My watch list  

Differential entropy

Differential entropy (also referred to as continuous entropy) is a concept in information theory which tries to extend the idea of (Shannon) entropy, a measure of average surprisal of a random variable, to continuous probability distributions.



Let X be a random variable with a probability density function f whose support is a set \mathbb X. The differential entropy h(X) or h(f) is defined as

h(X) = -\int_\mathbb{X} f(x)\log f(x)\,dx.

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. 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, Uniform(0,1/2) has differential entropy \int_0^\frac{1}{2} -2\log2\,dx=-\log2\,.

The definition of differential entropy above can be obtained by partitioning the range of X into bins of length Δ with associated sample points iΔ within the bins, for X Riemann integrable. This gives a quantized version of X, defined by XΔ = iΔ if i\Delta \leq X \leq (i+1)\Delta. Then the entropy of XΔ is

-\sum_i f(i\Delta)\Delta\log f(i\Delta) - \sum f(i\Delta)\Delta\log \Delta.

The first term approximates the differential entropy, while the second term is approximately − log(Δ). Note that this procedure suggests that the differential entropy of a discrete random variable should be -\infty.

Note that the continuous mutual information 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 X and Y as these partitions become finer and finer. Thus it is invariant under quite general transformations of X and Y, and still represents the amount of discrete information that can be transmitted over a channel that admits a continuous space of values.

Properties of differential entropy

  • For two densities f and g, D(f||g) \geq 0 with equality if f = g almost everywhere. Similarly, for two random variables X and Y, I(X;Y) \geq 0 and h(X|Y) \leq h(X) with equality if and only if X and Y are independent.
  • The chain rule for differential entropy holds as in the discrete case
h(X_1, \ldots, X_n) = \sum_{i=1}^{n} h(X_i|X_1, \ldots, X_{i-1}) \leq \sum h(X_i).
  • Differential entropy is translation invariant, ie, h(X + c) = h(X) for a constant c.
  • Differential entropy is in general not invariant under arbitrary invertible maps. In particular, for a constant a, h(aX) = h(X) + \log \left| a \right|. For a vector valued random variable X and a matrix A, h(A\mathbf{X}) = h(\mathbf{X}) + \log(\det A).
  • If a random vector \mathbf{X} \in \mathbb{R}^{n} has mean zero and covariance matrix K, h(\mathbf{X}) \leq \frac{1}{2} \log[(2\pi e)^n \det{K}] with equality if and only if X is jointly gaussian.

Example: Exponential distribution

Let X be an exponentially distributed random variable with parameter λ, that is, with probability density function

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

Its differential entropy is then

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

Here, he(X) was used rather than h(X) to make it explicit that the logarithm was taken to base e, to simplify the calculation.

Differential entropies for various distributions

In the table below, \Gamma(x) = \int_0^{\infty} e^{-t} t^{x-1} dt (the gamma function), \psi(x) = \frac{d}{dx} \Gamma(x), B(p,q) = \frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}, and γ is Euler's constant.

Table of differential entropies.
Distribution Name Probability density function (pdf) Entropy in nats
Uniform f(x) = \frac{1}{b-a} for a \leq x \leq b \ln(b - a) \,
Normal f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) \ln\left(\sigma\sqrt{2\,\pi\,e}\right)
Exponential f(x) = \lambda \exp\left(-\lambda x\right) 1 - \ln \lambda \,
Rayleigh f(x) = \frac{x}{b^2} \exp\left(-\frac{x^2}{2b^2}\right) 1 + \ln \frac{\beta}{\sqrt{2}} + \frac{\gamma}{2}
Beta f(x) = \frac{x^{p-1}(1-x)^{q-1}}{B(p,q)} for 0 \leq x \leq 1 \ln B(p,q) - (p-1)[\psi(p) - \psi(p + q)] - (q-1)[\psi(q) - \psi(p + q)] \,
Cauchy f(x) = \frac{\lambda}{\pi} \frac{1}{\lambda^2 + x^2} \ln(4\pi\lambda) \,
Chi f(x) = \frac{2}{2^{n/2} \sigma^n \Gamma(n/2)} x^{n-1} \exp\left(-\frac{x^2}{2\sigma^2}\right) \ln{\frac{\sigma\Gamma(n/2)}{\sqrt{2}}} - \frac{n-1}{2} \psi\left(\frac{n}{2}\right) + \frac{n}{2}
Chi-squared f(x) = \frac{1}{2^{n/2} \sigma^n \Gamma(n/2)} x^{\frac{n}{2} - 1} \exp\left(-\frac{x}{2\sigma^2}\right)

\ln 2\sigma^{2}\Gamma\left(\frac{n}{2}\right) - \left(1 - \frac{n}{2}\right)\psi\left(\frac{n}{2}\right) + \frac{n}{2}

Erlang f(x) = \frac{\beta^n}{(n-1)!} x^{n-1} \exp(-\beta x) (1-n)\psi(n) + \ln \frac{\Gamma(n)}{\beta} + n
F 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}}} \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) -

\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)

Gamma f(x) = \frac{x^{\alpha - 1} \exp(-\frac{x}{\beta})}{\beta^\alpha \Gamma(\alpha)} \ln(\beta \Gamma(a)) + (1 - \alpha)\psi(\alpha) + \alpha \,
Laplace f(x) = \frac{1}{2\lambda} \exp(-\frac{|x - \theta|}{\lambda}) 1 + \ln(2\lambda) \,
Logistic f(x) = \frac{e^{-x}}{(1 + e^{-x})^2} 2 \,
Lognormal f(x) = \frac{1}{\sigma x \sqrt{2\pi}} \exp\left(-\frac{(\ln x - m)^2}{2\sigma^2}\right) m + \frac{1}{2} \ln(2\pi e \sigma^2)
Maxwell-Boltzmann f(x) = 4 \pi^{-\frac{1}{2}} \beta^{\frac{3}{2}} x^{2} \exp(-\beta x^2) \frac{1}{2} \ln \frac{\pi}{\beta} + \gamma - 1/2
Generalized normal f(x) = \frac{2 \beta^{\frac{\alpha}{2}}}{\Gamma(\frac{\alpha}{2})} x^{\alpha - 1} \exp(-\beta x^2) \ln{\frac{\Gamma(\alpha/2)}{2\beta^{\frac{1}{2}}}} - \frac{\alpha - 1}{2} \psi\left(\frac{\alpha}{2}\right) + \frac{\alpha}{2}
Pareto f(x) = \frac{a k^a}{x^{a+1}} \ln \frac{k}{a} + 1 + \frac{1}{a}
Student's t f(x) = \frac{(1 + x^2/n)^{-\frac{n+1}{2}}}{\sqrt{n}B(\frac{1}{2},\frac{n}{2})} \frac{n+1}{2}\psi\left(\frac{n+1}{2}\right) - \psi\left(\frac{n}{2}\right) + \ln \sqrt{n} B\left(\frac{1}{2},\frac{n}{2}\right)
Triangular f(x) = \begin{cases} \frac{2x}{a} & 0 \leq x \leq a\\ \frac{2(1-x)}{1-a} & a \leq x \leq 1 \end{cases} \frac{1}{2} - \ln 2
Weibull f(x) = \frac{c}{\alpha} x^{c-1} \exp\left(-\frac{x^c}{\alpha}\right) \frac{(c-1)\gamma}{c} + \ln \frac{\alpha^{1/c}}{c} + 1
Multivariate normal f_X(x_1, \dots, x_N) = \frac{1} {(2\pi)^{N/2} \left|\Sigma\right|^{1/2}} \exp \left( -\frac{1}{2} ( x - \mu)^\top \Sigma^{-1} (x - \mu) \right) \frac{1}{2}\ln\{(2\pi e)^{N} \det(\Sigma)\}

See also


  • Thomas M. Cover, Joy A. Thomas. Elements of Information Theory New York: Wiley, 1991. ISBN 0-471-06259-6
  • Lazo, A. and P. Rathie. On the entropy of continuous probability distributions Information Theory, IEEE Transactions on, 1978. 24(1): p. 120-122.
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Differential_entropy". A list of authors is available in Wikipedia.
Your browser is not current. Microsoft Internet Explorer 6.0 does not support some functions on Chemie.DE