Cosmological parameters encoding our understanding of the expansion history of the Universe can be constrained by the accurate estimation of time delays arising in gravitationally lensed systems. We propose TD-CARMA, a Bayesian method to estimate cosmological time delays by modelling the observed and irregularly sampled light curves as realizations of a Continuous Auto-Regressive Moving Average (CARMA) process. Our model accounts for heteroskedastic measurement errors and microlensing, an additional source of independent extrinsic long-term variability in the source brightness. The semi-separable structure of the CARMA covariance matrix allows for fast and scalable likelihood computation using Gaussian Process modeling. We obtain a sample from the joint posterior distribution of the model parameters using a nested sampling approach. This allows for ``painless'' Bayesian Computation, dealing with the expected multi-modality of the posterior distribution in a straightforward manner and not requiring the specification of starting values or an initial guess for the time delay, unlike existing methods. In addition, the proposed sampling procedure automatically evaluates the Bayesian evidence, allowing us to perform principled Bayesian model selection. TD-CARMA is parsimonious, and typically includes no more than a dozen unknown parameters. We apply TD-CARMA to six doubly lensed quasars HS 2209+1914, SDSS J1001+5027, SDSS J1206+4332, SDSS J1515+1511, SDSS J1455+1447, SDSS J1349+1227, estimating their time delays as $-21.96 \pm 1.448$, $120.93 \pm 1.015$, $111.51 \pm 1.452$, $210.80 \pm 2.18$, $45.36 \pm 1.93$ and $432.05 \pm 1.950$ respectively. These estimates are consistent with those derived in the relevant literature, but are typically two to four times more precise.

翻译：宇宙学参数编码了我们对宇宙膨胀历史的理解，可通过精确估计引力透镜系统中产生的时间延迟加以约束。我们提出TD-CARMA，一种贝叶斯方法，通过将观测到的非均匀采样光变曲线建模为连续自回归滑动平均过程的实现来估计宇宙学时间延迟。我们的模型考虑了异方差测量误差和微引力透镜效应（一种源亮度中额外独立的长期外在变异性来源）。CARMA协方差矩阵的半可分离结构允许利用高斯过程建模实现快速且可扩展的似然计算。我们采用嵌套采样方法从模型参数的联合后验分布中获取样本。这使得「无痛」贝叶斯计算成为可能，能以直接方式处理后验分布预期的多模态性，且无需像现有方法那样指定初始值或时间延迟的初始猜测。此外，所提出的采样过程自动评估贝叶斯证据，使我们能够进行原理性的贝叶斯模型选择。TD-CARMA具有简约性，通常包含不超过十几个未知参数。我们将TD-CARMA应用于六个双透镜类星体HS 2209+1914、SDSS J1001+5027、SDSS J1206+4332、SDSS J1515+1511、SDSS J1455+1447和SDSS J1349+1227，估计其时间延迟分别为$-21.96 \pm 1.448$、$120.93 \pm 1.015$、$111.51 \pm 1.452$、$210.80 \pm 2.18$、$45.36 \pm 1.93$和$432.05 \pm 1.950$。这些估计值与相关文献中的结果一致，但精度通常提高两到四倍。