> [!tldr] > A collection of probabilistic distributions parametrized by $\theta$ is an **exponential family** if all its members has the form $f(x;\theta)=h(x)\cdot \exp[\pmb{\eta}(\theta)\cdot \mathbf{T}(x)-B(\theta)]$where $\pmb{\eta},\mathbf{T}$ are vector-valued functions with some shared length $k$. For more detailed account, see [[Exponential Families]]. ### The Cumulant Generating Function For the exponential family, their means $\mu$ are related to the parameter $\theta$ as in $\mu=(\gamma')^{-1}(\theta)$where $\gamma$ is the cumulant generating function of the distribution. This relation defines the canonical link function of the (subfamily) of distributions when used in [[Generalized Linear Models|GLMs]].