A CHARACTERIZATION OF THE BIVARIATE WISHART DISTRIBUTION
Dan Geiger David Heckerman
Abstract: We provide a characterization of the bivariate Wishart and normal-Wishart distributions.
Assume that has a non-singular bivariate normal pdf with
unknown mean vector and unknown precision matrix Let
where and
Similarly, define using the factorization
Assume and have a strictly positive joint pdf
Then is a normal-Wishart pdf if and only if global independence holds,
namely,
and
local independence holds, namely,
(where denotes the standardized r.v. and stands for independence). We also
characterize the bivariate pdfs that satisfy global independence alone. Such pdfs are termed
hyper-Markov laws and they are used for a decomposable prior-to-posterior analysis of
Bayesian networks.
has a non-singular bivariate normal pdf 
with
unknown mean vector 
and unknown precision matrix 
Let 
where 
and 
Similarly, define 
using the factorization
Assume 
and 
have a strictly positive joint pdf 
Then 
is a normal-Wishart pdf if and only if global independence holds,
namely,
denotes the standardized r.v. 
and 
stands for independence). We also
characterize the bivariate pdfs that satisfy global independence alone. Such pdfs are termed
hyper-Markov laws and they are used for a decomposable prior-to-posterior analysis of
Bayesian networks.