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Information

Module by: Anders Gjendemsjø

Summary: Information

In this module we introduce the concept of self information for an outcome of a stochastic variable.

Example 1 
Bergen, Norway is a rainy city. If the locals are "lucky" there is "only" 200 rainy days in a particular year. Let the random variable Z take the two values: "Rain", "No rain". Assuming 200 rainy days a year, we get PrZ=Rain=200365 Z Rain 200 365 and PrZ=No Rain=165365 Z No Rain 165 365 . We state that Z=No Rain Z No Rain carries more information than Z=Rain Z Rain , the reason is that the inhabitans of Bergen expect rain, so whenever it's not raining they are (more) surprised. An intuitive definition of an information measure should be larger when the probability is small.
Example 2 
The information content in a statement about the temperature and new lottery millionaires in Verdal,Norway on a given saturday should be the sum of the information on temperature on the particular saturday in Verdal and the information of the number of new lucky lottery winners, (under the assumption that these observations are independent). Let I denote the information of an event, then
Itemperaturelottery winners=Itemperature+Ilottery winners I temperature lottery winners I temperature I lottery winners (1)

The self information formula

An intuitive and meaningful measure of self information in an event should have the following properties:
  1. The more uncertain you, in advance, are about the outcome, the more new information you get by observing the actual outcome, or equivalently an event with low probability, pn pn, has high self information Ipn I pn . Ipn I pn should be a monotonically decreasing function of pnpn.
  2. Oberserving an event with certain outcome, i.e pn=1 pn 1 , should give zero information. The event pnpn is then said to have zero self information. Since Ipn I pn is monotonically decreasing for pn∈01 pn 0 1 this implies that the self information can never be less than zero, the observer can never lose information by observing an outcome.
  3. If we receive independent messages, the information should accumulate. This means that the measure must be additive.
It can be shown that there only exists one function satisfying the above conditions.
Self Information: Ipn=logb1pn=-logbpn I pn b 1 pn b pn
In the above equation the logarithm base can be chosen arbitrary. Usually b=2 b 2 is chosen so that the denomination is information bit. The choice b=2 b 2 is made to adapt to a digital "world", that is to facilitate electronic storage and transmission.

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