# nLab collision entropy

Contents

### Context

#### Measure and probability theory

measure theory

probability theory

# Contents

## 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.

order$\phantom{\to} 0$$\to 1$$\phantom{\to}2$$\to \infty$
Rényi entropyHartley 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)