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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


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




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 $P=\Sigma^{-1}$
    Inv_sig~dwish(R, m)
  • 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 $m$
2010/09/17 15:36 · 0 Comments · 0 Linkbacks

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