Tracing Individual Causal Effects With Petri Nets

From population averages to individual questions

In their November 3, 2026 preprint, “From Type to Token: Petri Nets, Individual Token Philosophy, and Individualized Causal Effects,” Fernando Delbianco, Fernando Tohmé and Gustavo Bodanza propose a two-layer framework for individualized causal reasoning. One layer represents mechanisms and interventions; the other uses reference data to quantify effects with uncertainty.

Their starting distinction is between claims about kinds of events and claims about particular events. A population-level relationship concerns an event type. An individual counterfactual concerns a specific case and what would have happened under different circumstances.

The authors acknowledge that structural causal models already express individual counterfactuals by fixing the background variables describing a particular unit. Their argument concerns emphasis: they contend that standard inference foregrounds population distributions, while their proposed process representation foregrounds individual states and histories.

Extreme treatment-effect heterogeneity motivates this argument. The authors maintain that population averages can become uninformative about particular individuals. This is a theoretical motivation, not a finding from an empirical comparison.

Making mechanisms executable

A Petri net describes a system through places, transitions and tokens. Places represent conditions or states. Transitions describe changes enabled by specified inputs. A marking records which tokens occupy which places.

Under the Individual Token Philosophy, tokens retain distinguishable identities and causal histories. The authors add an explicit representational postulate: an individual is represented by a marking, whose tokens track that individual’s features and the histories producing them. Token identity alone does not establish this connection to a real person or system.

The proposed translation starts with a Bayesian network of binary variables. A token represents an active variable. Transitions encode minimal combinations of active parents that permit a downstream variable to become active. Parent tokens are tested without being consumed, allowing one condition to support several consequences; this uses a contextual extension of ordinary Petri nets.

The construction assumes monotone support: once a parent configuration permits activation, switching additional parents on cannot remove that possibility. For the mechanisms subject to this condition, activation is also excluded when all parents are inactive.

What the authors claim about correspondence

The preprint presents two correspondence theorems. The first relates Bayesian-network d-separation, a graphical criterion for conditional independence, to the authors’ definition of flow-separation. Its assumptions include edge-effectiveness: every graphical edge must participate in a minimal activation configuration.

The second claims that positive-probability outcomes under a Bayesian-network intervention correspond to reachable outcome tokens in the translated net. This is intended as a qualitative possibility claim, not a calculation of outcome probabilities or treatment-effect magnitudes.

However, the intervention definition leaves a concrete difficulty unresolved. It changes the treatment place while preserving every other token in the individual’s marking. Consider a binary mechanism in which the outcome equals the treatment, with both initially equal to one. Setting treatment to zero leaves the existing outcome token untouched. The net therefore still records the outcome as reachable, although recomputing the mechanism under treatment zero would give outcome zero.

This example follows from the stated construction; it is not an experimental result. It means the claimed intervention correspondence should not be presented as an established equivalence for individual counterfactuals.

Selecting reference cases and assessing uncertainty

For each individual, the authors define a local network of reachable places and potentially enabled transitions. They propose selecting reference individuals whose local networks exactly match the query individual’s network, rather than relying only on similarity in observed features.

Conformal prediction supplies the proposed quantitative layer. It uses reference observations and nonconformity scores to construct prediction sets. The paper claims coverage of at least one minus a chosen error level for the individual causal effect—the difference between that individual’s outcomes under treatment and no treatment.

That coverage is not demonstrated by the paper’s argument. Its local-ergodicity assumption says individuals sharing a local mechanism have the same conditional outcome distribution. The proof then infers exchangeability of calibration and query scores. Equal marginal distributions, however, do not by themselves establish joint exchangeability: they do not specify how observations depend on one another.

The argument also does not explain how treatment-specific prediction scores produce the claimed coverage for the difference of two potential outcomes. Consequently, matching local networks and assuming local ergodicity should not be described as sufficient, established guarantees for an individual-effect interval.

A theoretical proposal, not a validated estimator

The study reports that no datasets were generated or analyzed. It therefore offers no empirical demonstration of improved individualized estimates.

Its proposed translation is binary and monotone. Continuous variables, non-monotone mechanisms, practical identification of matching local networks and verifiable conditions supporting calibration remain extension directions. Although Petri nets can represent cycles, the translation begins with acyclic Bayesian networks.

The central proposal is a division of labor: use identity-preserving process structure to describe individual causal possibilities, then use reference data to quantify uncertainty. The unresolved correspondence and coverage arguments limit what this preprint establishes about that proposal.