What is evaluation under covariate shift and how is importance weighting used?

Covariate shift evaluation adjusts metrics when P(X) changes between train and test using importance weighting.

Covariate shift occurs when the input distribution P(X) changes but the conditional distribution P(Y|X) remains stable. Standard evaluation becomes biased because the test distribution no longer matches training. Importance weighting corrects this by reweighting each sample using w(x)=P_test(x)/P_train(x). This allows unbiased estimation of performance under the new distribution, but requires accurate density ratio estimation, which is often the hardest part in practice.