[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]

The ridge estimator of parameter [math]\bbeta[/math] of the logistic regression model is the maximizer of the ridge penalized loglikelihood:

[[math]] \begin{eqnarray*} \mathcal{L}^{\mbox{{\tiny pen}}}(\mathbf{Y}, \mathbf{X}; \bbeta, \lambda) & = & \sum\nolimits_{i=1}^n \big\{ Y_i \mathbf{X}_i \bbeta - \log [ 1 + \exp(\mathbf{X}_i \bbeta) ] \big\} - \tfrac{1}{2} \lambda \| \bbeta \|_2^2. \end{eqnarray*} [[/math]]

The maximizer is found numerically by the iteratively reweighted least squares (IRLS) algorithm which is outlined in Section Ridge estimation . Modify the algorithm, as is done in , to find the generalized ridge logistic regression estimator of [math]\bbeta[/math] defined as:

[[math]] \begin{eqnarray*} \mathcal{L}^{\mbox{{\tiny pen}}}(\mathbf{Y}, \mathbf{X}; \bbeta, \lambda) & = & \sum\nolimits_{i=1}^n \big\{ Y_i \mathbf{X}_i \bbeta - \log [ 1 + \exp(\mathbf{X}_i \bbeta) ] \big\} - \tfrac{1}{2} (\bbeta - \bbeta_0)^{\top} \mathbf{\Delta} (\bbeta - \bbeta_0), \end{eqnarray*} [[/math]]

where [math]\bbeta_0[/math] and [math]\mathbf{\Delta}[/math] are as in Chapter Generalizing ridge regression .