Cross-validation: what does it estimate and how well does it do it?
This addresses a fundamental misunderstanding in statistical learning for researchers and practitioners, clarifying the limitations of cross-validation and improving error estimation reliability.
The paper proves that cross-validation does not estimate the prediction error for the specific model fit to the training data but rather the average error across models from other training sets, and it introduces a nested cross-validation scheme to correct confidence intervals that often have poor coverage due to correlations.
Cross-validation is a widely-used technique to estimate prediction error, but its behavior is complex and not fully understood. Ideally, one would like to think that cross-validation estimates the prediction error for the model at hand, fit to the training data. We prove that this is not the case for the linear model fit by ordinary least squares; rather it estimates the average prediction error of models fit on other unseen training sets drawn from the same population. We further show that this phenomenon occurs for most popular estimates of prediction error, including data splitting, bootstrapping, and Mallow's Cp. Next, the standard confidence intervals for prediction error derived from cross-validation may have coverage far below the desired level. Because each data point is used for both training and testing, there are correlations among the measured accuracies for each fold, and so the usual estimate of variance is too small. We introduce a nested cross-validation scheme to estimate this variance more accurately, and we show empirically that this modification leads to intervals with approximately correct coverage in many examples where traditional cross-validation intervals fail.