====== Wishart prior in WinBUGS/OpenBUGS ======

For a multilevel model, we specify the random coefficients as a multivariate normal distribution such as
\[
\left(\begin{array}{c}
L_{i}\\
S_{i}\end{array}\right)\sim MN\left[\left(\begin{array}{c}
\beta_{L}\\
\beta_{S}\end{array}\right),\Sigma^{-1}\right]
\]

The prior for the precision matrix $\Sigma^{-1}$ is given a Wishart
distribution in BUGS. 


\[\Sigma^{-1}\sim W(R[1:2,1:2],m)\]

Here $m$ is the degrees of freedom (scalar) that is equivalent prior
sample size and must be greater than or equal to dimension of matrix
in order for Wishart to be proper. $m$ determines the degree of certainty
you have about the mean.

In WinBUGS, the specification indicate that the mean of the covariance matrix (not the precision matrix) is

\[E(\Sigma^{-1})=mR^{-1}\]

In WinBUGS notation, if the precision matrix $P=\Sigma^{-1}$ follows
a Wishart distribution $W(R,m)$, then

\[E(P_{ij})=mR_{ij}^{-1}\]

\[Var(P_{ij})=m(R_{ij}^{-2}+R_{ii}^{-1}R_{jj}^{-1})\]

\[Cov(P_{ij},P_{kl})=m(R_{ik}^{-1}R_{jl}^{-1}+R_{il}^{-1}R_{jk}^{-1})\]

==== An intuitive way to specify the Wishart prior ====
  * Let $S$ equal the prior guess for the mean of the $p\times p$ variance/covariance matrix $\Sigma$.
  * Choose a degrees-of-freedom parameter $m (> p + 1)$ that roughly represents an “equivalent prior sample size” – your belief in $S$ as the value of $\Sigma$ is as strong as if you had seen $m$ previous vectors with sample covariance matrix $S$.
  * Define a matrix $R=(m-p-1)S$.
  * In WinBUGS, put the following Wishart prior on the corresponding precision matrix $P=\Sigma^{-1}$  <code>Inv_sig~dwish(R, m)</code>
  * Based on this specification, 
     * $E(\Sigma) = S $ 
     * $E(\Sigma^{-1}) = \frac{m}{m-p-1}S^{-1}$
     * the variance of the prior will be decreasing in $m$


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