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# Entropy

Summary: Entropy

The self information gives the information in a single outcome. In most cases, e.g in data compression, it is much more interesting 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 probability distribution. This average of self information is called the source entropy.

### Definition of entropy

Definition 1: Entropy
1. The entropy (average self information) of a discrete random variable XX is a function of its probability mass function and is defined as
HX= i =1N p X x i log p X x i H X i 1 N p X x i p X x i
(1)
where NN is the number of possible values of XX and P X x i =PrX= x i P X x i X x i . If log is base 2 then the unit of entropy is bits per (source)symbol. Entropy is a measure of uncertainty in a random variable and a measure of information it can reveal.
2. If symbol has zero probability, which means it never occurs, it should not affect the entropy. Letting 0×log0=0 0 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 HX H X and Hp 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 most calculators support. Fortunately it is easy to convert between different bases.

Assume you want to calculate log2x 2 x , where x>0 x 0 . Then log2x=y 2 x y implies that 2y=x 2 y x . Taking the natural logarithm on both sides we obtain

#### Logarithm conversion:

log2x=lnxln2 2 x x 2

### Examples

#### Example 1

When throwing a dice, one may ask for the average information conveyed in a single throw. Using the formula for entropy we get HX=i=16pXxilogpXxi=log6 bits/symbol H X i 1 6 pX xi pX xi 6 bits/symbol

#### Example 2

If a soure produces binary information 01 0 1 with probabilities pp and 1p 1 p . The entropy of the source is

HX=((plog 2 p))(1p)log 2 (1p) H X p 2 p 1 p 2 1 p
(2)
If p=0 p 0 then HX=0 H X 0 , if p=1 p 1 then HX=0 H X 0 , if p=1/2 p 12 then HX=1 H X 1 . The source has its largest entropy if p=1/2 p 12 and the source provides no new information if p=0 p 0 or p=1 p 1 .

#### Example 3

An analog source is modeled as a continuous-time random process with power spectral density bandlimited to the band between 0 and 4000 Hz. The signal is sampled at the Nyquist rate. The sequence of random variables, as a result of sampling, are assumed to be independent. The samples are quantized to 5 levels -2-1012 -2 -1 0 1 2 . The probability of the samples taking the quantized values are 121418116116 1 2 1 4 1 8 1 16 1 16 , respectively. The entropy of the random variables are

HX=i=15pXxilogpXxi=12+12+38+14+14=158bits/sample H X i 1 5 pX xi pX xi 1 2 1 2 3 8 1 4 1 4 15 8 bits/sample
(3)
There are 8000 samples per second. Therefore, the source produces 8000×158=15000 8000 15 8 15000 bits/sec of information.

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

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

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