nLab collision entropy

Definition

(…)

The collision entropy of a probability distribution on a finite set is the Rényi entropy at order 2:

$$S_2(p) \;=\; -\log \left( \sum_{i=1}^n p_i^2 \right) \,,$$

hence is the negative logarithm of the “collision probability”, i.e., of the probability that two independent random variables, both described by $p$, will take the same value.

Related concepts

order		$\phantom{\to} 0$		$\to 1$		$\phantom{\to}2$		$\to \infty$
Rényi entropy		Hartley entropy	$\geq$	Shannon entropy	$\geq$	collision entropy	$\geq$	min-entropy

References

• G. M. Bosyk, M. Portesi, A. Plastino, Collision entropy and optimal uncertainty, Phys. Rev. A 85 (2012) 012108 (arXiv:1112.5903)

See also:

• Wikipedia, Collision entropy