Differential entropy

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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.[citation needed] 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.

Definition[edit]

Let be a random variable with a probability density function whose support is a set . The differential entropy or is defined as[1]:243

.

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 [2]:54–59

.

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 .[3]:183-184 For example, the differential entropy of a quantity measured in millimeters will be log(1000) more than the same quantity measured in meters; a dimensionless quantity will have differential entropy of log(1000) more than the same quantity divided by 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, Uniform(0,1/2) has negative differential entropy

.

Thus, differential entropy does not share all properties of discrete entropy.

Note that the continuous mutual information 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 and as these partitions become finer and finer. Thus it is invariant under non-linear homeomorphisms (continuous and uniquely invertible maps) ,[4] including linear [5] transformations of and , and still represents the amount of discrete information that can be transmitted over a channel that admits a continuous space of values.

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

Properties of differential entropy[edit]

  • For probability densities and , the Kullback–Leibler divergence is greater than or equal to 0 with equality only if almost everywhere. Similarly, for two random variables and , and with equality if and only if and are independent.
  • The chain rule for differential entropy holds as in the discrete case[1]:253
.
  • Differential entropy is translation invariant, i.e. for a constant .[1]:253
  • Differential entropy is in general not invariant under arbitrary invertible maps.
In particular, for a constant
For a vector valued random variable and an invertible (square) matrix
[1]:253
  • In general, for a transformation from a random vector to another random vector with same dimension , the corresponding entropies are related via
where is the Jacobian of the transformation [6]. The above inequality becomes an equality if the transform is a bijection. Furthermore, when is a rigid rotation, translation, or combination thereof, the Jacobian determinant is always 1, and .
  • If a random vector has mean zero and covariance matrix , with equality if and only if is jointly gaussian (see below).[1]:254

However, differential entropy does not have other desirable properties:

  • 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).

Maximization in the normal distribution[edit]

Theorem[edit]

With a normal distribution, differential entropy is maximized for a given variance. A Gaussian random variable has the largest entropy amongst all random variables of equal variance, or, alternatively, the maximum entropy distribution under constraints of mean and variance is the Gaussian.[1]:255

Proof[edit]

Let be a Gaussian PDF with mean μ and variance and an arbitrary PDF with the same variance. Since differential entropy is translation invariant we can assume that has the same mean of as .

Consider the Kullback–Leibler divergence between the two distributions

Now note that

because the result does not depend on other than through the variance. Combining the two results yields

with equality when following from the properties of Kullback–Leibler divergence.

Alternative proof[edit]

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

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 and the requirement of fixed variance , are both satisfied, then a small variation δg(x) about g(x) will produce a variation δL about L which is equal to zero:

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

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

Example: Exponential distribution[edit]

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

Its differential entropy is then

Here, was used rather than to make it explicit that the logarithm was taken to base e, to simplify the calculation.

Relation to estimator error[edit]

The differential entropy yields a lower bound on the expected squared error of an estimator. For any random variable and estimator the following holds:[1]

with equality if and only if is a Gaussian random variable and is the mean of .

Differential entropies for various distributions[edit]

Warning: many of the results in the table below are known to be wrong; this was shown by Darbellay & Vajda, IEEE Trans Inf Theory 46 (2000) 709.

In the table below is the gamma function, is the digamma function, is the beta function, and γE is Euler's constant.[7]:219-230

Table of differential entropies
Distribution Name Probability density function (pdf) Entropy in nats Support
Uniform
Normal
Exponential
Rayleigh
Beta for
Cauchy
Chi
Chi-squared
Erlang
F
Gamma
Laplace
Logistic
Lognormal
Maxwell–Boltzmann
Generalized normal
Pareto
Student's t
Triangular
Weibull
Multivariate normal

Many of the differential entropies are from.[8]:120-122

Variants[edit]

As described above, differential entropy does not share all properties of discrete entropy. For example, the differential entropy can be negative; also it is not invariant under continuous coordinate transformations. Edwin Thompson Jaynes showed in fact that the expression above is not the correct limit of the expression for a finite set of probabilities.[9]:181–218

A modification of differential entropy adds an invariant measure factor to correct this, (see limiting density of discrete points). If is further constrained to be a probability density, the resulting notion is called relative entropy in information theory:

The definition of differential entropy above can be obtained by partitioning the range of into bins of length with associated sample points within the bins, for Riemann integrable. This gives a quantized version of , defined by if . Then the entropy of is[1]

The first term on the right approximates the differential entropy, while the second term is approximately . Note that this procedure suggests that the entropy in the discrete sense of a continuous random variable should be .

See also[edit]

References[edit]

  1. ^ a b c d e f g h , Cover, Thomas M.; Thomas, Joy A. (1991). Elements of Information Theory (PDF). New York: Wiley. ISBN 0-471-06259-6.
  2. ^ Vasicek, Oldrich (1976), "A Test for Normality Based on Sample Entropy", Journal of the Royal Statistical Society, Series B, 38 (1), JSTOR 2984828.
  3. ^ 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.
  4. ^ 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.
  5. ^ Fazlollah M. Reza (1994) [1961]. An Introduction to Information Theory. Dover Publications, Inc., New York. ISBN 0-486-68210-2.
  6. ^ https://math.stackexchange.com/questions/1745670/proof-of-upper-bound-on-differential-entropy-of-fx
  7. ^ Park, Sung Y.; Bera, Anil K. (2009). "Maximum entropy autoregressive conditional heteroskedasticity model" (PDF). Journal of Econometrics. Elsevier. Retrieved 2011-06-02.
  8. ^ Lazo, A. and P. Rathie (1978). "On the entropy of continuous probability distributions". IEEE Transactions on Information Theory. 24 (1). doi:10.1109/TIT.1978.1055832.
  9. ^ Jaynes, E.T. (1963). "Information Theory And Statistical Mechanics" (PDF). Brandeis University Summer Institute Lectures in Theoretical Physics. 3 (sect. 4b).

External links[edit]