Information criteria for ridge regression

Hi -- is there any analogue or adjustment of, say, Schwartz Bayesian (or other) information criterion that would be applicable to model selection in ridge regression with a given ridge parameter $\eta$, i.e. $\hat{\beta}_\eta = (X'X+\eta I)^{-1}X'Y$?

Thanks!!

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The ridge estimator corresponds to the posterior mean under a Normal linear regression model with a conjugate Normal-inverse-gamma prior on the regression coefficients: $\beta \mid \sigma^2, \lambda \sim \mbox{N}(0, \lambda^{-1}\sigma^2 \mbox{I})$ and $\sigma^2 \sim \mbox{IG}(a,b)$ for known hyperparameters $a$ and $b$. One may additionally put a prior distribution over $\lambda$. If you consider a discrete number of possible values for $\lambda$ then one may compute posterior probabilities for each of these values or compute Bayes factors to compare different values.
Thanks! Do I understand it right that in your notation $\lambda$ is the ridge parameter ($\eta$ in the question)? I am a bit confused as to how to use these priors to construct some analogue of AIC or BIC, I am actually concerned with selecting a subset of $X$'s to use. It does look like cross-validation is the way to go, as you suggest. –  laxxy Nov 15 '10 at 2:25