From Type to Token: Petri Nets, Individual Token Philosophy, and Individualized Causal Effects

Estimating causal effects from observational network data requires accounting for both individual treatment and neighborhood exposure. Existing outcome models commonly incorporate these intervention variables into a joint predictor and estimate causal effects through prediction contrasts, leaving the corresponding response components implicit. We propose the $\underline{N}$etwork $\underline{E}$ffect $\underline{D}$ecomposition $\underline{Net}$work (NEDNet), which parameterizes the conditional outcome response as a query-invariant pre-treatment baseline plus separate treatment- and exposure-response pathways. Path-specific neighborhood contextualization supplies local pre-treatment information to the two pathways. We characterize the fitted response through a unique reference-normalized decomposition, which is equivalent to a zero mixed treatment--exposure contrast. Experiments on semi-synthetic benchmarks derived from BlogCatalog and Flickr show that NEDNet achieves the lowest PEHE among the compared methods for individual, peer, and total effects across homogeneous and covariate-heterogeneous response settings.

Individual treatment effects are not point-identified from data. The Probability of Necessity and Sufficiency (PNS) circumvents this limitation by characterizing individual-level causality through intersection bounds derived from combined experimental and observational data. In finite samples, however, standard plug-in estimators systematically fail: they violate structural probability constraints and suffer from extremum bias induced by max-min operators, yielding spuriously narrow intervals. We propose a neural framework for finite-sample PNS estimation that resolves both pathologies. We introduce an anchored neural architecture that guarantees structural constraint satisfaction by construction. To correct extremum bias, we employ precision-corrected intersection-bound inference, leveraging Epistemic Neural Networks for scalable, high-dimensional uncertainty quantification. Empirical evaluations confirm that this approach maintains nominal coverage and exact constraint validity in high-dimensional regimes where standard estimators systematically undercover.

In this paper, the relationship between probabilistic graphical models, in particular Bayesian networks, and causal diagrams, also called structural causal models, is studied. Structural causal models are deterministic models, based on structural equations or functions, that can be provided with uncertainty by adding independent, unobserved random variables to the models, equipped with probability distributions. One question that arises is whether a Bayesian network that has obtained from expert knowledge or learnt from data can be mapped to a probabilistic structural causal model, and whether or not this has consequences for the network structure and probability distribution. We show that linear algebra and linear programming offer key methods for the transformation, and examine properties for the existence and uniqueness of solutions based on dimensions of the probabilistic structural model. Finally, we examine in what way the semantics of the models is affected by this transformation. Keywords: Causality, probabilistic structural causal models, Bayesian networks, linear algebra, experimental software.