On distributions whose conditional distributions are (multivariate) normal with applications : a vector space approach

Kurzfassung/Abstract

Let $X$ and $Y$ be two random vectors taking values in the real finite-dimensional inner product spaces $V$ and $W$, respectively. We determine the class of all possible joint distributions of $X$ and $Y$ on the vector space $V\oplus W$ such that conditional distributions of $X$ given $Y=w$ for all $w\in W$ and $Y$ given $X=v$ for all $v\in V$ are normal. Herefrom we can prove characterizations of the multivariate normal distribution on $V \oplus W$ by its conditional distributions. Further the result is applied to Bayesian analysis: We determine the largest class of conjugate priors under which the sampling distributions and the posterior distributions are normal. Moreover, exact formulas are given, showing how the posterior distribution depends on the sampling distributions and on the prior. The solutions of a functional equation form the basis of the results given in this paper.