A CHARACTERIZATION OF SOME CONJUGATE PRIOR DISTRIBUTIONS FOR EXPONENTIAL-FAMILIES

The random variable X has a continuous distribution in a one-parameter exponential family. After a random sample of observations is collecte d, the next X will be predicted. A twofold constraint on the prior dis tribution exists: first, the posterior predictive mean must lie betwee n the sample and prior predictive means. and second, for any set of da ta resulting in an updated prior, the first constraint must once again apply. Subject to regularity conditions, this constraint implies that the prior distribution is conjugate.