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.

我们旨在基于多元 Hawkes 过程明确地建模滞后的 Granger 因果效应。通过模拟滞后参数的后验分布来推断时间滞后，我们进一步研究了一个复杂环境下的模型估计方法，该方法对于事件预测和时间滞后推断的准确性在合成和真实数据上取得了良好的结果。