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Re-examining Granger Causality with Causal Bayesian Networks and Reichenbachs Principles

This paper addresses the lack of a rigorous causal foundation in traditional Granger causality by reinterpreting it through Reichenbach's principles and causal Bayesian networks to propose "causalized Granger causality" (c-GC), a theoretically grounded algorithm that offers a more principled framework for causal discovery in observational time series data.

Original authors: S. A. Adedayo

Published 2026-06-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: S. A. Adedayo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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 who is the boss in a chaotic office where everyone is talking at once. You have a recording of their conversations over time. You want to know: Did Person A's comment actually cause Person B's reaction, or did they just happen to talk at the same time because of a third person, Person C?

This is the core problem of Causal Discovery. For decades, scientists have used a tool called Granger Causality (GC) to solve this. Think of GC as a "predictive crystal ball." It asks: "If I know what Person A said yesterday, can I predict what Person B will say today better than if I only knew what Person B said yesterday?" If the answer is yes, GC says, "A causes B."

The Problem:
The paper argues that the traditional crystal ball has a flaw. It's too easily tricked.

The "Common Cause" Trap: If a boss (Person C) yells at both A and B, A and B might start talking at the same time. GC might wrongly think A caused B, when really, they were both reacting to C.
The "Spurious Link" Trap: Sometimes, two people are independent, but if you look at a specific moment where they both reacted to a third event, they might look connected.

The author, S.A. Adedayo, says: "GC is good at predicting, but it's bad at proving true cause-and-effect because it lacks a solid philosophical foundation."

The Solution: A New Detective Team
The author proposes a fix by teaming up GC with two other concepts: Causal Bayesian Networks (a map of relationships) and Reichenbach's Principles (a rulebook from a philosopher named Reichenbach).

Reichenbach's rulebook says: "If two things are connected, it's either because one caused the other, or they share a common cause that happened earlier."

The author created a new algorithm called Causalised Granger Causality (c-GC). Think of c-GC not as a single detective, but as a two-person detective squad that must agree before they arrest a suspect (declare a cause).

How the "Two-Person Squad" Works

The squad uses two different interrogation techniques on every pair of variables (A and B):

Detective BVGC (The "Bivariate" Detective):
- The Method: This detective looks only at A and B. "Did A's past help predict B's future?"
- The Flaw: This detective is gullible. If a third person (C) is influencing both, this detective will falsely accuse A of causing B.
- The Analogy: It's like seeing two people running and assuming one pushed the other, without checking if a dog chased them both.
Detective MVGC (The "Multivariate" Detective):
- The Method: This detective is very thorough. It looks at A and B, but it also checks everyone else in the room (C, D, E...). It asks: "Does A still help predict B even after we account for what everyone else is doing?"
- The Flaw: This detective is too strict in a specific way. If A and B are both causes of a third person (C), and we look at C, this detective might get confused and think A and B are connected when they aren't.
- The Analogy: It's like checking if two people are connected by looking at the person they both pushed. Sometimes, looking at the victim makes the two pushers look like they were working together, even if they weren't.

The Magic Move: The "AND" Operation
The paper's big breakthrough is realizing that these two detectives have opposite blind spots.

BVGC sees too many connections (False Positives).
MVGC sometimes misses connections or sees fake ones in specific "V-shape" scenarios.

So, the author says: "Only declare a cause if both detectives agree."

If BVGC says "Yes" and MVGC says "Yes," then it's a True Cause.
If one says "Yes" and the other says "No," they cancel each other out, and we assume it's a false alarm.

This "AND" rule acts like a filter that removes the noise, leaving only the genuine cause-and-effect relationships.

The Results: Did the New System Work?

The author tested this new "c-GC" system in three ways:

Synthetic Data (The Simulation Lab):
They created fake worlds with known rules (like a video game where they know exactly who causes what). They added "chaos" and "noise" to make it hard.
- Result: c-GC was excellent at finding the true connections and ignoring the fake ones. It performed better than or equal to other popular methods, especially in tricky situations where variables loop back on themselves.
The Sachs Protein Dataset (The Biology Test):
They used real data from human immune cells (proteins signaling to each other). This is a famous benchmark in science.
- Result: The algorithm successfully mapped out almost all the known biological pathways. It found the connections that scientists already knew existed, proving it works on real, messy biological data.
The Lorenz-96 Dataset (The Chaos Test):
They used a famous mathematical model that simulates chaotic weather systems.
- Result: Even in a highly chaotic system where things change rapidly, c-GC accurately identified the underlying structure.

The Bottom Line

The paper claims that by re-interpreting Granger Causality through the lens of Reichenbach's philosophy and Causal Networks, they have fixed its biggest weakness: the lack of a true "causal" foundation.

By forcing the "Bivariate" and "Multivariate" tests to agree, they created a tool (c-GC) that is much harder to trick. It doesn't just predict the future; it gives a much stronger, more reliable answer to the question: "Did this actually cause that?"

The author concludes that this method is ready for use on observational data (data we just watch, without running experiments) and is a significant step forward for understanding complex systems like the brain or the economy.

Technical Summary: Re-examining Granger Causality with Causal Bayesian Networks and Reichenbach's Principles

Problem Statement
Granger Causality (GC) remains a dominant computational tool for identifying causal relationships in time series data across fields such as econometrics and neuroscience. However, it faces persistent criticism for lacking a rigorous causal foundation, often being viewed as merely a predictive tool rather than a causal one. Furthermore, standard GC implementations are susceptible to latent confounders and rely heavily on linear assumptions. While extensions exist to handle nonlinearity or multivariate conditioning, the fundamental issue of grounding GC in a strict causal framework remains unresolved. This paper addresses the need to reinterpret GC to provide it with a robust causal interpretation under specific assumptions.

Methodology
The authors propose a reformulation of Granger Causality, termed causalized Granger Causality (c-GC), by integrating three theoretical frameworks:

Causal Bayesian Networks (CBNs): The authors interpret GC as a conditional (in)dependence test within the CBN framework. Specifically, they analyze the residuals of regression models to determine statistical dependence.
Reichenbach's Common Cause Principles (RCCP): The method leverages Reichenbach's logical notion that no effect precedes its cause and that dependence implies a common cause or direct causal link.
Synthesis of Bivariate and Multivariate GC: The core methodological innovation is the combination of Bivariate Granger Causality (BVGC) and Multivariate Granger Causality (MVGC) to satisfy RCCP requirements.

The algorithm operates on three propositions:

Proposition 1 (Unconditional Dependence): BVGC tests for marginal dependence between variable pairs without conditioning on confounders. This corresponds to Reichenbach's principle that dependent events must have a common cause or direct link. This step identifies potential links but may yield false positives due to unobserved confounders (e.g., in chain or fork structures).
Proposition 2 (Conditional Dependence): MVGC tests for dependence while conditioning on all other variables in the system. This step adheres to the Causal Markov Condition, correctly identifying independence in chains and forks but potentially inducing spurious dependence in collider structures (V-structures) when conditioned upon.
Proposition 3 (Logical Conjunction): The authors argue that neither BVGC nor MVGC alone is sufficient for causal discovery. By taking the logical AND of the results from both tests, the algorithm enforces all Reichenbach's principles. This combination filters out spurious links: BVGC detects colliders (where MVGC fails), while MVGC filters out spurious links in chains and forks (where BVGC fails).

The resulting algorithm, c-GC, constructs a conditioning set ( $Z$ ) based on past states of variables to satisfy d-separation and back-door criteria. An extension, c-GC*, utilizes a more conservative conditioning set that includes the cause's future states (an artifact of data manipulation in the discovery process), though the theoretical justification for this specific inclusion remains an open question. The implementation uses permutation-based Pearson's correlation for independence testing to maintain non-parametric flexibility.

Key Contributions

Theoretical Reinterpretation: The paper provides a rigorous justification for combining BVGC and MVGC, grounding the practice in CBNs and Reichenbach's principles to endow GC with a causal interpretation.
Algorithm Development: The introduction of c-GC and c-GC*, which operationalize this theoretical framework.
Handling Complex Structures: The method is designed to handle feedback loops, cyclicity, and bidirectional interactions without enforcing acyclicity, making it applicable to complex systems like biological neural networks.

Results
The authors evaluated c-GC and c-GC* on synthetic and benchmark datasets:

Synthetic Data: In simulations with known ground-truth matrices (including cycles and bidirectional links), both c-GC and c-GC* achieved high accuracy, precision, and recall. In linear and minimal chaotic systems, c-GC correctly inferred all links with minimal false negatives, while c-GC* showed slightly more conservative behavior (one additional false positive).
Comparison with State-of-the-Art: When compared against popular Causal Structure Learning (CSL) algorithms (TCDF, NTS-NOTEARS, Tigramite) on various simulation terrains, c-GC and c-GC* demonstrated robustness. Notably, while Tigramite (FullCI) achieved perfect recall, it exhibited lower precision (0.51–1.00) compared to c-GC* (0.95–1.00), indicating c-GC* is more conservative in edge detection.
Sachs Protein Dataset: Applied to the real-world Sachs signaling dataset, the method successfully inferred nearly all expected causal connections, including the unique directional link $PKA \to Jnk$ , which was only detected in specific data trials.
Lorenz-96 Dataset: On the chaotic Lorenz-96 system ( $p=40$ ), both algorithms converged to similar performance metrics, with slightly better results observed at lower chaos levels ( $F=10$ ) compared to higher levels ( $F=40$ ).

Significance and Claims
The paper claims that by reinterpreting GC through the lenses of CBNs and Reichenbach's principles, it provides a "principled framework for causal discovery in observational datasets." The authors assert that this reformulation resolves the criticism of GC lacking a causal foundation, provided specific assumptions (causal sufficiency, faithfulness, and causal Markov condition) are met. The work demonstrates that the combination of bivariate and multivariate tests is not merely heuristic but is theoretically necessary to satisfy the conditions for inferring true causal links. The authors conclude that c-GC yields satisfactory results in both linear and nonlinear domains, including cases with cycles and bidirectionality, offering a more robust alternative for causal discovery in complex dynamical systems. Future work is modestly scoped to scaling the algorithm for high-dimensional datasets.

How the "Two-Person Squad" Works

The Results: Did the New System Work?

The Bottom Line

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