A Characterization of the Bivariate Normal-Wishart Distribution
Dan Geiger, David E. Heckerman · 1995
We provide a new characterization of the Bivariate normal-Wishart distribution. Let ~x = fx1�x2g have a non-singular Bivariate normal pdf f(~x) =N(~ � W) with unknown mean vector ~ and unknown precision matrix W.Letf(~x) =f(x1)f(x2jx1) where f(x1) = N(m1 � 1=v1) and f(x2jx1)=N(m2j1+b12x1 � 1=v2j1). Similarly, de ne fv2�v1j2�b21�m2�m1j2g using the factorization f(~x) =f(x2)f(x1jx2). 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, fv1�m1g?fv2j1�b12�m2j1g and fv2�m2g?fv1j2�b21�m1j2g and local independence holds, namely,?fv1�m1g,?fv �b 2j1 12�m g and?fv