Point estimate of

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Point estimate of $\beta _g$

We approximate $\widehat{\beta}_g$ using the point estimate $\widehat{\sigma_g^2}$ and $\vec{\tau_K=1}$ :

$\displaystyle \widehat{\beta}_g$

$\displaystyle =$

$\displaystyle \arg \max_{\beta_g} p(\beta_g\vert Y,\sigma_g^2=\widehat{\sigma_g^2},\vec{\tau_K=1})$

(46)

where $p(\beta_g\vert Y,\sigma_g^2=\widehat{\sigma_g^2},\vec{\tau_K=1})$ is equation 15 with $\sigma_g^2=\widehat{\sigma_g^2}$ and $\vec{\tau_K=1}$ . The solution to this is:

$\displaystyle \widehat{\beta}_g$

$\displaystyle =$

$\displaystyle (X_{G}^T U^{-1} X_{G})^{-1}X_{G}^TU^{-1}\vec{\mu_{\beta_K}}$

(47)

with

as in equation 15, but with $S_k = (\sigma_{\beta_k}^2\Sigma_{\beta_k})+\widehat{\sigma_g^2}I$ .