Exponential Family

Model $\mathcal{P}\{P_\eta: \eta\in\Xi\}$ is an s-parameter exponential family iff its defined by a family of densities of the form:

$$p_\eta(x)=e^{\eta'T(x)-A(\eta)}h(x)$$

all wrt. a common domination measure $\mu$, a measure such that $P_\eta\ll\mu$ for all $\eta\in\Xi$.

• $T: \chi\rightarrow\mathbb{R}^s$ : sufficient statistic
• $h: \chi\rightarrow [0, \infty)$ : carrier density
• $\eta: \Xi\rightarrow\mathbb{R}^s$ : natural parameter
• $T: \Xi\rightarrow\mathbb{R}$ : log-partition function