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.
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[3]: 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 .[4]: 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, the uniform distribution has negative differential entropy
being less than that of which has zero 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),[5] including linear[6] 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.
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 .[7] 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 .
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
Theorem
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.[2]: 255
Proof
Let be a GaussianPDF 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 .
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
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
The differential entropy yields a lower bound on the expected squared error of an estimator. For any random variable and estimator the following holds:[2]
with equality if and only if is a Gaussian random variable and is the mean of .
Many of the differential entropies are from.[9]: 120–122
Variants
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.[10]: 181–218
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[2]
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 .
^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.