One-Dimensional Exponential Family of Densities

When we have a random sample of observations 2/1, -. -, j/n from a density f y ) and we have a prior density g(0), the posterior distribution is given by Bayes theorem 

Understanding Computational Bayesian Statistics. By William M. Bolstad Copyright 2010 John Wiley Sons. Inc. 

The density f y ff) is a member of the one-dimensional exponential family of densities if and only if the observation density function can be written 

The conjugate family. The likelihood function has the same formula as the observation density, only with the observation held fixed and the parameter varying over all possible values. For a one-dimensional exponential family likelihood, we can absorb the factor B y) into the constant of proportionality since it is only a scale factor and does not affect the shape. The conjugate family of priors for a member of the one-dimensional exponential family of densities has the same form as the likelihood. It is given by 

We recognize this is the member of the conjugate family with constants k and I where the new values of the constants are given by 

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Thus when the observations are from a one-dimensional exponential family, and the prior is from the conjugate family, the posterior is easily found without any need for integration. All that is needed is the simple updating formulas for the constants. We will now look at some common distributions that are members of the one-dimensional exponential family of densities. 

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