 Wishart distribution

For integer $\nu$, the sum ${\sum }_{j=1}^{\nu }{X}_j{X}_j^T$ of independent $N_p(0,V)$ vectors has a Wishart distribution. Its positive-definite density extends to real $\nu >p-1$ and is proportional to $|W{|}^{(\nu -p-1)/2}\exp [-\mathrm{tr}(V^{-1}W)/2]$. It is a conjugate prior for the precision matrix of a multivariate normal distribution. Its mean is $\nu V$.

The WinBUGS declaration dwish(R, nu) uses $R=V^{-1}$; the density convention is documented in the NIMBLE BUGS-language manual. For $\nu =p=3$ it is proper, while the expected inverse matrix does not exist: that expectation requires $\nu >p+1$. Specifying a Wishart prior on precision is therefore a substantive covariance assumption.