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Interpretable Causal Graphical Models for Equilibrium Systems with Confounding

This paper introduces interpretable anterial graphical models to enable valid causal inference and flexible covariate adjustment in equilibrium systems affected by confounding, while establishing connections to existing counterfactual and intervention graph frameworks.

Original authors: Kai Z. Teh, Kayvan Sadeghi, Terry Soo

Published 2026-03-27

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Kai Z. Teh, Kayvan Sadeghi, Terry Soo

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out why a cup of coffee is hot. You know the coffee maker turned it on (the cause), but you also know the room is cold (a confounder that affects both the machine's efficiency and how fast the coffee cools). In the real world, things don't just happen in a straight line; they often settle into a balance or equilibrium (like a thermostat keeping a room at a steady temperature), and hidden factors often mess with our ability to see the true cause-and-effect relationship.

This paper is like a new, super-smart map-making kit for scientists who want to draw these complex situations. Here is the breakdown using everyday analogies:

1. The Problem: The "Messy Room" Map

Traditional maps (called DAGs in statistics) are great for simple, one-way streets. If A causes B, and B causes C, you draw an arrow A → B → C.

The Issue: Real life is messier. Sometimes variables influence each other in a loop (equilibrium), like a thermostat and a heater. Sometimes, two things are linked because of a hidden third thing (confounding), like how ice cream sales and shark attacks both go up in summer (hidden cause: hot weather).
The Old Tools: Previous maps could handle loops or hidden links, but not both at the same time clearly. They were like trying to draw a 3D city on a flat piece of paper without getting lost.

2. The Solution: The "Anterial" Map

The authors introduce a new type of map called an Anterial Graph. Think of this as a 3D architectural blueprint that can handle:

One-way streets (Arrows: A causes B).
Two-way traffic (Bidirected lines: A and B are secretly linked by a hidden friend).
Roundabouts (Undirected lines: A and B are part of a balanced system where they adjust to each other).

This new map is special because it doesn't just show you the current traffic; it allows you to simulate what would happen if you changed the rules.

3. The "Time Travel" Feature: Counterfactuals

In science, we often ask: "What would have happened if I had taken a different path?" This is called a counterfactual.

The Analogy: Imagine you are playing a video game. You want to know: "If I had pressed 'Jump' instead of 'Run', would I have avoided the pit?"
The Paper's Trick: The authors created a way to draw a single map that shows both the world where you pressed 'Run' (Observation) and the world where you pressed 'Jump' (Intervention) side-by-side.
The "Single-World" View: They also figured out how to simplify this. Instead of showing two parallel universes, they show just one version of the world where the "Jump" button is forced, but the rest of the world reacts naturally. This prevents the map from getting too cluttered with "what-ifs" that can't actually happen at the same time.

4. The "Confounder Filter": Picking the Right Ingredients

When scientists want to prove that "Coffee causes Alertness," they have to control for other factors (like "Did the person sleep well?"). They need to pick the right list of variables to check, called an Adjustment Set.

The Constraint: Sometimes, you must include certain variables (e.g., a doctor must check blood pressure) or must not include others (e.g., you can't measure something that costs too much money).
The Algorithm: The paper provides a step-by-step recipe (an algorithm) to find the perfect, smallest list of variables to check that fits your rules.
- Imagine you are packing a suitcase. You have a rule: "Must include a toothbrush" and "Must not include a piano." The algorithm helps you pack the lightest possible suitcase that still has your toothbrush and no piano, ensuring you don't miss any other essential items.

5. Why This Matters

For Doctors: It helps figure out if a new drug works, even when patients have complex, balanced body systems and hidden lifestyle factors.
For Policymakers: It helps predict the outcome of a new law (intervention) while accounting for economic balances and hidden social factors.
For AI: It helps machines understand cause and effect, not just patterns, making them smarter at making decisions in complex environments.

Summary

The authors built a universal translator for complex systems. They took the messy reality of equilibrium (things balancing out) and confounding (hidden links) and created a clear, visual language (Anterial Graphs) to:

Draw the system.
Simulate changes (interventions).
Figure out exactly what data you need to collect to be sure about the cause-and-effect, even when you have strict rules on what you can or cannot measure.

It's like upgrading from a flat, 2D road map to a dynamic, 3D GPS that can show you traffic jams, hidden shortcuts, and exactly what route to take if you decide to change your destination mid-journey.

1. Problem Statement

Causal inference often requires modeling systems that reach an equilibrium (e.g., economic markets, biological feedback loops) or contain confounding (unobserved common causes). Standard causal graphical models, specifically Directed Acyclic Graphs (DAGs) used in Structural Causal Models (SCMs), are insufficient for these settings because:

DAGs cannot represent symmetric relationships induced by equilibrium dynamics or bidirectional confounding.
Chain Graphs and Ancestral Graphs can represent equilibrium and confounding respectively, but they lack a unified framework that handles both simultaneously while remaining closed under marginalization and conditioning.
Existing methods for selecting adjustment sets (covariates to control for confounding) often struggle when specific inclusion/exclusion constraints are required (e.g., due to ethical costs or regulatory mandates) and when the underlying graph structure is complex.

The authors aim to provide a unified, interpretable graphical framework for equilibrium systems with confounding and develop an algorithm to select adjustment sets under arbitrary constraints.

2. Methodology

A. Anterial Graphs as the Core Framework

The authors utilize Anterial Graphs (Lauritzen and Sadeghi, 2018), a generalization of DAGs, Ancestral Graphs, and Chain Graphs. These graphs contain three edge types:

Directed ( $\to$ ): Causal influence.
Undirected ($-$): Equilibrium relationships (symmetric dependencies within a chain component).
Bidirected ( $\leftrightarrow$ ): Confounding (shared unobserved errors).

Anterial graphs are unique because they are closed under marginalization and conditioning, meaning that if a distribution is Markovian to an anterial graph, its marginals and conditionals are also Markovian to some anterial graph. This property is crucial for handling latent variables and selection bias without losing the graphical representation.

B. Structural Equilibrium Models (SEMs)

The authors define a Structural Equilibrium Model (Definition 7) as a generalization of SCMs.

Variables are partitioned into ordered sets (chain components) $\tau_1, \dots, \tau_n$ .
Each component $\tau_i$ is determined by a function $f_{\tau_i}$ of its parents and an error term $\epsilon_{\tau_i}$ .
Equilibrium: Within a chain component, variables are determined by a dynamical process (modeled as a Gibbs Sampler) that reaches a conditional equilibrium distribution given the parents.
Confounding: Errors between different components can be dependent, represented by bidirected edges.

Key Theoretical Result (Theorem 1): If the induced distribution is a compositional graphoid, the joint distribution is Markovian to the corresponding anterial graph constructed via Algorithm 1. This graph construction relies on conditional independence tests within the equilibrium distributions.

C. Graphical Interventions and Counterfactuals

The paper extends causal inference to interventions on these equilibrium systems:

Intervention Graphs ( $do_C(G)$ ): Using Algorithm 2, the authors define how to modify the graph when a treatment set $C$ is intervened upon. This involves removing incoming edges to $C$ , converting undirected edges within $C$ to directed edges (if the target is outside $C$ ), and removing bidirected edges connected to $C$ .
Counterfactual Graphs ( $\phi(G; C)$ ): Using Algorithm 3, they construct a graph representing both observational variables ( $X$ ) and intervened variables ( $X^{do(C)}$ ). This graph encodes the coupling between the two via shared errors (bidirected edges).
Single-World Anterial Interventional Graphs (SWAIGs, $\phi'(G; C)$ ): To avoid violations of the faithfulness assumption caused by deterministic relationships in the full counterfactual graph, they propose marginalizing out "post-treatment" variables to create a single-world representation, analogous to SWIGs in DAGs.

D. Constrained Confounder Selection

The authors address the problem of finding a minimal adjustment set $S$ such that $L \subseteq S \subseteq U$ , where $L$ are covariates that must be included and $U$ are covariates that can be included.

Algorithm 4: An element-wise procedure that iteratively removes nodes from the upper bound $U$ while checking if the separation condition ( $X_O^{do(C)} \perp X_C | S$ ) holds in the maximized graph.
Unlike standard approaches that enumerate all sets and filter, this algorithm incorporates constraints directly into the search process, ensuring efficiency and correctness under faithfulness assumptions.

3. Key Contributions

Unified Graphical Model: Introduction of Anterial Graphs as the natural language for modeling systems with both equilibrium dynamics and confounding, unifying previous frameworks (Chain Graphs, Ancestral Graphs, DAGs).
Structural Equilibrium Models: Formalization of SCMs for equilibrium systems where variables within a component reach a Gibbs-sampler-like equilibrium.
Graphical Intervention Calculus: Definition of Algorithm 2 for intervening on anterial graphs and Algorithm 3 for constructing counterfactual graphs, proving that these operations preserve the Markov property (Theorems 2 and 3).
Constrained Adjustment Algorithm: Development of Algorithm 4, a provably correct method for selecting minimal adjustment sets with hard inclusion/exclusion constraints, which is particularly relevant for clinical trials and algorithmic fairness.
Interpretability: Demonstrating that causal notions (interventions, counterfactuals) can be expressed purely through local graph manipulations without needing to know the specific coupling of error terms, provided the graph is anterial.

4. Results and Validation

Theoretical Proofs: The paper provides rigorous proofs for the Markov properties of structural equilibrium models (Theorem 1), intervened models (Theorem 2), and counterfactual interpretations (Theorem 3).
Simulations: The authors simulated data from structural equilibrium models corresponding to three different anterial graphs (Figures 4, 5, 6).
- They sampled data using both direct equilibrium distributions and Gibbs samplers.
- Fisher-Z tests confirmed that the pairwise conditional independencies implied by the Markov property of the anterial graphs held true in the simulated data (p-values were consistent with the theoretical separation criteria).
Algorithm Performance: The constrained selection algorithm was illustrated on a counterfactual graph (Figure 10), demonstrating how it successfully identifies a minimal set satisfying $L \subseteq S \subseteq U$ by iteratively marginalizing nodes.

5. Significance and Future Work

Significance:

This work bridges a critical gap in causal inference by providing tools for equilibrium systems, which are common in real-world applications but previously lacked a unified causal graphical framework.
It offers a practical solution for constrained causal inference, addressing the reality that researchers often cannot collect or must exclude certain variables due to cost, ethics, or regulation.
The framework maintains interpretability, allowing causal effects to be derived via graph operations rather than complex algebraic manipulations of distributions.

Future Work:

Relaxing Assumptions: Investigating if the "compositional graphoid" assumption can be relaxed by defining a local Markov property specific to anterial graphs.
Algorithm Optimization: Exploring the use of conditioning operations ( $\alpha_c$ ) alongside marginalization in the selection algorithm to potentially speed up convergence while maintaining minimality.
General Counterfactuals: Extending the framework to handle multiple intervention values (not just fixed $a_C$ ) using "templates" similar to Richardson and Robins (2013).
Multivariate Extensions: Adapting Algorithm 4 to handle cases where the treatment and outcome sets are subsets of nodes rather than singletons.

In summary, Teh et al. provide a robust theoretical and algorithmic foundation for causal inference in complex, equilibrium-driven systems with confounding, offering both a new graphical language and practical tools for constrained variable selection.