Localized Conformal Prediction: A Generalized Inference Framework for Conformal Prediction

We propose a new inference framework called localized conformal prediction. It generalizes the framework of conformal prediction and offers a single-test-sample adaptive construction by emphasizing a local region around it. Although there have been methods constructing heterogeneous prediction intervals for $Y$ by designing better conformal score functions, to our knowledge, this is the first work that introduces an adaptive nature to the inference framework itself. We prove that our proposal leads to an assumption-free and finite sample marginal coverage guarantee, as well as an approximate conditional coverage guarantee. Our proposal achieves asymptotic conditional coverage under suitable assumptions. The localized conformal prediction can be combined with many existing works in conformal prediction, including different types of conformal score constructions. We will demonstrate how to change from conformal prediction to localized conformal prediction in these related works and a potential gain via numerical examples.