Minimal exponential family
= Minimal exponential family
An order-$p$ <exponential family> is minimal if no nonzero linear combination of its $p$ canonical <sufficient statistics> is almost surely constant under the base measure. Equivalently, its sufficient-statistic support lies in no proper affine hyperplane. Its <cumulant function> then has positive-definite <Hessian matrix> on the interior natural domain, because that Hessian is a <covariance matrix>.