Precise measurements of the black hole mass are essential to gain insight on the black hole and host galaxy co-evolution. A direct measure of the black hole mass is often restricted to nearest galaxies and instead, an indirect method using the single-epoch virial black hole mass estimation is used for objects at high redshifts. However, this method is subjected to biases and uncertainties as it is reliant on the scaling relation from a small sample of local active galactic nuclei. In this study, we propose the application of conformalised quantile regression (CQR) to quantify the uncertainties of the black hole predictions in a machine learning setting. We compare CQR with various prediction interval techniques and demonstrated that CQR can provide a more useful prediction interval indicator. In contrast to baseline approaches for prediction interval estimation, we show that the CQR method provides prediction intervals that adjust to the black hole mass and its related properties. That is it yields a tighter constraint on the prediction interval (hence more certain) for a larger black hole mass, and accordingly, bright and broad spectral line width source. Using a combination of neural network model and CQR framework, the recovered virial black hole mass predictions and uncertainties are comparable to those measured from the Sloan Digital Sky Survey. The code is publicly available at https://github.com/yongsukyee/uncertain_blackholemass.

翻译：黑洞质量的精确测量对于理解黑洞与宿主星系的协同演化至关重要。直接测量黑洞质量通常仅限于最近的星系，而对于高红移天体，则采用基于单历元维里黑洞质量估计的间接方法。然而，该方法依赖于一小部分本地活动星系核的标度关系，因此存在偏差和不确定性。在本研究中，我们提出应用共形分位数回归（CQR）来量化机器学习场景中黑洞预测的不确定性。我们将CQR与各种预测区间技术进行比较，证明CQR能够提供更有用的预测区间指标。与预测区间估计的基线方法相比，我们展示了CQR方法提供的预测区间会随黑洞质量及其相关性质进行调整。即，对于更大的黑洞质量，以及相应的明亮、宽光谱线源，该方法对预测区间施加了更严格的约束（从而更具确定性）。通过结合神经网络模型和CQR框架，恢复的维里黑洞质量预测及其不确定性与斯隆数字巡天测量的结果相当。代码已公开在https://github.com/yongsukyee/uncertain_blackholemass。