I am more unpredictable than you are ! (Entropy measure with an difference)

"I am moody, I'm unpredictable". Well this article is for all those who are often characterized as moody and unpredictable. Well its a into information theory for measuring the moodiness/entropy. Answering "Who is more moody than other ?". If not interested please don't proceed beyond this. Why would you like such a measure ? Quantitative measures are good aren't they ? 0.9 of an feminist is a better than a 0.5 feminist isn't it ?

If give a circumstance X you tend to behave differently in |Y| number of ways. For example Given a women issue a person comes up with the a stand favoring women or a stand that is not quite liked by women. So here

X -> circumstance -> A women issue.
Y -> number of stands you can take -> 2
(Well this is an assumption we make - we are only dealing with dichotomies ie. Y/N outcomes)

"Well we will answer who is more predictable given an circumstance on a scale of 0-1". That last sentence i made was quite an sentence if you get it. It says atleast more than what is clearly understood.

1. We are not answering who is a feminist and who is Mysogynist. To answer this the problem has to modeled in a different way :)
2. We are answering who is a more stable feminist or a Mysogynist.

That is, when posed with a feminist dichotomy (question) Virginia Wolf's (a well known feminist) stand can be more predictable than my answer. Now can we have a measure of this unpredictability ? (Its closely related to the corollary of probability i.e p-value. The chance that you are hit my misfortune). Below is a sample table.

Legend
Answer Y=Supporting feminist views N=No support feminist issues.
X=random women question

A staunch feminist
Answer Y Y Y Y N Y Y Y
QuestionX Q1 Q2 ..... Q8

A weak feminist.
Answer Y Y N Y N N Y Y
QuestionX Q1 Q2 ..... Q8

Entropy/Randomness Measure = Summation_over_outcomes ( -Probability(outcome) * log2(Probability(outcome)) )



Here we have only 2 outcome Y/N. (cleverly crafted na ? See iam smart). So we have to sum only over 2 outcomes.


Randomness in our case is just the summation of 2 terms =
-Probab(SupportFeminist) * log2(Probab(SupportFeminist))
-Probab(NotSupportFeminist)*log2(Probab(NotSupportFeminist))


K. Now the randomess/craziness/unpredictability of an staunch feminist is.
= - (7/8) log 2 (7/8) - (1/8) log 2 (1/8) .... (Because 7 out 8 she supported feminist)
= 0.168 + 0.375
= 0.543

K. Now the randomess/craziness/unpredictability of an weak feminist is.
= - (5/8) log 2 (5/8) - (3/8) log 2 (3/8) .... (Because 5 out 8 she supported feminist))
= 0.423 + 0.5306
= 0.9539

Well we calculated a measure of unpredictability. The measure of predictability is simply (1-unpredictability).

So people here we have the numbers on a scale of 0...1 (nice scale) as to who is more unpredictable than the other. A staunch feminist is unpredictable (0.543) and a weak feminist is (0.9533).

Well i will some food for thought.

When is unpredictability highest/lowest i.e 1 and 0. (in the dichotomy case)?
Why is the log to the base 2 ?
Many dollar Question. Is the weak feminist in our example a misogynist ?

Applications of the above include ?
- Qualitative analysis of 2 random number generators ?
- Manipulating sample spaces of an Exit polls in an election, thereby biasing the the most unpredictable voters towards a desired candidate.
- Understanding the other side of probability. i.e what are the chances that you will be hit by a result not obvious from an probability distribution.
- Appreciating unpredictability of a person. (nothing bad about the person he just has an higher index for randomness)

P.S None of the formulas are invented by me (it was Shannon), just to be clear about it. Example is mine and so are the applications.