# measure of information

We know that $l_i=\log \frac{1}{p_i}$ is the solution to the Shannon’s source compression problem: $\arg \min_{\{l_i\}} \sum p_i l_i$ where the minimization is over all possible code length assignments $\{l_i\}$ satisfying the Kraft inequality $\sum 2^{-l_i}\le 1$.

Also $H(p)=\log \frac{1}{p}$ is additive in the following sense. If $E$ and $F$ are two independent events with probabilities $p$ and $q$ respectively, then $H(pq)=H(p)+H(q)$.

As far as I know, mainly for these two reasons $H(p)=\log \frac{1}{p}$ is considered as a measure of information contained in a random event $E$ with probability $p>0$.

On the other hand, if we average the exponentiated lengths, $\sum p_i2^{tl_i}, t>0$, subject to the same Kraft inequality constraints, the optimal solution is $l_i=\log \frac{1}{p_i'}$ where $p_i'=\frac{p_i^{\alpha}}{\sum_k p_k^{\alpha}}, \alpha=\frac{1}{1+t}$, called Campbell’s problem.

Now $H_{\alpha}(p_i)=\log \frac{1}{p_i'}$ is also additive in the sense that $H_{\alpha}(p_i p_j)=H_{\alpha}(p_i)+H_{\alpha}(p_j)$. Moreover $H_{\alpha}(1)=0$ as in the case of Shannon’s measure.

Also note that, when $\alpha=1$, $H_1(p_i)=\log \frac{1}{p_i}$ we get back Shannon’s measure.

My question is, are these reasons suffice to call $H_{\alpha}(p_i)=\log \frac{1}{p_i'}$ a (generalized) measure of information?

I don’t know whether the dependence of measure of information of an event also on the probabilities of the other events make sense.

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That’s exactly the extension known as Renyi entropy with a normalization factor of $\frac{1}{1-\alpha}$.

$$H_\alpha(X) = \frac{1}{1-\alpha}\log\Bigg(\sum_{i=1}^n p_i^\alpha\Bigg)$$

• Rényi, Alfréd (1961). “On measures of information and entropy”. Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability 1960. pp. 547–561.