---

title: Exponential Family

date: '2026-07-20 18:18:00'

tags:

- Statistics

cover: https://img.yulecna.top/posts/yn-background.jpg

description: ''

---
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