Maximum-Likelihood Estimation of Hyperedge-Triggered Hawkes Processes via a Closed-Form EM Algorithm

Hypergraph effects in event streams are difficult to estimate because a group-level burst can often be explained either by direct higher-order excitation or by a collection of ordinary pairwise Hawkes interactions.

This paper studies maximum-likelihood estimation for a hyperedge-triggered Hawkes process, in which the conditional intensity is excited both by individual past events and by the completion of a multi-node firing pattern within a short temporal window.

We derive a closed-form EM algorithm based on latent branching responsibilities and a piecewise compensator for the most-recent-anchor hyperedge mechanism.

The compensator corrects the naive integral that overcounts superseded pattern completions.

For independently parameterised candidate hyperedges, the EM updates are closed form; when a low-rank CP parameterisation is imposed, the hyperedge factors are updated by block-coordinate ascent on the same expected complete-data objective, yielding a generalised EM implementation.

Synthetic experiments show near-unbiased recovery under a time-rescaling-validated simulator, stable EM convergence, identifiable trigger-window structure, and the expected O(n^2) event-count scaling of the prototype implementation.

The main statistical limitation is not numerical optimisation but identifiability: when pairwise and hyperedge components are supported on the same co-firing events, likelihood gains can be hard to attribute.

Held-out analyses on retina and primary visual-cortex spike-train datasets show stable positive candidate-count BIC differences for the two cortical datasets and more fragile evidence for the retina dataset as the candidate set expands.

Code and reproducibility scripts are available at this https URL.