5.3 Entropy

The self information gives the information in a single outcome. In most cases, e.g in data compression, it is much moreinteresting to know the average information content of a source. This average is given by the expected value of the self information with respect to the source's probabilitydistribution. This average of self information is called the source entropy.

Definition of entropy

Entropy

If symbol has zero probability, which means it never occurs, it should not affect the entropy. Letting $0\lg 0=0$ , we have dealt with that.

In texts you will find that the argument to the entropy function may vary. The two most common are $H(X)$ and $H(p)$ . We calculate the entropy of a source X, but the entropy is,strictly speaking, a function of the source's probabilty function p. So both notations are justified.

Calculating the binary logarithm

Most calculators does not allow you to directly calculate the logarithm with base 2, so we have to use a logarithm base that mostcalculators support. Fortunately it is easy to convert between different bases.

Assume you want to calculate $\log_{2}x$ , where $x> 0$ . Then $\log_{2}x=y$ implies that $2^y=x$ . Taking the natural logarithm on both sides we obtain

Examples

Entropy is closely tied to source coding. The extent to which a source can be compressed is related to its entropy.There are many interpretations possible for the entropy of a random variable, including

• (Average)Self information in a random variable
• Minimum number of bits per source symbol required to describe the random variable without loss
• Description complexity
• Measure of uncertainty in a random variable

References

• ien, G.E. and Lundheim,L. (2003) Information Theory, Coding and Compression , Trondheim: Tapir Akademisk forlag.