Cross validation

September 5, 2016 — May 13, 2021

estimator distribution
linear algebra
model selection
Figure 1

On substituting simulation for analysis in model selection, in e.g. choosing the “right” regularisation parameter for sparse regression.

The computationally expensive default option when your model doesn’t have any obvious short cuts for complexity regularization, for example when AIC cannot be shown to work.

To learn: how this interacts with Bayesian inference.

1 Basic Cross Validation


2 Generalised Cross Validation

Why the name? It’s specialised cross-validation, AFAICS (Andrews 1991; Golub, Heath, and Wahba 1979; Li 1987).

🏗 Hat matrix, smoother matrix. Note comparative computational efficiency. Define hat matrix.

3 Bayesian Cross validation


4 What even is cross validation?

I always thought the answer here was simple: It is asymptotically equivalent to generalised Akaike information criteria. (e.g. Stone (1977)) Related to bootstrap in various ways.

But there is other stuff going on. Here is an interesting sampling of opinions: Rob Tibshirani, Yuling Yao, and Aki Vehtari on cross validation.

4.1 Testing leakage

The vtreat introduction mentions their why you need hold-out article and also (Perlich and Świrszcz 2011):

Cross-methods such as cross-validation, and cross-prediction are effective tools for many machine learning, statistics, and data science related applications. They are useful for parameter selection, model selection, impact/target encoding of high cardinality variables, stacking models, and super learning. As cross-methods simulate access to an out of sample data set the same the original data, they are more statistically efficient, lower variance, than partitioning training data into calibration/training/holdout sets. However, cross-methods do not satisfy the full exchangeability conditions that full hold-out methods have. This introduces some additional statistical trade-offs when using cross-methods, beyond the obvious increases in computational cost.

Specifically, cross-methods can introduce an information leak into the modeling process.

5 References

Andrews. 1991. Asymptotic Optimality of Generalized CL, Cross-Validation, and Generalized Cross-Validation in Regression with Heteroskedastic Errors.” Journal of Econometrics.
Bates, Hastie, and Tibshirani. n.d. “Cross-Validation: What Does It Estimate and How Well Does It Do It?”
Bürkner, Gabry, and Vehtari. 2020. Approximate Leave-Future-Out Cross-Validation for Bayesian Time Series Models.” Journal of Statistical Computation and Simulation.
———. 2021. Efficient Leave-One-Out Cross-Validation for Bayesian Non-Factorized Normal and Student-t Models.” Computational Statistics.
Fong, and Holmes. 2020. On the Marginal Likelihood and Cross-Validation.” Biometrika.
Giordano, Jordan, and Broderick. 2019. A Higher-Order Swiss Army Infinitesimal Jackknife.” arXiv:1907.12116 [Cs, Math, Stat].
Giordano, Stephenson, Liu, et al. 2019. A Swiss Army Infinitesimal Jackknife.” In AISTATS.
Golub, Heath, and Wahba. 1979. Generalized Cross-Validation as a Method for Choosing a Good Ridge Parameter.” Technometrics.
Hall, Racine, and Li. 2004. Cross-Validation and the Estimation of Conditional Probability Densities.” Journal of the American Statistical Association.
Li. 1987. Asymptotic Optimality for \(C_p, C_L\), Cross-Validation and Generalized Cross-Validation: Discrete Index Set.” The Annals of Statistics.
Perlich, and Świrszcz. 2011. On Cross-Validation and Stacking: Building Seemingly Predictive Models on Random Data.” ACM SIGKDD Explorations Newsletter.
Polley. 2010. Super Learner In Prediction.” U.C. Berkeley Division of Biostatistics Working Paper Series.
Sivula, Magnusson, and Vehtari. 2020a. Unbiased Estimator for the Variance of the Leave-One-Out Cross-Validation Estimator for a Bayesian Normal Model with Fixed Variance.” arXiv:2008.10859 [Stat].
———. 2020b. Uncertainty in Bayesian Leave-One-Out Cross-Validation Based Model Comparison.” arXiv:2008.10296 [Stat].
Stone. 1977. An Asymptotic Equivalence of Choice of Model by Cross-Validation and Akaike’s Criterion.” Journal of the Royal Statistical Society. Series B (Methodological).
van der Laan, Polley, and Hubbard. 2007. Super Learner.” Statistical Applications in Genetics and Molecular Biology.
Wood. 1994. Monotonic Smoothing Splines Fitted by Cross Validation.” SIAM Journal on Scientific Computing.
Yao, Vehtari, Simpson, et al. 2018. Using Stacking to Average Bayesian Predictive Distributions.” Bayesian Analysis.