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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 18, Fasc. 1,
pages 119 - 131
 

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 x = (x ,x)
      1  2 has a non-singular bivariate normal pdf f(x) = N (m,W ) with unknown mean vector m and unknown precision matrix W. Let f (x) = f(x)f(x |x),
          1   2  1 where f (x ) = N(m  ,1/v )
   1        1   1 and f(x | x ) = N (m + b x ,1/v  ).
   2 1       2|1   12 1    2|1 Similarly, define (v,v  ,b ,m  ,m  )
 2  1|2  21  2  1|2 using the factorization f(x) = f (x )f(x | x ).
         2    1 2 Assume m and W have a strictly positive joint pdf f   (m,W ).
 mW Then f
 m,W  is a normal-Wishart pdf if and only if global independence holds, namely,

(v,m  ) _L  (v ,b  ,m   )  and  (v,m  ) _L  (v ,b ,m   ),
  1  1     2|1  12  2|1         2   2     1|2  21  1|2
and local independence holds, namely,

    *  *      *   *   *             *  *      *   *   *
 _L  (v1,m1), _L  (v2|1,b12,m 2|1) and   _L  (v2,m2), _L  (v1|2,b21,m 1|2),
(where  *
x denotes the standardized r.v. x and  _L  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.

2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;

Key words and phrases: -

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