Self-information

Within the context of information theory, self-information is defined as the amount of information that knowledge about (the outcome of) a certain event, adds to someone's overall knowledge. The amount of self-information is expressed in the unit of information: a bit.

By definition, the amount of self-information contained in a probabilistic event depends only on the probability $$p$$ of that event. More specifically: the smaller this probability is, the larger is the self-information associated with receiving information that the event indeed occurred.

Further, by definition, the measure of self-information has the following property. If an event C is composed of two mutually independent events A and B, then the amount of information at the proclamation that C has happened, equals the sum of the amounts of information at proclamations of event A and event B respectively.

Taking into account these properties, the self-information $$I(A_n)$$ (measured in bits) associated with outcome $$A_n$$ whose outcome has probability $$p$$ is  defined as:


 * $$I(A_n) = \log_2 \left(\frac{1}{p(A_n)} \right) = - \log_2(p(A_n)) $$

This definition, using the binary logarithm function, complies with the above conditions.

This measure has also been called surprisal, as it represents the "surprise" of seeing the outcome (a certain outcome is not surprising). This term was coined by Myron Tribus in his 1961 book Thermostatics and Thermodynamics. Some claim it is more accurate than "self-information", but it has not been widely used.

Information Entropy
The concept is related to that of information entropy; the information entropy of a random event $$A$$ is the expected value of its self-information:


 * $$H(A)=\sum_{i=1}^np(A_i)\log_2 \left(\frac{1}{p(A_i)}\right)$$

Examples

 * On tossing a coin, the chance of 'tail' is 0.5. When it is proclaimed that indeed 'tail' occurred, this amounts to
 * I('tail') = log2 (1/0.5) = log2 2 = 1 bits of information.


 * When throwing a die, the probability of 'four' is 1/6. When there is proclaimed that 'four' has been thrown, the amount of self-information is
 * I('four') = log2 (1/(1/6)) = log2 (6) = 2.585 bits.


 * When, independently, two dice are thrown, the amount of information associated with {throw 1 = 'two' & throw 2 = 'four'} equals
 * I('throw 1 is two & throw 2 is four') = log2 (1/Pr(throw 1 = 'two' & throw 2 = 'four')) = log2 (1/(1/36)) = log2 (36) = 5.170 bits. This outcome equals the sum of the individual amounts of self-information associated with {throw 1 = 'two'} and {throw 2 = 'four'}; namely 2.585 + 2.585 = 5.170 bits.