= Wishart distribution
{c}
{title2=$W_p(\nu,V)$}
{wiki}
For integer $\nu$, the sum $\sum_{j=1}^{\nu}X_jX_j^T$ 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 $|W|^{(\nu-p-1)/2}\exp[-\operatorname{tr}(V^{-1}W)/2]$. 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 https://r-nimble.org/manual/cha-writing-models.html[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.
Back to article page
