We aim to explicitly model the delayed Granger causal effects based on multivariate Hawkes processes. The idea is inspired by the fact that a causal event usually takes some time to exert an effect. Studying this time lag itself is of interest. Given the proposed model, we first prove the identifiability of the delay parameter under mild conditions. We further investigate a model estimation method under a complex setting, where we want to infer the posterior distribution of the time lags and understand how this distribution varies across different scenarios. We treat the time lags as latent variables and formulate a Variational Auto-Encoder (VAE) algorithm to approximate the posterior distribution of the time lags. By explicitly modeling the time lags in Hawkes processes, we add flexibility to the model. The inferred time-lag posterior distributions are of scientific meaning and help trace the original causal time that supports the root cause analysis. We empirically evaluate our model's event prediction and time-lag inference accuracy on synthetic and real data, achieving promising results.

翻译：我们旨在基于多元霍克斯过程显式建模延迟的葛兰杰因果效应。该思路源于因果事件通常需要一定时间才能产生效应这一事实，而研究这一时间延迟本身具有重要价值。针对所提出的模型，我们首先证明在温和条件下延迟参数的可辨识性。进一步，我们在复杂设置下探究模型估计方法，即希望推断时延的后验分布，并理解该分布如何随不同情景变化。我们将时延视为潜变量，并设计变分自编码器（VAE）算法来近似时延的后验分布。通过在霍克斯过程中显式建模时延，我们增加了模型的灵活性。推断得到的时延后验分布具有科学意义，有助于追溯原始因果时间，从而支持根因分析。我们在合成数据与真实数据上对模型的事件预测及时延推断精度进行了实证评估，取得了令人满意的结果。