# Hartley function

Definition

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

 $H(A):=\log_{b}(|A|).$

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

Remark:

The Hartley function is a special case of Shannon’s entropy  . Each element in the sample space $A$ is associated with probability $p=1/|A|$. For an element $\omega\in A$, the Hartley of the event $\{\omega\}$ is $-\log(p)=\log(|A|)$, which is constant over $\omega\in A$. 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 function  on natural numbers  . 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.

$H(m)\leq H(m+1)$    (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 HartleyFunction 2013-03-22 14:31:41 2013-03-22 14:31:41 kshum (5987) kshum (5987) 15 kshum (5987) Definition msc 94A17 ShannonsTheoremEntropy EntropyOfAPartition Hartley entropy Hartley information