Normalized Exponential Probability Distributions
Definition
Suppose $\mathcal{X} $ is a finite set.
A distribution $p: \mathcal{X} \to [0,1]$ is a
normalized exponential
distribution (also Gibbs
distribution, Boltzmann
distribution) if there exists a function
$F: \mathcal{X} \to \R $ so that
\[
p(x) = \frac{\exp(-F(x))}{\sum_{\xi \in \mathcal{X} }
\exp(-F(\xi ))} \quad \text{for all } x \in \mathcal{X}
\]