Deep learning models have significantly improved prediction accuracy in various fields, gaining recognition across numerous disciplines. Yet, an aspect of deep learning that remains insufficiently addressed is the assessment of prediction uncertainty. Producing reliable uncertainty estimators could be crucial in practical terms. For instance, predictions associated with a high degree of uncertainty could be sent for further evaluation. Recent works in uncertainty quantification of deep learning predictions, including Bayesian posterior credible intervals and a frequentist confidence-interval estimation, have proven to yield either invalid or overly conservative intervals. Furthermore, there is currently no method for quantifying uncertainty that can accommodate deep neural networks for survival (time-to-event) data that involves right-censored outcomes. In this work, we provide a valid non-parametric bootstrap method that correctly disentangles data uncertainty from the noise inherent in the adopted optimization algorithm, ensuring that the resulting point-wise confidence intervals or the simultaneous confidence bands are accurate (i.e., valid and not overly conservative). The proposed ad-hoc method can be easily integrated into any deep neural network without interfering with the training process. The utility of the proposed approach is illustrated by constructing simultaneous confidence bands for survival curves derived from deep neural networks for survival data with right censoring.

深度学习模型在各个领域显著提高了预测准确性，并在许多学科中得到认可。然而，有一个深度学习方面仍未充分解决的问题是预测不确定性的评估。本研究提供了一种有效的非参数自助法，能够正确区分数据不确定性和所采用的优化算法中固有的噪音，确保生成的逐点置信区间或同时置信带是准确的（即有效且不过于保守）。该提议的方法可以轻松整合到任何深度神经网络中，而无需干扰训练过程。该方法的实用性通过为具有右删失的生存数据的深度神经网络构建同时置信带来进行了说明。

基于深度学习的置信区间和同时置信带