How PC-based Methods Err: Towards Better Reporting of Assumption Violations and Small Sample Errors

arXiv:2502.14719 · stat.ML, cs.LG · Submitted 2026-03-18 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Next we'll be talking about the paper "How PC-based Methods Err: Towards Better Reporting of Assumption Violations and Small Sample Errors".

Jane: The paper was written by Sofia Faltenbacher, Jonas Wahl, Rebecca Herman and Jakob Runge from University of Potsdam and German Research Center for Artificial Intelligence (DFKI).

Tom: Stay tuned as we take you through the paper and discuss its implications.

Title: Tom: We’re diving into "How PC-based Methods Err" now, and it’s a real eye-opener. It basically explains that when things go wrong in the PC algorithm—which is a very popular method for finding causal links—we often end up with graphs that are just plain wrong.

Jane: It's not just about the final graph being incorrect, Tom; they detail *how* it’s wrong, which is something we rarely see discussed before.

Meng: So, if I use PC to map how my business processes affect each other, this paper suggests my map might be misleading me even if the software says it's correct?

Lu: That’s the danger. The authors show that errors can manifest in three distinct ways—orientation conflicts, incoherencies, and graph-type mismatches.

Tom: These aren't just random glitches; they are systematic failures based on how the CI tests interact with the structural assumptions of a specific type of failure.

Jane: It’s important to understand that these issues happen regardless of whether we have a perfect ground truth or not, which is where this paper really shines.

Meng: If I can't trust the output because it might be flawed in one of those ways, how do we even begin to fix it?

Summary: Tom: The paper summarizes these errors by showing that they aren't monolithic; they are distinct failure modes. For example, you have orientation conflicts where the method tries to point an edge both left and right simultaneously.

Jane: But there’s also the concept of "incoherency," which is a truly mind-bending concept for many listeners.

Lu: It’s when the test results—the CI test outcomes—contradict what is implied by the output graph, even if there are no visual conflicts.

Meng: So, I get a result where my system says X and Y are independent, but the graph structure implies they must be connected? That's a contradiction.

Jane: Precisely. The authors categorize these failures into different sets that they call D1 through D3 based on whether or not a distribution could actually represent those contradictory results.

Tom: And G1 through G3 describes the resulting output, which is either visually conflicted, internally incoherent, or looking perfectly fine on the surface.

Lu: It’s a complete classification system for failure modes that previously just didn't exist in our error analysis tools.

Improvements: Tom: This is where the "Coherency Score" comes in, and it is a massive methodological leap. The authors introduce this score to quantify how well the CI tests match the graph structure.

Jane: It’s like a built-in self-check for your PC method, Tom, that tells you if you're contradicting yourself internally.

Meng: What I really care about is that this score is computationally cheap, right? We are talking about something that takes seconds to calculate on a eight-node graph.

Lu: That’s the genius of it. It provides a global view of consistency without the massive computational burden of methods like Answer Set Programming (ASP).

Tom: The authors show that this score is not just some random metric; they prove it serves as a heuristic proxy for the Structural Hamming Distance to the unknown ground truth.

Jane: That’s incredible, Lu. It means we can get an idea of how far off our results are from reality, even when we don't know what reality looks like.

Meng: So, if my score is low, I can tell me that my AI model is likely making assumptions or tests that are fundamentally inconsistent with the real world?

Conclusion: Tom: We’ve spent a lot of time on this paper, "How PC-based Methods Err," and the biggest message I take away is the importance of being skeptical.

Jane: It’s not a silver bullet, Tom; they are very clear that this score is a heuristic, not proof. There are still cases where errors are totally undetectable.

Lu: But having all these tools—the incoherency checks, the coherency scores—gives us a whole new toolkit for vetting AI models.

Meng: I think my implementation teams will be really interested in how we can integrate this low-cost sanity check into our existing pipelines to flag potential issues before they become production problems.

Tom: It's a huge win for better error reporting, acknowledging the inherent limitations of the academic tools we’ve been using.

Jane: It’s exciting to see an AI tool that doesn' is not just a black box, but one that can tell us when it might be failing in a real-world scenario.

Tom: We have so much more to talk about next time with the next paper, but for now, let's give it up for Tom, Jane, Lu, Meng and Lalam!

Sofia Faltenbacher, Jonas Wahl, Rebecca Herman, Jakob Runge

University of Potsdam · German Research Center for Artificial Intelligence (DFKI)

stat.ML, cs.LG

Submitted: 2026-03-18

Updated: 2026-08-20

Comments: under review

Journal ref: Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence, PMLR 337:1498-1519, 2026

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 78/100

The gist: Causal discovery methods, such as those based on the PC algorithm, are proven to be sound only in an "idealized setting" where all structural assumptions are fulfilled and all conditional

Key concepts

PC-based Methods: A popular algorithm used for finding causal links or mapping how business processes affect each other. The paper warns that the resulting graphs can be systematically wrong, even if the software indicates they are correct.
Incoherency: A failure mode where the test results (CI test outcomes) contradict what is implied by the output graph. This contradiction exists even if there are no visible conflicts in the graph structure.
Coherency Score: A new, computationally cheap metric introduced to quantify how well the CI tests match the graph structure. It serves as a self-check for PC methods, providing a global view of consistency without massive computational burden.
Structural Hamming Distance: The unknown measure of how far off an algorithm's results are from the true reality or 'ground truth.' The Coherency Score is shown to act as a heuristic proxy for this distance.

Terminology

Causal Discovery — 46× in this paper · explained in 2 episodes
Causal Inference — 5× in this paper · explained in 4 episodes
Robustness — 4× in this paper · explained in 5 episodes
Accuracy — 3× in this paper · explained in 1 episode
Answer Set Programming — 3× in this paper
Axioms — 3× in this paper · explained in 2 episodes
Conference on Uncertainty in Artificial Intelligence — 3× in this paper
Hamming Distance — 3× in this paper · explained in 2 episodes
Structural Hamming Distance — 3× in this paper
Answer Set Programming (ASP) — 2× in this paper · explained in 1 episode
Benchmarking — 2× in this paper · explained in 3 episodes
Consistency — 2× in this paper · explained in 3 episodes
Failure Modes — 2× in this paper · explained in 3 episodes
Theoretical Guarantees — 2× in this paper · explained in 2 episodes
alpha — 2× in this paper · explained in 1 episode
Aggregation — 1× in this paper · explained in 1 episode
Causal Reasoning — 1× in this paper · explained in 2 episodes
Core — 1× in this paper · explained in 2 episodes
Downstream Tasks — 1× in this paper · explained in 2 episodes
Elias — 1× in this paper · explained in 2 episodes
Grounding — 1× in this paper · explained in 3 episodes

Summary

Causal discovery methods, such as those based on the PC algorithm, are proven to be sound only in an idealized setting where all structural assumptions are fulfilled and all conditional independence tests are correct. The paper addresses the reality that this idealized setting is rarely given in real data, leading to potential inconsistencies like orientation conflicts. The primary goal of this work is to introduce a computationally cheap method for detecting and quantifying these errors without requiring knowledge of the ground truth causal graph (G true).

Theoretical Framework: Errors, Incoherence, and Classification

The paper establishes a hierarchy of errors:

Erroneous Tuples: A tuple (X, Y, S) is erroneous if the CI test result for (X, Y, S) of a method M does not match the separation statement of (X, Y, S) in the ground truth causal graph G true.
CI-Deviance: A method M is CI-deviant if there is at least one erroneous tuple.
Incoherence (Definition 3): Incoherence is defined as a state where the CI test result for (X, Y, S) of a method M does not match the separation statement of (X, Y, S) in the method’s output graph G out. This means that X G out Y S but X about(T, alpha) Y S, or vice versa.

The paper distinguishes three manifestations of CI-deviances or graph-type mismatches:

Case-D1: No distribution P X can represent the observed CI test results.
Case-D2: A distribution P X exists, but none is Markovian and faithful to G out.
Case-D3: A distribution P X exists that is Markovian and faithful to G out.

These are classified at the graph level (G1-G3):

Case-G1: Marked conflicts or ambiguities exist in G out.
Case-G2: No conflicts or ambiguities are marked, but the results are incoherent.
Case-G3: No conflicts or ambiguities are marked, and the results are coherent.

A key theoretical result is Proposition 1: "Erroneous outputs of a method M have no marked conflicts and ambiguities and are coherent with the CI test results T ind T dep (Case-G3) if and only if there is a distribution that can represent the conditional independencies T ind and is Markovian and faithful to G out (Case-D3). This implies that in Case-G3, the output is undetectable" using only the list of CI tests and the output graph.

Methodology: The Coherency Score

To address the need for error detection without ground truth, the paper introduces a Coherency Score.

The score is defined as:

sc(w, M) = 1 - sum(X,Y,S) in T w(X, Y, S) inc(X, Y, S) over sum(X,Y,S) in T w(X, Y, S)

where inc(X, Y, S) = iota G out (X, Y, S) - iota(T, alpha) (X, Y, S). The total coherency score is obtained by setting the weight function w(X, Y, S) = 1.

The paper demonstrates that this score serves as a heuristic proxy for the Structural Hamming Distance (SHD) to the ground truth graph.

Experimental Validation and Findings

The experiments validate the utility of this approach:

Auto-MPG Dataset: The analysis showed that there are incoherencies (total coherency score 0.917) that do not show as orientation conflicts, implying a CI-deviance or a graph-type mismatch, even when using the default PC algorithm.
Simulational Studies (SHD Proxy): When generating random DAGs, the results suggest that the total coherency score on average serves as a proxy for the SHD to the ground truth when the ground truth is not available.
Assumption Violations: The scores are used to analyze how different degrees of hidden confounding affect results. For instance, in cases of faithfulness violations (Example 3), the coherency score is high for the dense confounding model, while constant low scores across sample sizes can indicate detectable assumption violations.
Causal Insufficiency: The paper demonstrates that even when a method is coherent (Case-G3), it can still be erroneous due to causal insufficiency, as shown in Example 6, where "there are no orientation conflicts and the method is not incoherent (G3). In this case, we have no chance to detect that in fact there was a faithfulness violation and the output graph G out is erroneous given the list of CI tests and the output graph."

In summary, the paper concludes that testing for incoherencies provides an exhaustive error detection approach given only the information collected in one run of the PC-based algorithms and without access to the ground truth, establishing a theoretical limit to error detection in this setting.

Improvements for AI systems

Core Architectural Improvement: Implementation of a Multi-Stage Causal Structure Validator and Refiner.

The improved system must move beyond simply outputting a graph based on conditional independence (CI) tests. It needs to incorporate explicit modules for validating the coherence and faithfulness of the resulting structure relative to known theoretical limitations.

Improvement: Implement a modular pipeline that integrates classical constraint-based methods (PC/FCI) with structural learning techniques (e.g., linear models, non-linear kernel methods) and explicitly tracks multiple necessary conditions for graph validity.

What the Improved System Can Do:

Robust Constraint Testing: Instead of relying solely on a single CI test result (T), the system must compute and store a confidence score for every independence test relative to its theoretical requirements (e.g., using penalized likelihood methods or bootstrapping across subsets) to quantify the risk associated with each T.
Automatic Conflict Detection: The system must integrate an explicit Collider Conflict Resolver. When multiple identified colliders (W 1 to X 3 from W 2 and W 3 to X 6 from W 4) impose contradictory directional constraints on shared edges (e.g., X3, X6), the system must flag a Structural Incoherence Warning and reject the current output graph until the conflict is resolved or downgraded to an ambiguity set.

Summary of Impact: The improved AI system transforms from a simple constraint graph generator into a Comprehensive Causal Hypothesis Engine. It does not just find a structure; it validates that structure against theoretical constraints (coherence, faithfulness) and provides quantifiable metrics regarding the reliability and limitations of its own findings.