Score-Based Causal Discovery with Temporal Background Information

Anne Helby Petersen; Claus Thorn Ekstr{\o}m; Tobias Ellegaard Larsen

REVIEW 3 major objections 6 minor 1 cited by

Score-Based Causal Discovery with Temporal Background Information

T0 review · 3 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read A temporal extension of score-based causal discovery, TGES, is proven to always return a tiered MPDAG and to recover the true restricted equivalence class in the large-sample limit.

desk verdict TGES is a genuinely new temporal extension of GES with a solid proof appendix; the main fixable problem is that the pseudocode leaves the DAG representative underlying the scoring step unspecified. read the letter →

arxiv 2502.06232 v2 pith:AV54QTJI submitted 2025-02-10 stat.ME

Tobias Ellegaard Larsen , Claus Thorn Ekstr{o}m , Anne Helby Petersen This is my paper · ORCID

classification stat.ME MSC 62D2062H05

keywords causaldiscoveryscore-basedlearningtemporalbackgroundknowledgetieredMPDAGGreedyEquivalenceSearchBayesianInformationCriterionMarkovclassobservationaldata

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper develops Temporal Greedy Equivalence Search (TGES), a score-based causal discovery algorithm that exploits temporal information encoded as tiers over variables: causes cannot point backward in time. The central claim is that TGES always terminates in a tiered maximally oriented partially directed acyclic graph (tiered MPDAG), and that in the large-sample limit this output is almost surely sound and complete, representing exactly the restricted Markov equivalence class containing the true data-generating DAG. The authors introduce the Temporal Bayesian Information Criterion (TBIC), a scoring criterion with the consistency, decomposability, and score-equivalence properties needed for the proof. The practical motivation is that temporal knowledge improves both orientation and adjacency recovery; simulations show TGES achieves better recall with comparable precision than the temporal PC algorithm, and a real-world example on life-course health data demonstrates its use.

What carries the argument

The machinery is the tiered MPDAG together with the TBIC score and a three-stage greedy step. Tiered background knowledge Kτ forbids directed edges from later to earlier tiers; restricting a CPDAG by Kτ and applying Meek's rule 1 yields a tiered MPDAG, a partially directed graph representing a restricted Markov equivalence class. Each TGES step scores neighboring CPDAGs with TBIC, orients all undirected cross-tier edges according to the tier ordering, then closes under Meek's rule 1. TBIC is decomposable, Kτ-score equivalent, and Kτ-locally consistent, so local score differences can be computed from parent sets of the current graph; these properties are what let the classical GES proof be carried over to the restricted setting.

What would settle it

On simulated Gaussian data from a known DAG with clear tiers, implement TGES exactly as in Algorithms 4–6 without adding any DAG-extension rule, and check whether the returned tiered MPDAG converges to the true restricted equivalence class as the sample size grows; if the output depends on the arbitrary representative used to read parent sets, or systematically misses the true class in the limit, the claimed guarantee fails for the published pseudocode.

Watch

Extended reading notes

Core claim

The paper's central result, Theorem 3, is that TGES using a score that is Kτ-score equivalent, decomposable, Kτ-consistent, and Kτ-locally consistent produces a tiered MPDAG, and as n → ∞ this tiered MPDAG is almost surely a sound and complete estimate of the restricted Markov equivalence class of the true data-generating DAG. Equivalent statements are proved for each phase: the forward phase adds edges until every graph in the class is an independence supergraph of the truth, the backward phase removes edges until class equality is reached, and the turning phase provably makes no harmful changes once the class is correct. The paper also proves that TBIC, which is BIC with a −∞ penalty for any graph contradicting the tier ordering, satisfies all required properties for Gaussian data.

Load-bearing premise

The proof assumes the greedy score differences are computed from parent sets that are well-defined for the current partially directed graph, but Algorithms 4–6 do not specify how a DAG representative of the tiered MPDAG is chosen for scoring, and different choices could change the greedy path.

Editorial extensions

If this is right

TGES always returns a tiered MPDAG, so the output represents a well-defined class of DAGs and supports downstream causal-inference machinery for MPDAGs.
In the large-sample limit, the algorithm recovers the restricted Markov equivalence class of the true DAG almost surely, matching GES's guarantee while exploiting temporal constraints.
Any consistent, decomposable, finite score criterion can be turned into a temporal one by assigning −∞ to graphs contradicting Kτ, so the result extends beyond Gaussian BIC to other score families.
Finite-sample simulations indicate TGES has considerably better recall of adjacencies than the temporal PC algorithm with comparable precision, giving a more balanced precision-recall trade-off.
On a life-course health dataset, TGES finds significantly more correct adjacencies than random guessing, with direction agreement on most of the shared edges.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the asymptotic guarantee holds, temporal tiers shrink the restricted equivalence class, which should make subsequent procedures such as causal-effect bound estimation tighter; the paper notes the class is smaller but does not quantify this gain.
The proof's reliance on reading parent sets from a partially directed graph leaves open a choice of DAG extension; different implementations of that choice may change finite-sample behavior even if the limit is unaffected.
The tier constraint affects only orientations, yet simulations show adjacency recovery improves as well; a possible explanation is that correct orientations prevent score-based searches from chasing wrong conditional-independence patterns, a mechanism worth testing directly.
Extending TGES to non-Gaussian or mixed data should be possible by using the paper's recipe with a suitable consistent score, which would make temporal score-based discovery applicable to the broader data types common in epidemiology.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

TGES is a genuine contribution: a score-based temporal extension of GES that incorporates tiered background knowledge, with a plausible asymptotic guarantee and a large simulation study. It should go to peer review, but the pseudocode needs to be cleaned up before the algorithm as written can be said to satisfy the theorem.

New and well done: TGES and its Temporal BIC (which sends any graph contradicting the tier ordering to −∞) are not in the earlier work by Bang & Didelez or Witte. The proof appendix adapts Chickering’s machinery to restricted Markov equivalence classes, and the theoretical derivation builds on external results rather than circling. The simulation study is substantial—10,000 DAGs—and shows TGES with a more balanced precision-recall trade-off than TPC, which matches the known GES-vs-PC pattern. The authors are also honest about limitations: Gaussian misspecification on the binary Metropolit data, the manual tuning of TPC and of the λ penalty, and the open question of general background knowledge are all acknowledged in Section 8.

One reader concern does not stand up. The alleged inequality error in Lemma 5 is not an error. Kτ-consistency (ii) gives a higher score to the graph with fewer edges; the proof uses that exactly, comparing the pre-addition graph against the final graph. So that part is fine.

The real problems are elsewhere. First, Algorithms 4–6 compute local score differences using parent sets of the current tiered MPDAG. A tiered MPDAG is only partially directed, so parent sets are not defined without choosing a DAG extension. The paper never states which extension is used or why the local differences are invariant. A literal implementation could make greedy moves that are not the ones analyzed in Lemma 4. This is fixable by specifying a representative or proving invariance, but as it stands there is a gap between the proof and the pseudocode. Second, Algorithm 1 looks like it has a sign error: when {A→B} is in the forbidden set F, the text says to replace {A—B} with {A→B}, which would orient the edge in the forbidden direction. The intended line must be {B→A}. This is an easy correction, but it is a concrete error.

The empirical comparisons are weakened by tuning: TGEStuned’s λ is chosen to match the expert graph’s edge count, and TPC’s α is chosen after seeing the metrics. The authors acknowledge this, so I read it as a moderate limitation rather than a deceptive one.

Who this is for: anyone doing score-based causal discovery with temporal or tiered structure, and applied researchers in life-course epidemiology who want an alternative to TPC. I would cite it and would bring it to a causal-inference reading group. It deserves a serious referee; the requested revisions should be about clarity and correctness of the pseudocode, not about reworking the core theory.

Referee Report

3 major / 6 minor

Summary. The paper develops Temporal Greedy Equivalence Search (TGES), a score-based causal discovery algorithm that incorporates tiered background knowledge (e.g., temporal ordering) into greedy equivalence search. The authors define tiered background knowledge Kτ, restricted Markov equivalence classes, and a Temporal Bayesian Information Criterion (TBIC) with properties of Kτ-consistency, Kτ-score equivalence, and decomposability. They prove that TGES always returns a tiered MPDAG and, in the large-sample limit, a sound and complete estimate of the restricted Markov equivalence class of the true DAG. They also propose a simpler post-hoc method (STGES), present simulations comparing TGES with GES, STGES, and TPC, and apply TGES to life-course health data.

Significance. If the issues identified below are addressed, the paper would be a useful contribution to causal discovery with background knowledge. It provides a score-based alternative to the constraint-based temporal PC algorithm, with explicit asymptotic guarantees for restricted equivalence classes. The definitions of Kτ-consistency and the extension of GES to tiered background knowledge are natural and could be reusable by other authors. The simulation study, based on 10,000 generated DAGs, suggests practical gains in recall and overall structural Hamming distance relative to TPC, and the code is stated to be available as supplementary material. However, the current manuscript has load-bearing problems: the algorithmic pseudocode is not well-defined for partially directed graphs, one of the key subroutines contains an orientation error, and some proof steps invoke greedy search without constructing the required neighboring class.

major comments (3)

[Appendix A.2, Algorithms 4–6; Section 5.2] The scoring steps in Algorithms 4–6 compute local score differences using parent sets Pa^W_i of the current tiered MPDAG W. A tiered MPDAG is partially directed, so the parent set of a node is not uniquely defined across DAGs in the restricted equivalence class. The theoretical analysis in Appendix B.6 works with DAGs G ∈ E^Kτ and scores S(G, D, Kτ), but the pseudocode scores objects that are not DAGs or relies on an unspecified choice of DAG extension. The paper neither states which DAG representative is used nor proves that the local score differences are invariant across extensions. Consequently, the greedy decisions made by an implementation that follows the pseudocode literally need not coincide with the restricted-class decisions analyzed in Lemma 4, and Theorem 3 is not established for the algorithm as written. The authors should specify how a DAG extension is selected and maintained (e.g., by keeping a current DAG in the class alongside the tiered MPDAG) and prove that the scoring comparisons are well-defined.
[Section 4, Algorithm 1] Algorithm 1 (Restrict PDAG according to Kτ) contains an orientation error: line 4 replaces an undirected edge {A—B} with {A→B} when {A→B} ∈ F. Since F contains exactly the edges with τ(A) > τ(B), the forbidden direction is A→B, so the undirected edge should be oriented as B→A. As written, Algorithm 1 returns a PDAG that contradicts Kτ. This subroutine is used in Stage (ii) of TGES and in STGES, so Lemma 3, which asserts that each TGES step results in a tiered MPDAG, relies on a procedure that, taken literally, fails to produce a graph in agreement with Kτ. This is likely a typo, but it must be corrected and the downstream effects checked.
[Appendix B.5, Lemma 2; Appendix B.6.2, Lemma 5] The proofs of Lemma 2 and Lemma 5 derive contradictions by comparing the current equivalence class with the true DAG G* and then invoking greediness. For example, in Lemma 2, the argument states that if the resulting class E′ is not in agreement with Kτ, then all DAGs in E′ contradict Kτ, so by Kτ-consistency (iii) S(G*,D,Kτ) > S(G′,D,Kτ), and 'since GES is greedy' this is a contradiction. However, G* is not necessarily a neighboring equivalence class of E′, and the greedy algorithm only moves to neighboring classes with higher scores. A valid proof must construct a specific neighboring class with a higher score, as is done for the unconstrained case in Chickering [2003]. The same issue appears in the backward-phase proof of Lemma 5 when it asserts that the score 'has decreased' without identifying the neighboring class. This gap affects Lemma 1 and the proof of Theorem 3. (On a related point, the reader's concern that Definition 10(ii) implies the opposite inequality in Lemma 5 does not land: with G′ in Definition 10(ii) assigned to the fewer-edge graph tilde G* and G assigned to G′, the inequality S(tilde G*) > S(G′) is exactly what Definition 10(ii) yields. The remaining problem is the missing neighbor construction, not the inequality direction.)

minor comments (6)

[Definition 1] The codomain of τ is written as {1,...,T}^d; it should be {1,...,T}.
[Theorem 2 and Appendix B.3] The statement of Theorem 2 says STGES results in a CPDAG and estimates the Markov equivalence class, but the proof concludes a tiered MPDAG and the restricted Markov equivalence class. Since a non-empty Kτ generally produces orientations beyond the CPDAG, the stated theorem is false as written. The proof indicates the intended correct statement, which should be reflected in the theorem statement.
[Section 7] The tuning of λ so that TGEStuned has exactly the same number of edges as the expert DAG uses the gold standard to select a hyperparameter. This makes the reported confusion matrices and p-values partly a product of fitting to the reference graph. The authors should clearly state that this is an illustrative calibration and not an unbiased evaluation of TGES.
[Abstract and Section 5.2] The abstract says the tiered MPDAG 'contains the true DAG' in the large sample limit, while Theorem 3 more precisely states that the tiered MPDAG is a sound and complete estimate of the restricted Markov equivalence class of the true DAG. The looser wording may mislead readers about the identifiability of individual DAGs.
[Algorithms 4–6, line 7] The instruction 'W ← CPDAG of W' is ambiguous when W is a tiered MPDAG rather than a DAG. The authors should specify whether the CPDAG is computed from a DAG extension of W and how that extension is chosen.
[Appendix B.6.1, Case D.2] The sentence 'we know that Kτ /∈ PaG_Y' appears to contain a typo and should read 'we know that K /∈ PaG_Y'.

Circularity Check

1 steps flagged · score 2.0 of 10

Core derivation is self-contained (built on Chickering 2003, Meek 1995, and Bang and Didelez 2023); the only circularity is in the data application, where the TBIC penalty lambda is tuned to match the expert DAG edge count and the expert DAG is then used as the evaluation target.

fitted input called prediction [Section 7 (Data Application), Tables 1 and 2, and Appendix D]
"We choose λ such that TGEStuned has the same number of edges (30) as the expert consensus DAG. ... TGEStuned finds 10 out of the 30 adjacencies in the expert DAG."

The penalty λ is not selected by an independent model-selection criterion; it is set so that the estimated graph has exactly the same number of edges as the expert consensus DAG, and that same expert DAG is then used as the reference for the confusion matrix and for the random-guessing p-values. Choosing λ to match the expert edge count fixes the marginal TP+FP = 30, so the reported agreement counts are an in-sample evaluation of a target-derived tuning quantity rather than an independent prediction. The specific set of 10 correct adjacencies is not fully forced by this choice, which is why this is a minor, partial circularity; it does not affect the theoretical soundness and completeness proof of TGES.

full rationale

The theoretical derivation of TGES is not circular. Definitions 1 and 2 and the claim that Meek's rule 1 yields a tiered MPDAG are imported from external work by Bang and Didelez (2023) and Meek (1995); the main proof of Theorem 3 adapts Chickering's (2003) GES proof to restricted equivalence classes, using properties that are stated independently of the target result. TBIC is defined by setting the score to -∞ for graphs contradicting Kτ, and the Kτ-consistency and Kτ-score equivalence results then follow from the corresponding BIC properties; this is a designed extension, not a circular reduction. The paper's self-citations (Petersen et al. 2023a for the expert DAG, Petersen 2024 for the random-guessing test) are not load-bearing for the theorem. The only circularity I identified is in the real-data illustration, where TGEStuned's penalty is tuned to match the edge count of the expert DAG that is later used as ground truth; this is a validation leakage rather than a forced derivation, so it warrants a low score. Separate from circularity, there are correctness concerns worth noting: Algorithms 4-6 evaluate local score differences using parent sets of a tiered MPDAG, whose parent sets are not uniquely defined without choosing a DAG extension, and Algorithm 1's line 4 appears to orient an undirected edge in the direction that is forbidden by Kτ; these are implementation or proof-coverage gaps, not circularity.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The theoretical core of TGES has no fitted parameters: TBIC is a fixed scoring criterion with no tuning constant. The free parameters listed appear only in the empirical evaluation: the tuned lambda in the real-data example and the selected alpha for the TPC baseline. The asymptotic claims rest on standard causal discovery assumptions (causal sufficiency, faithfulness, Gaussianity) plus the correctness of the tiered background knowledge, and on external results from Chickering, Meek, and Bang and Didelez. No new physical or ontological entities are introduced.

free parameters (2)

lambda penalty scaling for TGEStuned = chosen so graph has 30 edges, value not reported
Section 7: TGEStuned uses a scaled penalty S_lambda_tau and lambda is tuned so the estimated graph has exactly 30 edges, matching the expert DAG. The confusion matrix and p-values depend on this choice.
TPC alpha significance level = 0.01
Section 6.1: alpha = 0.01 was selected because it performed best on the evaluation metrics. This is a competitor hyperparameter, but its post-hoc selection affects the TGES versus TPC comparison.

assumptions (7)

domain assumption Causal sufficiency: no unobserved confounders and no selection
Stated in Section 2. Without this, the estimated graph may not correspond to the true causal DAG and the asymptotic guarantee can fail.
domain assumption The distribution is multivariate Gaussian, Markov and faithful to the true DAG; data are i.i.d.
Section 2. Needed for BIC consistency, local consistency, and the equivalence between d-separation and conditional independence used throughout the proofs.
domain assumption The true data-generating DAG encodes the tiered background knowledge K_tau
Section 4 states 'We assume throughout that the background knowledge is encoded by the true distribution p'. If a tier assignment is wrong, the restricted search space can exclude the true DAG and TGES is inconsistent by construction.
standard math Chickering 2003 Theorem 4 and GES valid-edge results hold and extend to restricted equivalence classes
Used throughout Appendix B.6 to construct sequences of edge additions and reversals and to justify the validity conditions in Algorithms 4-6.
standard math Applying Meek's rule 1 alone to a CPDAG restricted by tiered background knowledge yields the tiered MPDAG
Invoked in Lemma 3 and stage (iii), citing Bang and Didelez 2023. The paper does not reproduce the proof of this characterization.
standard math BIC is consistent, locally consistent, decomposable, and score equivalent for Gaussian data
Proposition 2 cites Haughton 1988 and Chickering 2003. TBIC inherits these properties for graphs that encode K_tau.
domain assumption The score criterion satisfies K_tau-consistency, K_tau-score equivalence, and decomposability
Theorem 3 is conditional on these properties. They are proven for TBIC on Gaussian data in Proposition 1 and Appendix B.4.

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Cite this review

Pith. "Pith review of Score-Based Causal Discovery with Temporal Background Information." pith.science (2026). https://pith.science/paper/AV54QTJI

@misc{pith2026250206232,
  author       = {Pith},
  title        = {Pith review of: Score-Based Causal Discovery with Temporal Background Information},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AV54QTJI}},
  note         = {Machine review of arXiv:2502.06232}
}

read the original abstract

Temporal background information can improve causal discovery algorithms by orienting edges and identifying relevant adjustment sets. We develop the Temporal Greedy Equivalence Search (TGES) algorithm and terminology essential for score-based causal discovery with tiered background knowledge. TGES learns a restricted Markov equivalence class of directed acyclic graphs (DAGs) using observational data and tiered background knowledge. To construct TGES we formulate a scoring criterion that accounts for tiered background knowledge. We establish theoretical results for TGES, stating that the algorithm always returns a tiered maximally oriented partially directed acyclic graph (tiered MPDAG) and that this tiered MPDAG contains the true DAG in the large sample limit. We present a simulation study indicating a gain from using tiered background knowledge and an improved precision-recall trade-off compared to the temporal PC algorithm. We provide a real-world example on life-course health data.

Figures

Figures reproduced from arXiv: 2502.06232 by the authors.

**Figure 3.** Boxplots showing adjacency, all direction, and [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 2.** Boxplots showing the standardized Structural [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: An overview of the steps in Algorithm 3 (STGES). [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: Representations of Meek’s rules 1-4 [Meek, 1995]. A dashed edge is inferred by the rule. The symbol [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: Illustration of Lemma 4 proof. Here π is chosen as a path from X to Y in G ∗ that is open given PaG X. This path exists by X ̸⊥⊥G∗ Y |PaG X. K is the node in the collider structure of C closest to Y on π. πV is the sub-path of π which follows π from X to V . 26 [PITH_…

**Figure 7.** Figure 7: Computational time for a given number of nodes from 3 to 20 nodes. For each number of nodes, the computation [PITH_FULL_IMAGE:figures/full_fig_p029_7.png]

**Figure 8.** Figure 8: Standardized Structural Hamming Distance for GES, STGES, TGES, TPC with [PITH_FULL_IMAGE:figures/full_fig_p030_8.png]

**Figure 9.** Figure 9: Metrics of adjacencies, all directions and in-tier directions for GES, STGES, TGES, TPC with [PITH_FULL_IMAGE:figures/full_fig_p031_9.png]

**Figure 10.** Figure 10: Tiered MPDAG fitted using TGES with score criterion TBIC with a tuned penalty parameter such that the number [PITH_FULL_IMAGE:figures/full_fig_p032_10.png]

**Figure 11.** Figure 11: Tiered MPDAG fitted using TGES with score criterion TBIC on the data from Petersen et al. [2023a]. [PITH_FULL_IMAGE:figures/full_fig_p033_11.png]

**Figure 12.** Figure 12: Expert consensus DAG from Petersen et al. [2023a]. [PITH_FULL_IMAGE:figures/full_fig_p033_12.png]

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Forward citations

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Integrating Background Knowledge for Scalable Causal Discovery
stat.ML 2026-07 accept novelty 5.0 of 10

Integrating adjacency, orientation, and gap background knowledge during PC, SNAP, MB-by-MB, LDECC, and LOAD reduces CI tests and improves causal-effect estimates under standard assumptions.

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