Hartley function


The Hartley function is a of uncertainty, introduced by Hartley in 1928. If we pick a sample from a finite setMathworldPlanetmath A uniformly at random, the revealed after we know the is given by the Hartley function


If the base of the logarithm is 2, then the uncertainty is measured in bits. If it is the natural logarithmMathworldPlanetmath, then the is nats. It is also known as the Hartley entropy.


The Hartley function is a special case of Shannon’s entropyMathworldPlanetmath. Each element in the sample space A is associated with probability p=1/|A|. For an element ωA, the Hartley of the event {ω} is -log(p)=log(|A|), which is constant over ωA. The average over the whole sample space is thus also equal to log(|A|).

The Hartley function only depends on the number of elements in a set, and hence can be viewed as a functionMathworldPlanetmath on natural numbersMathworldPlanetmath. Rényi showed that the Hartley function in base 2 is the only function mapping natural numbers to real numbers that

  1. 1.

    H(mn)=H(m)+H(n)    (),

  2. 2.

    H(m)H(m+1)    (monotonicity), and

  3. 3.

    H(2)=1    (normalization).

Condition 1 says that the uncertainty of the Cartesian product of two finite sets A and B is the sum of uncertainties of A and B. Condition 2 says that a larger set has larger uncertainty.

Title Hartley function
Canonical name HartleyFunction
Date of creation 2013-03-22 14:31:41
Last modified on 2013-03-22 14:31:41
Owner kshum (5987)
Last modified by kshum (5987)
Numerical id 15
Author kshum (5987)
Entry type Definition
Classification msc 94A17
Related topic ShannonsTheoremEntropy
Related topic EntropyOfAPartition
Defines Hartley entropy
Defines Hartley information