Statistica Sinica 11(2001), 409-418

CHARACTERIZATION OF CONJUGATE PRIORS

FOR DISCRETE EXPONENTIAL FAMILIES

Jine-Phone Chou

Academia Sinica, Taipei

Abstract: Let $X$ be a nonnegative discrete random variable distributed according to an exponential family with natural parameter $\theta \in \Theta$. Subject to some regularity we characterize conjugate prior measures on $\Theta$ through the property of linear posterior expectation of the mean parameter of $X:E\{E(X\vert\theta)\vert X=x\}=ax+b$. We also delineate some necessary conditions for the hyperparameters $a$ and $b$, and find a necessary and sufficient condition that $0<a<1$. Besides the power series distribution with parameter space bounded above (for example, the negative binomial distribution and the logarithmic series distribution) and the Poisson distribution, we apply the result to the log-zeta distribution and all hyper-Poisson distributions.

Key words and phrases: Bounded analytic functions, characterization theorems, conjugate priors, discrete exponential families, posterior expectations.