High-probability large-sample coverage for DNN-based calibration prediction intervals

Establish a high-probability large-sample conditional coverage guarantee for calibration prediction intervals constructed with deep neural network estimators: specifically, prove that there exists an integer N such that for all sample sizes n ≥ N, with probability approaching 1 over the training sample, the DNN-based calibration prediction intervals achieve conditional coverage at least 1−α for any fixed covariate value X_f = x_f, analogous to the guarantee proved for the kernel-based calibration prediction intervals.

Background

The paper develops calibration prediction intervals (cPI) using both deep neural network (DNN) estimators and kernel estimators. It provides asymptotic validity for both approaches and proves a non-asymptotic, high-probability large-sample coverage guarantee for the kernel-based cPI.

For the DNN-based cPI, the authors do not provide a rigorous proof of a corresponding high-probability large-sample guarantee. Instead, they conjecture that the same property should hold for DNN-based cPI, motivated by its role as an intermediate guarantee between the established finite-sample results (under strong conditions) and the asymptotic validity that they prove.

Resolving this conjecture would close the theoretical gap between the kernel and DNN variants by providing a non-asymptotic coverage guarantee for the DNN-based cPI comparable to the kernel-based result.