Cumulant function of an exponential family

The cumulant function, or log-partition function, is the normalizing termwith a sum in the discrete case.

Differentiating the normalization identity gives

The mean parameter corresponding to the natural parameter $\theta$ is

For a finite support whose convex hull has full dimension, every interior target mean is realized by a unique natural parameter. Minimize the coercive function $\log {\sum }_n{e}^{\theta \cdot n}-\theta \cdot \mu$; its gradient is the mean discrepancy and its Hessian matrix is the positive-definite covariance matrix of the support.