Sign up & Download
Sign in

Heuristics of instability and stabilization in model selection

by Leo Breiman
The Annals of Statistics ()


In model selection, usually a "best" predictor is chosen from a collection μ^(⋅,s){\hat{\mu}(\cdot, s)} of predictors where μ^(⋅,s)\hat{\mu}(\cdot, s) is the minimum least-squares predictor in a collection Us\mathsf{U}_s of predictors. Here s is a complexity parameter; that is, the smaller s, the lower dimensional/smoother the models in Us\mathsf{U}_s. If L\mathsf{L} is the data used to derive the sequence μ^(⋅,s){\hat{\mu}(\cdot, s)}, the procedure is called unstable if a small change in L\mathsf{L} can cause large changes in μ^(⋅,s){\hat{\mu}(\cdot, s)}. With a crystal ball, one could pick the predictor in μ^(⋅,s){\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 μ′^(⋅,s){\hat{\mu'}(\cdot, s)} and then averaging over many such predictor sequences.

Cite this document (BETA)

Readership Statistics

122 Readers on Mendeley
by Discipline
by Academic Status
44% Ph.D. Student
8% Post Doc
7% Researcher (at an Academic Institution)
by Country
15% United States
2% France
2% Sweden

Sign up today - FREE

Mendeley saves you time finding and organizing research. Learn more

  • All your research in one place
  • Add and import papers easily
  • Access it anywhere, anytime

Start using Mendeley in seconds!

Sign up & Download

Already have an account? Sign in