Prediction optimism ($\mathrm{Opt}={\mathrm{Err}}_{\mathrm{in}}-\mathbb{E}\mathrm{err}$)

Prediction optimism is the difference between expected squared prediction error on new independent responses at the training covariates and expected training error. Reusing responses to fit and assess a model tends to make training error too favorable. The covariance formula for prediction optimism measures this reuse without assuming a linear fitting rule.

// Target: probability-and-statistics.bigb

With fixed covariates and independent new responses having the same marginal distributions as the training responses, expanding squared losses cancels their second moments. Independence makes $\mathbb{E}(Y_i^{\mathrm{new}}{\hat{Y}}_i)=\mathbb{E}{Y}_i\mathbb{E}{\hat{Y}}_i$, yielding the displayed identity. For a fixed linear smoother $\hat{Y}=HY$ with covariance ${\sigma }^{2}I$, it reduces to $2{\sigma }^2\mathrm{tr}(H)/n$, linking prediction optimism to effective degrees of freedom.

// Target: analysis.bigb