A Characterization of the Bivariate Normal-Wishart Distribution

We provide a new characterization of the Bivariate normal-Wishart distribution. Let ~ x = fx 1 x 2 g have a non-singular Bivariate normal pdf f(~ x) = N(~ W) with unknown mean vector ~ and unknown precision matrix W. L e t f(~ x) = f(x 1)f(x 2 jx 1) where f(x 1) = N(m 1 1=v 1) and f(x 2 jx 1) = N(m 2j1 +b 12 x 1 1=v 2j1). Similarly, deene fv 2 v 1j2 b 21 m 2 m 1j2 g using the factorization f(~ x) = f(x 2)f(x 1 jx 2). Assume ~ and W have a strictly positive joint pdf f ~ W (~ W). Then f ~ W is a normal-Wishart pdf if and only if global independence holds, namely, fv 1 m 1 g?fv 2j1 b 12 m 2j1 g and fv 2 m 2 g?fv 1j2 b 21 m 1j2 g and local independence holds, namely, ?fv 1 m 1 g, ?fv 2j1 b 12 m 2j1 g and ?fv 2 m 2 g, ?fv 1j2 b 21 m 1j2 g (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.