Bayesian Statistics: An Introduction, 4th Edition

2.11 The exponential family

2.11.1 Definition

It turns out that many of the common statistical distributions have a similar form. This leads to the definition that a density is from the one-parameter exponential family if it can be put into the form

or equivalently if the likelihood of n independent observations from this distribution is

It follows immediately from Neyman’s Factorization Theorem that is sufficient for θ given X.

2.11.2 Examples

Normal mean. If with known then

which is clearly of the above form.

Normal variance. If with θ known then we can express the density in the appropriate form by writing

Poisson distribution. In the Poisson case, we can write

Binomial distribution. ...