The Uniqueness of the Information Function

Remember that we proposed that the Information should satisfy the following list of properties.

We want to show that For our purposes we will content ourselves with the following

Theorem

Let be a differentiable function such that

1. $\QTR{bs}{I(1)=0}$
Then

Proof
:

Considering to be a constant and differentiating 3. we have

or

and setting $\QTR{bf}{x=1}$ we have

Proof

Proof

Proof

The result now follow by Integrating both sides of the equation:

MATH or

Since $\QTR{bs}{I(1)=0}$ we have our result.

$\vspace{1pt}$

In point of fact we should be showing the uniqueness of the Entropy function, but we will simply define

If

One intuitive concequence of the result of the last lecture is that we get the most information from an equiprobable event. That is if

since

MATH

MATH