The Annals of Statistics

Heuristics of instability and stabilization in model selection

Leo Breiman

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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.

Article information

Ann. Statist., Volume 24, Number 6 (1996), 2350-2383.

First available in Project Euclid: 16 September 2002

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Zentralblatt MATH identifier

Primary: 62H99: None of the above, but in this section

Regression subset selection predictive loss cross-validation prediction error


Breiman, Leo. Heuristics of instability and stabilization in model selection. Ann. Statist. 24 (1996), no. 6, 2350--2383. doi:10.1214/aos/1032181158.

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