Let $H$ be a function of finite sequences of probabilities (non-negative numbers summing up to 1) into real numbers, such that:

- $H$ is continuous,
- $H$ is symmetric w.r.t. the order of its arguments,
- $H$ is
*additive*, i.e. $H(P \otimes Q) = H(P) + H(Q)$,

where $P = \{p_i\}_i$, $Q = \{q_i\}_i$, $P \otimes Q = \{p_i q_j\}_{i,j}$. (A physicist would call the last property *extensive*.)

One class of examples is Rényi entropy $H_q$, along with linear combinations of $H_q$ (for different $q$s), what was already pointed out in the original paper:

- Alfréd Rényi, On Measures of Entropy and Information (1961) [pdf here]

Which other, if any, functions $H$ fulfill the above postulates?

seamedto me that is need to have form $\prod_q \sum_i p_i^q$ (to make it symmetric and additive), but I don't know how to actually prove it. – Piotr Migdal Mar 1 at 15:48