= Prediction-risk bound for kernel ridge regression
For the unscaled objective $\sum_i(Y_i-f(x_i))^2+\lambda\|f\|_{\mathcal H}^2$, the <kernel-ridge hat matrix> is $H=K(K+\lambda I)^{-1}$. Its prediction variance is $\sigma^2\operatorname{tr}(H^2)/n$. For eigenvalues $d_i\geq0$, use $d_i^2/(d_i+\lambda)^2\leq\min\{d_i/(4\lambda),1\}$ and $\lambda^2d_i/(d_i+\lambda)^2\leq\lambda/4$ to bound the variance and the squared bias. The latter also uses the <reproducing property> and <Bessel inequality>, and does not require invertibility of $K$.
> EXISTING BODY EXPANSION, not a new concept: replace the present one-direction definition of Faithfulness of a directed acyclic graph in probability-and-statistics.bigb, under Causal directed acyclic graph, with the following body. Retain its existing header and parent.
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