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Information

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

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 Z Rain 200 365 and Z No Rain 165 365 .We state that Z No Rain carries more information than Z Rain , the reason is that the inhabitans of Bergenexpect rain, so whenever it's not raining they are (more) surprised. An intuitive definition of an information measure should be largerwhen the probability is small.

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The information content in a statement about the temperature and new lottery millionaires in Verdal,Norway on a given saturdayshould be the sum of the information on temperature on the particular saturday in Verdal and the information of the numberof new lucky lottery winners, (under the assumption that these observations are independent). Let I denote the information of an event, then

I temperature lottery winners I temperature I lottery winners

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The self information formula

An intuitive and meaningful measure of self information in an event should have the following properties:

  • The more uncertain you, in advance, are about the outcome, the more new information you get by observing the actual outcome, or equivalently an eventwith low probability, p n , has high self information I p n . I p n should be a monotonically decreasing function of p n .
  • Oberserving an event with certain outcome, i.e p n 1 , should give zero information. The event p n is then said to have zero self information. Since I p n is monotonically decreasing for p n 0 1 this implies that the self information can never be less than zero, the observer can never lose informationby observing an outcome.
  • 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.

I p n b logbase --> 1 p n b logbase --> p n

In the above equation the logarithm base can be chosen arbitrary. Usually b 2 is chosen so that the denomination is information bit . The choice b 2 is made to adapt to a digital "world", that is to facilitate electronic storage and transmission.

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Source:  OpenStax, Information and signal theory. OpenStax CNX. Aug 03, 2006 Download for free at http://legacy.cnx.org/content/col10211/1.19
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