[math] \require{textmacros} \def \bbeta {\bf \beta} \def\fat#1{\mbox{\boldmath$#1$}} \def\reminder#1{\marginpar{\rule[0pt]{1mm}{11pt}}\textbf{#1}} \def\SSigma{\bf \Sigma} \def\ttheta{\bf \theta} \def\aalpha{\bf \alpha} \def\ddelta{\bf \delta} \def\eeta{\bf \eta} \def\llambda{\bf \lambda} \def\ggamma{\bf \gamma} \def\nnu{\bf \nu} \def\vvarepsilon{\bf \varepsilon} \def\mmu{\bf \mu} \def\nnu{\bf \nu} \def\ttau{\bf \tau} \def\SSigma{\bf \Sigma} \def\TTheta{\bf \Theta} \def\XXi{\bf \Xi} \def\PPi{\bf \Pi} \def\GGamma{\bf \Gamma} \def\DDelta{\bf \Delta} \def\ssigma{\bf \sigma} \def\UUpsilon{\bf \Upsilon} \def\PPsi{\bf \Psi} \def\PPhi{\bf \Phi} \def\LLambda{\bf \Lambda} \def\OOmega{\bf \Omega} [/math]

Consider the linear regression model [math]\mathbf{Y} = \mathbf{X} \bbeta + \vvarepsilon[/math] with [math]\vvarepsilon \sim \mathcal{N}(\mathbf{0}_p, \sigma^2 \mathbf{I}_{pp})[/math]. Assume [math]\bbeta \sim \mathcal{N}(\bbeta_0, \sigma^2 \mathbf{\Delta}^{-1})[/math] with [math]\bbeta_0 \in \mathbb{R}^p[/math] and [math]\mathbf{\Delta} \succ 0[/math] and a gamma prior on the error variance. Verify (i.e., work out the details of the derivation) that the posterior mean coincides with the generalized ridge estimator defined as:

[[math]] \begin{eqnarray*} \hat{\bbeta} & = & (\mathbf{X}^{\top} \mathbf{X} + \mathbf{\Delta})^{-1} (\mathbf{X}^{\top} \mathbf{Y} + \mathbf{\Delta} \bbeta_0). \end{eqnarray*} [[/math]]