We identify a fundamental pathology in the likelihood for time delay inference which challenges standard inference methods. By analysing the likelihood for time delay inference with Gaussian process light curve models, we show that it generically develops a boundary-driven "W"-shape with a global maximum at the true delay and gradual rises towards the edges of the observation window. This arises because time delay estimation is intrinsically extrapolative. In practice, global samplers such as nested sampling are steered towards spurious edge modes unless strict convergence criteria are adopted. We demonstrate this with simulations and show that the effect strengthens with higher data density over a fixed time span. To ensure convergence, we provide concrete guidance, notably increasing the number of live points. Further, we show that methods implicitly favouring small delays, for example optimisers and local MCMC, induce a bias towards larger $H_0$. Our results clarify failure modes and offer practical remedies for robust fully Bayesian time delay inference.

翻译：我们识别出时间延迟推断中似然函数的一个基本病理特性，该特性挑战了标准推断方法。通过分析采用高斯过程光变曲线模型的时间延迟推断似然，我们证明了它会普遍发展出由边界驱动的“W”形结构，其全局最大值位于真实延迟处，并向观测窗口边缘逐渐抬升。这一现象源于时间延迟估计本质上的外推性质。在实际应用中，嵌套采样等全局采样器会趋向于虚假的边缘模态，除非采用严格的收敛准则。我们通过模拟验证了这一点，并表明在固定时间跨度内数据密度越高，该效应越显著。为确保收敛，我们提供了具体指导，特别是增加活动点数。此外，我们证明了隐式偏向小延迟的方法（例如优化器和局部马尔可夫链蒙特卡洛）会导致对哈勃常数 $H_0$ 的高估偏差。我们的研究结果明确了失效模式，并为稳健的全贝叶斯时间延迟推断提供了实用补救措施。