Axiom:Axioms of Uncertainty/Axiom 4

Axiom

Let $Z$ be a random variable.

Let $Z$ take the values $a_i$ with probability $p_i$, where $i \in \set {1, 2, \ldots, n}$.

Let $H_n: Z^n \to \R$ be a mapping which is to be defined as the uncertainty of $Z$.

${H_n}$ fulfils the following axiom:

$\map {H_{n + 1} } {p_1, p_2, \ldots, p_n, 0} = \map {H_n} {p_1, p_2, \ldots, p_n}$

For example: a $7$-sided die which has no chance of showing a $7$, but is otherwise fair, has the same uncertainty as a fair $6$-sided die.

Sources

• 1988: Dominic Welsh: Codes and Cryptography ... (previous) ... (next): $\S 1$: Entropy = uncertainty = information: $1.1$ Uncertainty