Exponential Family

Def Exponential Family A parametric family of univariate continuous distributions is said to be an exponential family if the probability density function of any member of the family can be written aswhere

• is the vector of natural parameters;
• is base measure;
• is natural parameters.
• is sufficient statistic.
• is log-partition. Equivalently, we can also write it as:

Thrm The natural parameter space is a convex domain, and is a convex function.

Def Dual Parameter Given an exponential family , is the dual/mean/expectation parameter of the exponential family.

Prop The exponential family is conjugate.

Prop The maximum likelihood estimator of an exponential family is simply arithmetic average: Prop The fisher information matrix of an exponential family is given by