A popular strategy is to choose a penalty parameter that yields a good but parsimonious model. Information criteria measure the balance between model fit and model complexity. One possibility is Aikaike's information criterion (AIC). The AIC measures model fit by the log-likelihood and model complexity is measured by the number of parameters used by the model. The number of model parameters in regular regression simply corresponds to the number of covariates in the model. Or, by the degrees of freedom consumed by the model, which is equivalent to the trace of the hat matrix. For ridge regression it thus seems natural to define model complexity analogously by the trace of the ridge hat matrix. This yields the AIC for the linear regression model with ridge estimates: $$ \begin{align*} \mbox{AIC}(\lambda) & = 2 \, p - 2 \log(\hat{L}) \\ & = 2 \, \mbox{tr} [\mathbf{H}(\lambda)] - 2 \log\{L[\hat{\beta}(\lambda), \hat{\sigma}^2(\lambda)]\} \\ & = 2 \, \sum_{j=1}^p \frac{d_{jj}^2}{d_{jj}^2 + \lambda} + 2 n \, \log[\sqrt{2 \, \pi} \, \hat{\sigma}(\lambda)] + \frac{1}{\hat{\sigma}^2(\lambda)} \sum_{i=1}^n [y_i - \mathbf{X}_{i, \ast} \, \hat{\beta}(\lambda)]^2. \end{align*} $$ The value of \( \lambda \) which minimizes \( \mbox{AIC}(\lambda) \) corresponds to the `optimal' balance of model complexity and overfitting.