Questions tagged [exponential-family]

An exponential family distribution has the following form, $p(x|\eta)=h(x)e^{\eta^\top t(x)-a(\eta)}$.

The different parts of this equation are:

• The natural parameter $\eta$
• The sufficient statistic $t(x)$
• The underlying measure $h(x)$, e.g., counting measure or Lebesgue measure
• The log normalizer $a(\eta)$, $a(\eta)=\log\int h(x)e^{\eta^\top t(x)}$.

The statistic $t(x)$ is called sufficient because the likelihood for $\eta$ only depends on $x$ through $t(x)$.

The exponential family has fundamental connections to the world of graphical models.