty g(x)(x-\mu)^2\,dx\right) }$

where g(x) is some function with mean μ. When the entropy of g(x) is at a maximum and the constraint equations, which consist of the normalization condition $\displaystyle{ \left(1=\int_{-\infty}^\infty g(x)\,dx\right) }$ and the requirement of fixed variance $\displaystyle{ \left(\sigma^2=\int_{-\infty}^\infty g(x)(x-\mu)^2\,dx\right) }$, are both satisfied, then a small variation δg(x) about g(x) will produce a variation δL about L which is equal to zero:

$\displaystyle{ 0=\delta L=\int_{-\infty}^\infty \delta g(x)\left (\ln(g(x))+1+\lambda_0+\lambda(x-\mu)^2\right )\,dx }$

Since this must hold for any small δg(x), the term in brackets must be zero, and solving for g(x) yields:

$\displaystyle{ g(x)=e^{-\lambda_0-1-\lambda(x-\mu)^2} }$

Using the constraint equations to solve for λ0 and λ yields the normal distribution:

$\displaystyle{ g(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} }$

--CecileLi(讨论) 【审校】补充翻译： 这个结果也可以用变分演算来证明。具有两个拉格朗日乘子的拉格朗日函数可定义为：

$\displaystyle{ L=\int{-\infty}^\infty g（x）\ln（g（x））\，dx-\lambda\u 0\左（1-\int{-\infty}^\infty g（x）\，dx\右）-\lambda\左（\sigma^2-\int{-\infty}^\infty g（x）（x-\mu）^2\，dx\右） }$

$\displaystyle{ 0=\delta L=\int{-\infty}^\infty\delta g（x）\left（\ln（g（x））+1+\lambda\u 0+\lambda（x-\mu）^2\ right）\，dx }$

$\displaystyle{ g（x）=e^{-\lambda\u 0-1-\lambda（x-\mu）^2} }$

$\displaystyle{ g（x）=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{（x-\mu）^2}{2\sigma^2}} }$

## Example: Exponential distribution

$\displaystyle{ f(x) = \lambda e^{-\lambda x} \mbox{ for } x \geq 0. }$

Its differential entropy is then

 $\displaystyle{ h_e(X)\, }$ $\displaystyle{ =-\int_0^\infty \lambda e^{-\lambda x} \log (\lambda e^{-\lambda x})\,dx }$ $\displaystyle{ = -\left(\int_0^\infty (\log \lambda)\lambda e^{-\lambda x}\,dx + \int_0^\infty (-\lambda x) \lambda e^{-\lambda x}\,dx\right) }$ $\displaystyle{ = -\log \lambda \int_0^\infty f(x)\,dx + \lambda E[X] }$ $\displaystyle{ = -\log\lambda + 1\,. }$

Here, $\displaystyle{ h_e(X) }$ was used rather than $\displaystyle{ h(X) }$ to make it explicit that the logarithm was taken to base e, to simplify the calculation.

--CecileLi(讨论) 【审校】补充翻译： 设$\displaystyle{ X }$指数分布随机变量，参数为$\displaystyle{ \lambda }$，即概率密度函数

$\displaystyle{ f（x）=\lambda e^{-\lambda x}\mbox{for}x\geq 0. }$

 $\displaystyle{ h\u e（X）\， }$ $\displaystyle{ =-\int\u 0^\infty\lambda e^{-\lambda x}\log（\lambda e^{-\lambda x}）\，dx }$ $\displaystyle{ =-\left（\int\u 0^\infty（\log\lambda）\lambda e^{-\lambda x}\，dx+\int\u 0^\infty（-\lambda x）\lambda e^{-\lambda x}\，dx\right） }$ $\displaystyle{ =-\log\lambda\int\u 0^\infty f（x）\，dx+\lambda E[x] }$ $\displaystyle{ =-\log\lambda+1\，. }$

## Relation to estimator error

The differential entropy yields a lower bound on the expected squared error of an estimator. For any random variable $\displaystyle{ X }$ and estimator $\displaystyle{ \widehat{X} }$ the following holds:[2]

$\displaystyle{ \operatorname{E}[(X - \widehat{X})^2] \ge \frac{1}{2\pi e}e^{2h(X)} }$

with equality if and only if $\displaystyle{ X }$ is a Gaussian random variable and $\displaystyle{ \widehat{X} }$ is the mean of $\displaystyle{ X }$.

--CecileLi(讨论) 【审校】补充翻译：

## 与估计器误差的关系

$\displaystyle{ \operatorname{E}[（X-\widehat{X}）^2]\ge\frac{1}{2\pi E}E^{2h（X）} }$

## Differential entropies for various distributions

In the table below $\displaystyle{ \Gamma(x) = \int_0^{\infty} e^{-t} t^{x-1} dt }$ is the gamma function, $\displaystyle{ \psi(x) = \frac{d}{dx} \ln\Gamma(x)=\frac{\Gamma'(x)}{\Gamma(x)} }$ is the digamma function, $\displaystyle{ B(p,q) = \frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)} }$ is the beta function, and γE is Euler's constant.[9]:219–230

Table of differential entropies
Distribution Name Probability density function (pdf) Entropy in nats Support
Uniform $\displaystyle{ f(x) = \frac{1}{b-a} }$ $\displaystyle{ \ln(b - a) \, }$ $\displaystyle{ [a,b]\, }$
Normal $\displaystyle{ f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) }$ $\displaystyle{ \ln\left(\sigma\sqrt{2\,\pi\,e}\right) }$ $\displaystyle{ (-\infty,\infty)\, }$
Exponential $\displaystyle{ f(x) = \lambda \exp\left(-\lambda x\right) }$ $\displaystyle{ 1 - \ln \lambda \, }$ $\displaystyle{ [0,\infty)\, }$
Rayleigh $\displaystyle{ f(x) = \frac{x}{\sigma^2} \exp\left(-\frac{x^2}{2\sigma^2}\right) }$ $\displaystyle{ 1 + \ln \frac{\sigma}{\sqrt{2}} + \frac{\gamma_E}{2} }$ $\displaystyle{ [0,\infty)\, }$
Beta $\displaystyle{ f(x) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)} }$ for $\displaystyle{ 0 \leq x \leq 1 }$ $\displaystyle{ \ln B(\alpha,\beta) - (\alpha-1)[\psi(\alpha) - \psi(\alpha +\beta)]\, }$
$\displaystyle{ - (\beta-1)[\psi(\beta) - \psi(\alpha + \beta)] \, }$
$\displaystyle{ [0,1]\, }$
Cauchy $\displaystyle{ f(x) = \frac{\gamma}{\pi} \frac{1}{\gamma^2 + x^2} }$ $\displaystyle{ \ln(4\pi\gamma) \, }$ $\displaystyle{ (-\infty,\infty)\, }$
Chi $\displaystyle{ f(x) = \frac{2}{2^{k/2} \Gamma(k/2)} x^{k-1} \exp\left(-\frac{x^2}{2}\right) }$ $\displaystyle{ \ln{\frac{\Gamma(k/2)}{\sqrt{2}}} - \frac{k-1}{2} \psi\left(\frac{k}{2}\right) + \frac{k}{2} }$ $\displaystyle{ [0,\infty)\, }$
Chi-squared $\displaystyle{ f(x) = \frac{1}{2^{k/2} \Gamma(k/2)} x^{\frac{k}{2}\!-\!1} \exp\left(-\frac{x}{2}\right) }$ $\displaystyle{ \ln 2\Gamma\left(\frac{k}{2}\right) - \left(1 - \frac{k}{2}\right)\psi\left(\frac{k}{2}\right) + \frac{k}{2} }$ $\displaystyle{ [0,\infty)\, }$
Erlang $\displaystyle{ f(x) = \frac{\lambda^k}{(k-1)!} x^{k-1} \exp(-\lambda x) }$ $\displaystyle{ (1-k)\psi(k) + \ln \frac{\Gamma(k)}{\lambda} + k }$ $\displaystyle{ [0,\infty)\, }$
F $\displaystyle{ f(x) = \frac{n_1^{\frac{n_1}{2}} n_2^{\frac{n_2}{2}}}{B(\frac{n_1}{2},\frac{n_2}{2})} \frac{x^{\frac{n_1}{2} - 1}}{(n_2 + n_1 x)^{\frac{n_1 + n2}{2}}} }$ $\displaystyle{ \ln \frac{n_1}{n_2} B\left(\frac{n_1}{2},\frac{n_2}{2}\right) + \left(1 - \frac{n_1}{2}\right) \psi\left(\frac{n_1}{2}\right) - }$
$\displaystyle{ \left(1 + \frac{n_2}{2}\right)\psi\left(\frac{n_2}{2}\right) + \frac{n_1 + n_2}{2} \psi\left(\frac{n_1\!+\!n_2}{2}\right) }$
$\displaystyle{ [0,\infty)\, }$
Gamma $\displaystyle{ f(x) = \frac{x^{k - 1} \exp(-\frac{x}{\theta})}{\theta^k \Gamma(k)} }$ $\displaystyle{ \ln(\theta \Gamma(k)) + (1 - k)\psi(k) + k \, }$ $\displaystyle{ [0,\infty)\, }$
Laplace $\displaystyle{ f(x) = \frac{1}{2b} \exp\left(-\frac{|x - \mu|}{b}\right) }$ $\displaystyle{ 1 + \ln(2b) \, }$ $\displaystyle{ (-\infty,\infty)\, }$
Logistic $\displaystyle{ f(x) = \frac{e^{-x}}{(1 + e^{-x})^2} }$ $\displaystyle{ 2 \, }$ $\displaystyle{ (-\infty,\infty)\, }$
Lognormal $\displaystyle{ f(x) = \frac{1}{\sigma x \sqrt{2\pi}} \exp\left(-\frac{(\ln x - \mu)^2}{2\sigma^2}\right) }$ $\displaystyle{ \mu + \frac{1}{2} \ln(2\pi e \sigma^2) }$ $\displaystyle{ [0,\infty)\, }$
Maxwell–Boltzmann $\displaystyle{ f(x) = \frac{1}{a^3}\sqrt{\frac{2}{\pi}}\,x^{2}\exp\left(-\frac{x^2}{2a^2}\right) }$ $\displaystyle{ \ln(a\sqrt{2\pi})+\gamma_E-\frac{1}{2} }$ $\displaystyle{ [0,\infty)\, }$
Generalized normal $\displaystyle{ f(x) = \frac{2 \beta^{\frac{\alpha}{2}}}{\Gamma(\frac{\alpha}{2})} x^{\alpha - 1} \exp(-\beta x^2) }$ $\displaystyle{ \ln{\frac{\Gamma(\alpha/2)}{2\beta^{\frac{1}{2}}}} - \frac{\alpha - 1}{2} \psi\left(\frac{\alpha}{2}\right) + \frac{\alpha}{2} }$ $\displaystyle{ (-\infty,\infty)\, }$
Pareto $\displaystyle{ f(x) = \frac{\alpha x_m^\alpha}{x^{\alpha+1}} }$ $\displaystyle{ \ln \frac{x_m}{\alpha} + 1 + \frac{1}{\alpha} }$ $\displaystyle{ [x_m,\infty)\, }$
Student's t $\displaystyle{ f(x) = \frac{(1 + x^2/\nu)^{-\frac{\nu+1}{2}}}{\sqrt{\nu}B(\frac{1}{2},\frac{\nu}{2})} }$ $\displaystyle{ \frac{\nu\!+\!1}{2}\left(\psi\left(\frac{\nu\!+\!1}{2}\right)\!-\!\psi\left(\frac{\nu}{2}\right)\right)\!+\!\ln \sqrt{\nu} B\left(\frac{1}{2},\frac{\nu}{2}\right) }$ $\displaystyle{ (-\infty,\infty)\, }$
Triangular $\displaystyle{ f(x) = \begin{cases} \frac{2(x-a)}{(b-a)(c-a)} & \mathrm{for\ } a \le x \leq c, \\[4pt] \frac{2(b-x)}{(b-a)(b-c)} & \mathrm{for\ } c \lt x \le b, \\[4pt] \end{cases} }$ $\displaystyle{ \frac{1}{2} + \ln \frac{b-a}{2} }$ $\displaystyle{ [0,1]\, }$
Weibull $\displaystyle{ f(x) = \frac{k}{\lambda^k} x^{k-1} \exp\left(-\frac{x^k}{\lambda^k}\right) }$ $\displaystyle{ \frac{(k-1)\gamma_E}{k} + \ln \frac{\lambda}{k} + 1 }$ $\displaystyle{ [0,\infty)\, }$
Multivariate normal $\displaystyle{ f_X(\vec{x}) = }$
$\displaystyle{ \frac{\exp \left( -\frac{1}{2} ( \vec{x} - \vec{\mu})^\top \Sigma^{-1}\cdot(\vec{x} - \vec{\mu}) \right)} {(2\pi)^{N/2} \left|\Sigma\right|^{1/2}} }$
$\displaystyle{ \frac{1}{2}\ln\{(2\pi e)^{N} \det(\Sigma)\} }$ $\displaystyle{ \mathbb{R}^N }$

Many of the differential entropies are from.[10]:120–122

--CecileLi(讨论) 【审校】补充翻译：

## 各种分布的微分熵

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|正态| |$\displaystyle{ f（x）=\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left（\frac{（x-\mu）^2}{2\sigma^2}\right） }$|$\displaystyle{ \ln left（\sigma\sqrt{2\，\pi\，e}\right） }$|$\displaystyle{ （\infty，\infty）\， }$

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|指数| |$\displaystyle{ f（x）=\lambda\exp\left（-\lambda x\right） }$| |$\displaystyle{ 1-\ln\lambda\， }$| |$\displaystyle{ [0，\infty）\， }$

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| Rayleigh| |$\displaystyle{ f（x）=\frac{x}{\sigma^2}\exp\left（-\frac{x^2}{2\sigma^2}\right） }$|$\displaystyle{ 1+\ln\frac{\sigma}{\sqrt{2}+\frac{\gamma E}{2} }$|{math>[0，\infty）\，[/itex]

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|数学>f（x）f（x））=\frac{{x数学>

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| Cauchy| |$\displaystyle{ f（x）=\frac{\gamma}{\pi}\frac{1}{\gamma^2+x^2} }$|$\displaystyle{ \ln（4\pi\gamma）\， }$|$\displaystyle{ （-infty，\infty）\， }$

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|[中国分布| Chi分布.[中国分布| Chi分布.[中国分布.[中国分布| Chi分布.]].[中国分布|中国分布| | |数学>f（x）x（x）的数学）=\分形{{2{{k/2{k/2}{k/2}γ（k/2）γ（k/2）}}x ^ k-1}x ^ x{2}[/itex]|[/itex][0，\infty）\，[/itex]

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|卡方| |$\displaystyle{ f（x）=\frac{1}{2^{k/2}\Gamma（k/2）}x^{\frac{k}{2}\！-\!1} \exp\left（-\frac{x}{2}\right） }${k}{2}\ln 2\Gamma\left（\frac{k}{2}\right）-\left（1-\frac{k}{2}\right）</psi\left（\frac{k}{2}\right）+\frac{k}{2}[/itex]{k}\infty）\，[/itex]

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| Erlang| |$\displaystyle{ f（x）=\frac{\lambda^k}{（k-1）！}x^{k-1}\exp（-\lambda x） }$| |$\displaystyle{ （1-k）\psi（k）+\ln\frac{\Gamma（k）}{\lambda}+k }$| |$\displaystyle{ [0，\infty）\， }$

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|（F）分销部门的分销ӝF]ӝ数学|数学|数学|数学（x）方面的统计{分销部门的分销ӝ分销保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保保n{u 1+n2}{2}}[/itex]{math>\ln\frac{n{u 1}{n}u 2}B\左（\frac{n{u 1}{2}），\frac{n{u 2}{2}\right）+\left（1-\frac{n{u 1}{2}\right）\psi\ left（\frac{n{u 1}{2}\right）[/itex]
$\displaystyle{ \ left（1+\frac{n{u 2}{2}\right）\psi\ left（\frac{n{u 2}\right）+\frac{n{u 1+n{u 2}\psi\ left（\frac{n{u 1}\right）！+\!n\u 2}{2}\右） }$| |$\displaystyle{ [0，\infty）\， }$

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| Gamma| |$\displaystyle{ f（x）=\frac{x^{k-1}\exp（-\frac{x}{\theta}）}{\theta^k\Gamma（k）} }$|$\displaystyle{ \ln（\theta\Gamma（k））+（1-k）\psi（k）+k\， }$|$\displaystyle{ [0，\infty）\， }$

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|拉普拉斯分布

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| Logistic| |$\displaystyle{ f（x）=\frac{e^{-x}{（1+e^{-x}）^2} }$|$\displaystyle{ 2\， }$|$\displaystyle{ （\infty，\infty）\， }$

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| Lognormal| |$\displaystyle{ f（x）=\frac{1}{\sigma x\sqrt{2\pi}}\exp\left（\frac{（\ln x-\mu）^2}{2\sigma^2}\right） }$| |$\displaystyle{ \mu+\frac{1}{2}\ln 2\pie\sigma^2） }$|{math>[0，infty）\，[/itex] |-

| Maxwell-Boltzmann| |$\displaystyle{ f（x）=\frac{1}{a^3}\sqrt{\frac{2}{\pi}}\，x^{2}\exp\左（\frac{x^2}{2a^2}\右） }$| |$\displaystyle{ \ln（a\sqrt{2\pi}）+\gamma u E-\frac{1}2}\lt /math |\lt math\gt [0，infty）\， }$>

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|[广义高斯分布[广义高斯分布[广义高斯分布[广义高斯分布[广义高斯分布[广义高斯分布[广义高斯分布[广义高斯分布{\alpha}{2}\右）+\frac{\alpha}{2}[/itex]| |$\displaystyle{ （-infty，\infty）\， }$

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| Pareto| |$\displaystyle{ f（x）=\frac{\alpha x{m^\alpha}{x^{\alpha+1} }$|{math>\ln\frac{x{m}{\alpha}+1+\frac{1}{\alpha}[/itex]|{math>[x{m，\infty）\，[/itex]

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| Student's t| |$\displaystyle{ f（x）=\frac{（1+x^2/\nu）^{-\frac{\nu+1}{2}}}{\sqrt{\nu}B（\frac{1}{2}，\frac{\nu 2}）} }${！+\!1} {2}\左（\psi\左（\frac{\nu\）！+\!1} {2}\对）\！-\!\psi\左（\frac{\nu}{2}\右）\right）\！+\!\ln\sqrt{\nu}B\左（\frac{1}{2}，\frac{\nu}{2}\右）[/itex]|[/itex]（\infty，\infty）\，[/itex]

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|三角| |$\displaystyle{ f（x）=\begin{cases} \frac{2（x-a）}{（b-a）（c-a）}&\mathrm{for\}a\le x\leq c，\\[4pt] \frac{2（b-x）}{（b-a）（b-c）}&\mathrm{for\}c\lt x\le b，\\[4pt] \结束{cases} }$|{math>\frac{1}{2}+\ln\frac{b-a}{2}[/itex]|{math>[0,1]\，[/itex]

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| Weibull| |$\displaystyle{ f（x）=\frac{k}{\lambda^k}x^{k-1}\exp\左（\frac{x^k}{\lambda^k}\右） }$|{math>\frac{（k-1）\gamma E}{k}+\ln frac{\lambda}{k}+1[/itex]{math>[0，\infty）\，[/itex]

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|多元正态| |<数学>

|-

|}

1. Jaynes, E.T. (1963). "Information Theory And Statistical Mechanics" (PDF). Brandeis University Summer Institute Lectures in Theoretical Physics. 3 (sect. 4b).
2. Cover, Thomas M.; Thomas, Joy A. (1991). Elements of Information Theory. New York: Wiley. ISBN 0-471-06259-6.
3. Vasicek, Oldrich (1976), "A Test for Normality Based on Sample Entropy", Journal of the Royal Statistical Society, Series B, 38 (1), JSTOR 2984828.
4. Gibbs, Josiah Willard (1902). Elementary Principles in Statistical Mechanics, developed with especial reference to the rational foundation of thermodynamics. New York: Charles Scribner's Sons.
5. Kraskov, Alexander; Stögbauer, Grassberger (2004). "Estimating mutual information". Physical Review E. 60: 066138. arXiv:cond-mat/0305641. Bibcode:2004PhRvE..69f6138K. doi:10.1103/PhysRevE.69.066138.
6. Fazlollah M. Reza (1994) [1961]. An Introduction to Information Theory. Dover Publications, Inc., New York. ISBN 0-486-68210-2.
7. "proof of upper bound on differential entropy of f(X)". Stack Exchange. April 16, 2016.
8. 引用错误：无效<ref>标签；未给name属性为“cover\u的引用提供文字
9. Park, Sung Y.; Bera, Anil K. (2009). "Maximum entropy autoregressive conditional heteroskedasticity model" (PDF). Journal of Econometrics. Elsevier. Archived from the original (PDF) on 2016-03-07. Retrieved 2011-06-02.
10. Lazo, A. and P. Rathie (1978). "On the entropy of continuous probability distributions". IEEE Transactions on Information Theory. 24 (1): 120–122. doi:10.1109/TIT.1978.1055832.
11. {引用期刊| author=Lazo，A.和P.Rathie | title=关于连续概率分布熵| journal=IEEE Transactions On Information Theory | year=1978 | volume=24 | issue=1 | doi=10.1109/TIT.1978.1055832 | pages=120–122}