将模型选择视为从复杂度参数 s 索引的预测器序列中挑选最小二乘预测器。定义不稳定性为数据微小变化导致预测器大幅变化，并用“预测损失”度量水晶球最优选择与统计学家实际选择之间的预测误差差。通过简单情形的解析和四种线性回归方法的模拟比较来展示不稳定性的影响，并提出扰动数据后平均多个预测器序列的稳定化策略。
In model selection, usually a "best" predictor is chosen from a collection ${\hat{\mu}(\cdot, s)}$ of predictors where $\hat{\mu}(\cdot, s)$ is the minimum least-squares predictor in a collection $\mathsf{U}_s$ of predictors. Here s is a complexity parameter; that is, the smaller s, the lower dimensional/smoother the models in $\mathsf{U}_s$. If $\mathsf{L}$ is the data used to derive the sequence ${\hat{\mu}(\cdot, s)}$, the procedure is called unstable if a small change in $\mathsf{L}$ can cause large changes in ${\hat{\mu}(\cdot, s)}$. With a crystal ball, one could pick the predictor in ${\hat{\mu}(\cdot, s)}$ having minimum prediction error. Without prescience, one uses test sets, cross-validation and so forth. The difference in prediction error between the crystal ball selection and the statistician's choice we call predictive loss. For an unstable procedure the predictive loss is large. This is shown by some analytics in a simple case and by simulation results in a more complex comparison of four different linear regression methods. Unstable procedures can be stabilized by perturbing the data, getting a new predictor sequence ${\hat{\mu'}(\cdot, s)}$ and then averaging over many such predictor sequences.
In model selection, usually a "best" predictor is chosen from a collection μ^(⋅,s) of predictors where μ^(⋅,s) is the minimum least-squares predictor in a collection Us of predictors. Here s is a complexity parameter; that is, the smaller s, the lower dimensional/smoother the models in Us. If L is the data used to derive the sequence μ^(⋅,s), the procedure is called unstable if a small change in L can cause large changes in μ^(⋅,s). With a crystal ball, one could pick the predictor in μ^(⋅,s) having minimum prediction error. Without prescience, one uses test sets, cross-validation and so forth. The difference in prediction error between the crystal ball selection and the statistician's choice we call predictive loss. For an unstable procedure the predictive loss is large. This is shown by some analytics in a simple case and by simulation results in a more complex comparison of four different linear regression methods. Unstable procedures can be stabilized by perturbing the data, getting a new predictor sequence μ′^(⋅,s) and then averaging over many such predictor sequences.