Question: Is there always an equality constraint implied by a missing edge? I think this probably isn't the case, because it's not true for regular ADMGs. Sorry for answering my own question there, but hopefully the follow up is also interesting -- is there a notion of maximality that can be defined here, similar to MAGs for ordinary conditional independence and MArGs for generalized equality constraints? The latter might be quite hard, but it may be worth thinking of a maximality condition for ordinary constraints (if there isn't one already)?
Evidence: A linear Hawkes process has a very simple 'mechanistic' interpretation which is evident from both the intensity representation in Definition 1 and the cluster representation (e.g., Figure). In the intensity process of β, a function φ βα concisely describes how an α-event increases the intensity with which future β-events occur. This makes Hawkes processes attractive in settings where an analyst is not merely trying to fit a process to an observed distribution but is also interested in understanding how events in some coordinate processes create events in other coordinate processes.
Examples include social media networks, systems of interacting neurons, and alarm networks in complex engineered systems. In many applications, the observed data only include events in a proper subset of the coordinate processes of the full system.
Local independence has been used for structure learning in stochastic process models, analogously to how conditional independence is used for constraint-based structure learning in classical models. Structure learning based on local independence typically assumes faithfulness, that is, that X B is locally independent of X A given X C if and only if B is µ-separated from A given C in G, the DMG representing the model (µ-separation, or δ-separation, is a concept analogous to m-, or d-separation). The Markov equivalence class of G is the set of DMGs that encode exactly the same set of µ-separations, hence local independencies. show that each Markov equivalence class contains a unique maximal element and this graph is therefore a natural learning target when using tests of local independence.
An obvious question is therefore whether there are constraints imposed by the structure that are not described by local independence, or µ-separation. This is indeed the case and such constraints can be used to distinguish between graphical structures that imply the same set of local independencies. In the next subsection, we show that one can use the integrated mean and covariance to find such constraints. In applications, this means that the set of graphical structures that can explain the observed data may be reduced, thus we may obtain strictly more information about how coordinate processes interact.
Mogensen and Hansen (2020) show that each Markov equivalence class contains a unique maximal element and this graph is therefore a natural learning target when using tests
of local independence.
Answer: Yes, there is such a notion in the case of the ordinary conditional independence relation (for these graphs, this is (conditional) local independence, a notion very similar to Granger causality). The paper by Mogensen and Hansen which is cited in the manuscript proves that there's a unique maximal element within each Markov equivalence class (using \mu-separation instead of m-separation) of DMGs. As argued in that paper, this solves the maximality question as learning based on tests of local independence can output that unique graph. The paper also contains a method for visualizing the entire Markov equivalence class starting from the maximal element of the class. For generalized constraints, we agree that it seems difficult and we probably need to first better understand the generalized constraints.