Granger Causal Inference in Multivariate Hawkes Processes by Minimum Message Length

TL;DR

This work targets Granger causal inference in multivariate Hawkes processes with exponential kernels by learning sparse connectivity graphs through a minimum message length (MML) principle. The authors derive a per-node MML criterion (MMLH) using the Wallace–Freeman approximation, incorporating priors on nonnegative parameters and a nodewise structure search that decouples across components. Through extensive synthetic experiments and real-world G7 bond data, MMLH demonstrates superior performance in sparse graphs on short time horizons and competitive results in mid-dense settings, often aligning with expert knowledge better than baselines like MDLH, ADM4, BIC, and AIC. The approach offers a principled balance between model complexity and data fit, with practical advantages in scenarios with limited data and potential to leverage prior structural knowledge; future work includes extending to other kernel forms and DAG-constrained structures.

Abstract

Multivariate Hawkes processes (MHPs) are versatile probabilistic tools used to model various real-life phenomena: earthquakes, operations on stock markets, neuronal activity, virus propagation and many others. In this paper, we focus on MHPs with exponential decay kernels and estimate connectivity graphs, which represent the Granger causal relations between their components. We approach this inference problem by proposing an optimization criterion and model selection algorithm based on the minimum message length (MML) principle. MML compares Granger causal models using the Occam's razor principle in the following way: even when models have a comparable goodness-of-fit to the observed data, the one generating the most concise explanation of the data is preferred. While most of the state-of-art methods using lasso-type penalization tend to overfitting in scenarios with short time horizons, the proposed MML-based method achieves high F1 scores in these settings. We conduct a numerical study comparing the proposed algorithm to other related classical and state-of-art methods, where we achieve the highest F1 scores in specific sparse graph settings. We illustrate the proposed method also on G7 sovereign bond data and obtain causal connections, which are in agreement with the expert knowledge available in the literature.

Figures (5)

• Figure 1: Blue dotted line: Upper bound for $\kappa_k$. Black solid line: Lower bound for $\kappa_k$. Red dashed line: Limit of the bounds for $k \to \infty$. Grey dots: Known values of $\kappa_k$.
• Figure 2: Cascade structure for $p=7$ and $T=200$. F1 score (blue lines) and TP-score (red lines) as functions of the uniform prior parameter $b$ (left panel) and exponential prior parameter $c$ (right panel). The x-axes are reported in log-scale.
• Figure 3: Bernoulli random structure for $p=7$ and $T=200$. F1 score (blue lines) and TP score (red lines) as functions of the uniform prior parameter $b$ (left panel) and exponential prior parameter $c$ (right panel). The vertical grey dashed lines indicate the investigated values $b=4$ and $c=0.3$, and the horizontal black dotted lines correspond to the F1 score for BIC. The x-axes are reported in log-scale.
• Figure 4: Connectivity graph derived from umar2022network (an edge is drawn if it appears in at least one of the three graphs shown in their Figure 1).
• Figure 5: Connectivity graph for ADM4 (top left panel), MDLH (top right panel), BIC (middle left panel), AIC (middle right panel), MMLH-e with $c=10^{-5}$ (bottom left panel), $c=0.3$ (bottom central panel), and $c=2.5$ (bottom right panel). The red connections are those that are in agreement with umar2022network.

Theorems & Definitions (5)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5