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Hartley function (Definition)

Definition

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

$\displaystyle H(A) := \log_b(\vert A\vert). $
If the base of the logarithm is 2, then the uncertainty is measured in bits. If it is the natural logarithm, then the unit 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/\vert A\vert$. For an element $ \omega\in A$, the Hartley information of the event $ \{\omega\}$ is $ -\log(p)=\log(\vert A\vert)$, which is constant over $ \omega\in A$. The average information over the whole sample space is thus also equal to $ \log(\vert A\vert)$.


Characterization

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 satisfies

  1. $ H(mn) = H(m)+H(n)$ (additivity),
  2. $ H(m) \leq H(m+1)$ (monotonicity), and
  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.



"Hartley function" is owned by kshum.
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See Also: Shannon's entropy, entropy of a partition

Also defines:  Hartley entropy, Hartley information

Attachments:
derivation of Hartley function (Derivation) by Mathprof
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Cross-references: sum, Cartesian product, monotonicity, real numbers, mapping, natural numbers, function, number, characterization, average, event, Shannon's entropy, natural logarithm, logarithm, base, finite set
There are 2 references to this entry.

This is version 12 of Hartley function, born on 2004-08-04, modified 2006-09-08.
Object id is 6070, canonical name is HartleyFunction.
Accessed 4085 times total.

Classification:
AMS MSC: 94A17 (Information and communication, circuits :: Communication, information :: Measures of information, entropy)
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