Predicting future events always comes with uncertainty, but traditional non-Bayesian methods cannot distinguish certain from uncertain predictions or explain the confidence in their predictions. In survival analysis, Bayesian methods applied to state-of-the-art solutions in the healthcare and biomedical field are still novel, and their implications have not been fully evaluated. In this paper, we study the benefits of modeling uncertainty in deep neural networks for survival analysis with a focus on prediction and calibration performance. For this, we present a Bayesian deep learning framework that consists of three Bayesian network architectures, which we train by optimizing the Cox partial likelihood and combining input-dependent aleatoric uncertainty with model-specific epistemic uncertainty. This enables us to provide uncertainty estimates as credible intervals when predicting the survival curve or as a probability density function over the predicted median survival times. For our empirical analyses, we evaluated our proposed method on four benchmark datasets and found that our method demonstrates prediction performance comparable to the state-of-the-art based on the concordance index and outperforms all other Cox-based approaches in terms of the mean absolute error. Our work explicitly compares the extent to which different Bayesian approximation techniques differ from each other and improves the prediction over traditional non-Bayesian alternatives.

翻译：预测未来事件必然伴随不确定性，但传统非贝叶斯方法无法区分确定性与不确定性预测，也无法解释预测结果的置信度。在生存分析领域，针对医疗与生物医学领域前沿解决方案的贝叶斯方法仍属新兴方向，其应用价值尚未得到充分评估。本文重点研究深度神经网络中不确定性建模对生存分析预测性能与校准能力的影响。为此，我们提出一个包含三种贝叶斯网络架构的深度学习框架，通过优化Cox部分似然函数进行训练，并将输入相关的偶然不确定性（aleatoric uncertainty）与模型特定的认知不确定性（epistemic uncertainty）相结合。这使得我们能够在预测生存曲线时提供可信区间形式的不确定性估计，或对预测中位生存时间生成概率密度函数形式的估计。在实证分析中，我们在四个基准数据集上评估了所提方法，发现该方法在一致性指数（concordance index）方面展现出与现有最优方法相当的预测性能，并在平均绝对误差（mean absolute error）指标上优于所有其他基于Cox的方法。本研究明确比较了不同贝叶斯近似技术之间的差异程度，并证明该预测效果优于传统非贝叶斯方法。