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A characterization of the bivariate wishart distribution

Geiger D., Heckerman D.

Probability and Mathematical Statistics

1998

Vol. 18, Fasc. 1

119--131

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{m_2|1 + b₁₂x₁ l/ν_2|1). Similarly, define {ν₂, b₂₁,m₂, m_1|2} using the factorization f(x)=f(x₂)f(x₁|x₂)- Assume μ and W have a strictly positive joint pdf f_μw(μW). Then f_μw 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.