Tytuł pozycji:
A characterization of the bivariate wishart distribution
We provide a characterization of the bivariate Wishart
and normal-Wishart distributions. Assume that x = {x1,x} has
a non-singular bivariate normal pdf f(x) = N (μ, W) with unknown
mean vector fi and unknown precision matrix W. Let f(x)= f(x1)f(x2|x), where f(x1) = N{m1 1/ν1 and f(x2 | x1) =
N{m2|1 + b12x1 l/ν2|1). Similarly, define {ν2, b21,m2, m1|2}
using the factorization f(x)=f(x2)f(x1|x2)- Assume μ and W have a strictly positive joint pdf fμw(μW). Then fμw is a normal-Wishart pdf if and only if global independence holds, namely,…[formula]
and local independence holds, namely,
[formula]
(where x* denotes the standardized r.v. x 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.
