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SNAP: Sequential Non-Ancestor Pruning for Targeted Causal Effect Estimation With an Unknown Graph

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

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Pith's one-line read The full causal graph is unnecessary for targeted effect estimation: a sequential pruning algorithm that keeps only possible ancestors of the targets recovers exactly the relevant CPDAG.

desk verdict Useful targeted causal discovery method, but the k=1 soundness proof has a concrete lemma-application gap that needs fixing before the prefiltering claims are credible. read the letter →

arxiv 2502.07857 v1 pith:ZJJFX3P3 submitted 2025-02-11 stat.ML cs.AIcs.LG

Mátyás Schubert , Tom Claassen , Sara Magliacane This is my paper · ORCID
classification stat.MLcs.AIcs.LG MSC 62D20
keywords targetedcausaleffectestimationdiscoverypossibleancestorsnon-ancestorpruningconditionalindependencetestsCPDAGadjustmentsetssufficiency
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

The paper takes on targeted causal effect estimation with an unknown graph: the user names a small set of target variables and wants the interventional effects between them, without learning the full causal graph over all variables. Its central claim is that only the possible ancestors of the targets — variables that could lie on a causal path into a target in some graph consistent with the data — are needed for this task, both to orient the causal relations among targets and to form statistically efficient adjustment sets. The paper proves that a possibly ancestral superset of these variables supports exactly the same restricted CPDAG and the same adjustment sets as the full graph, and it contributes SNAP, an algorithm that alternates low-order conditional-independence search with v-structure orientation and prunes definite non-ancestors at every step. In experiments with oracle and finite-sample tests, SNAP cuts the number of independence tests and the computation time by large factors while keeping causal effect estimates comparable to global discovery.

What carries the argument

The central object is the set of possible ancestors of the targets. In a CPDAG $G$, a node $X$ is a possible ancestor of $Y$ when a possibly directed path connects them — a path made of undirected edges and edges pointing toward $Y$ — which is equivalent to $X$ being an ancestor of $Y$ in at least one DAG of the Markov equivalence class. The load-bearing identity is Lemma 3.1: a possibly ancestral set $V^*$ that contains $\mathrm{PossAn}(T)$ induces the same CPDAG as the full graph restricted to $V^*$, so discovery can be confined to $V^*$ without losing any informative adjustment set. The algorithm's mechanism is a loop that runs a PC-style skeleton search with conditioning sets of size $i$, orients v-structures (using PC rules for $i \in \{0,1\}$ and RFCI-style rules with extra dependency checks for $i \geq 2$, because low-order tests can create conflicting v-structures even when the underlying edge is real), and then removes every node with no possibly directed path to a target. The safety of the cheap PC-style orientation at order 1 rests on a d-separation lemma about treks (Lemma B.3), applied repeatedly.

What would settle it

Enumerate all labeled DAGs on four nodes and check Lemma B.3 directly: any quadruple $(W, X, Y, Z)$ in which $X$ and $Y$ are d-connected given the empty set, $X \perp\!\!\!\perp_d Y \mid Z$ and $Z \perp\!\!\!\perp_d Y \mid W$ both hold, but $X \perp\!\!\!\perp_d Y \mid W$ fails, is a counterexample that collapses the order-1 soundness proof. A complementary end-to-end check is to run SNAP(1) with oracle d-separation tests over all DAGs up to six nodes and all possible target sets, recording the first graph in which a true possible ancestor of a target is pruned.

Watch

Extended reading notes

Core claim

The paper establishes that definite non-ancestors of the targets are unnecessary for learning causal relations among the targets and for identifying efficient adjustment sets. Theorem 3.1 states that, given oracle conditional independence tests, every iteration of SNAP(k) retains all possible ancestors of the targets and leaves a possibly ancestral set, one that contains all of its own possible ancestors. Lemma 3.1 then guarantees that any possibly ancestral set containing the targets' possible ancestors yields the same canonical, parental, ancestor, and asymptotically optimal adjustment sets as the full CPDAG. Theorem 3.2 completes the picture: SNAP(∞), the standalone variant, returns exactly $\hat{G} = G|_{\mathrm{PossAn}(T)}$, the induced subgraph of the full CPDAG over the possible ancestors of the target set, and is sound and complete over that subgraph. The upshot is that recovering the complete CPDAG over all variables is not required for targeted causal effect estimation; the relevant subgraph suffices, and pruning can happen during discovery rather than after it.

Load-bearing premise

The soundness guarantee assumes oracle conditional-independence tests, and within that idealization the fragile step is Lemma B.3, a d-separation implication about treks that lets SNAP orient v-structures cheaply at order 1; if that implication fails in any configuration, a true possible ancestor of the targets could be mislabeled as a definite non-ancestor and pruned.

Editorial extensions

If this is right

  • Stopping SNAP(k) at any order and running a sound-and-complete discovery algorithm on the survivors returns the induced subgraph of the full CPDAG restricted to those variables, so all informative adjustment sets for the target effects are preserved (Corollary 3.1).
  • Even SNAP(0), which prunes purely on marginal-independence information, already cuts independence tests and runtime by large factors as a prefilter in the reported experiments, and it consistently improved PC, MARVEL, and MB-by-MB* across settings.
  • SNAP(∞) as a standalone method recovers exactly the induced CPDAG over $\mathrm{PossAn}(T)$, and its worst-case complexity of $O(|V|^{d_{\max}+2} + |V|^4)$ matches PC's $O(|V|^{d_{\max}+2})$ for graphs with maximum degree at least 2.
  • SNAP's cost is nearly constant as the number of targets grows, because all targets are handled jointly, whereas local methods that process targets one at a time degrade substantially with more targets.
  • Because SNAP concentrates its independence tests at low orders, it avoids high-order conditional-independence tests, which are statistically weaker for kernel-based and $\chi^2$ tests.

Reading between the lines

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

  • My inference: the 'possible ancestors are all you need' principle should carry over to causally insufficient settings, where the analogous object is the set of nodes possibly ancestral to the targets in a partial ancestral graph; the paper names this as future work, and nothing in Lemma 3.1 depends essentially on causal sufficiency.
  • My inference: SNAP's prefiltering is agnostic to the downstream method, so it should combine equally well with score-based discovery, where the reduced variable set also shrinks the search space over DAGs.
  • My inference: the paper's own finite-sample diagnostics (App. D.9) find more missing than extra edges and attribute them to pruning the wrong nodes, which suggests that a confidence-aware pruning rule — keeping nodes whose possible-ancestor status is borderline — could close the gap between the oracle soundness guarantee and finite-sample behaviour; the paper does not pursue this.
  • My inference: Lemma B.3 is a purely discrete statement about d-separation, so exhaustive enumeration over all four-node DAGs would certify or refute it outright, a check that costs almost nothing and would settle the most fragile step of the proof.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper addresses targeted causal effect estimation with an unknown graph: rather than learning a full CPDAG over all variables, it proposes to learn only a subgraph over a possibly ancestral superset of the possible ancestors of a small target set T. The central claim is that definite non-ancestors of T are unnecessary both for identifying causal relations among targets and for finding valid and efficient adjustment sets. The proposed SNAP(k) algorithm interleaves a PC-style skeleton search with orientation of v-structures and iterative pruning of nodes that have no possibly directed path to T; SNAP(∞) adds Meek rule completion and is claimed to return exactly the induced CPDAG over PossAn(T). The paper gives oracle soundness/completeness theorems (Theorems 3.1 and 3.2), a worst-case complexity analysis, and extensive experiments on synthetic data and two real benchmark networks.

Significance. If the oracle claims are correct, the paper makes a useful step: it formalizes a targeted discovery task and shows that only possible ancestors of the target set need to be retained, building on known ancestral-set preservation results of Lauritzen and Guo et al. The paper is strong on framing, has extensive experiments with several CI test families, reports both computational and effect-estimation metrics, and releases code. No fitted constants or circularity in the main theorems was found. However, the k=1 soundness proof contains a concrete gap in the appendix, and Theorem 3.2 and the prefiltering Corollary 3.1 rely on that k=1 case. Because SNAP(1) and SNAP(∞) are central to the paper's practical claims, this gap is load-bearing and prevents accepting the paper in its current form.

major comments (3)
  1. [Appendix B, Lemma B.4] The proof of the k=1 case applies Lemma B.3 with a substitution that does not fit the lemma's statement. Lemma B.3 states: if X and Y are d-connected given the empty set, X⊥d Y | Z, and Z⊥d Y | W, then X⊥d Y | W. In Lemma B.4, to conclude X⊥d Z | {V}, the proof requires the third antecedent to be W⊥d Z | {V}. Instead, the displayed implication uses X⊥d W | {V}. Thus the claimed derivation is not an instance of Lemma B.3, and the same mismatch recurs in the 'two successive applications' argument later in the same proof. As written, the k=1 case of Theorem 3.1 is not established, and since Theorem 3.2 and Corollary 3.1 depend on Theorem 3.1, the central soundness claim that SNAP(1) prunes only definite non-ancestors is not backed by the supplied proof.
  2. [Appendix B, Lemma B.4 and Theorem 3.1] The repeated-application argument in Lemma B.4 is an inductive chain over non-adjacent nodes, but each step invokes the same unmatched form of Lemma B.3. Even if the intended implication 'X⊥d Z|{W}, X⊥d W|{V}, X not⊥d Z implies X⊥d Z|{V}' were true, it is not proved in the paper; the proof as written is therefore incomplete for every iteration in the chain, not just the first step. The authors should either state and prove the correct auxiliary lemma, or modify the algorithm so that the RFCI dependency checks are also used at iteration k=1.
  3. [Appendix D.9] The paper itself reports that, in finite samples, missing edges in SNAP(∞)'s output arise not from type II CI-test errors but from pruning the wrong nodes. This is an honest limitation statement, but it should be weighed explicitly against the abstract's unqualified claim that pruning is done without compromising estimation quality. The oracle theorems are not contradicted by this finite-sample behavior, but the main text should state that the 'prunes only definite non-ancestors' guarantee holds only with oracle CI tests, and that finite-sample SNAP can remove true possible ancestors, with the experimental support for the no-compromise claim resting on intervention distance rather than structural fidelity.
minor comments (3)
  1. [Theorem 3.1] The theorem says a set is possibly ancestral, 'i.e., PossAn_G(\hat V_i) ⊆ \hat V_{i+1}'. The displayed condition is stronger than the definition of \hat V_{i+1} being possibly ancestral, which should be PossAn_G(\hat V_{i+1}) ⊆ \hat V_{i+1}. The proof establishes the stronger statement, but the wording is imprecise.
  2. [Section 3.2 / Section 6] The conclusion refers to 'SNAP(1), a stand-alone sound and complete discovery algorithm', while the rest of the paper defines SNAP(∞) as the sound and complete stand-alone algorithm. This appears to be a typo and should be corrected.
  3. [Appendix B.3] In the proof of Corollary 3.1, the sentence 'from the application of Theorem 3.1, that shows that a possibly ancestral set...' should presumably refer to Lemma 3.1, not Theorem 3.1.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the soundness theorems follow from d-separation arguments and externally cited lemmas, with no fitted parameters, self-citation chain, or definitional reduction.

full rationale

The paper's central claim is that definite non-ancestors of target variables are unnecessary for targeted causal effect estimation, and that SNAP(k) prunes only such nodes. This claim is supported by Theorem 3.1, whose proof rests on Lemma B.2 and Lemma B.4, both of which are d-separation arguments about the ground-truth DAG, and on Lemma B.3, a standalone graphical claim. The key graph-restriction result, Lemma 3.1, is imported from Lauritzen (1996) and Guo et al. (2023), which are external sources, not self-citations. The algorithm contains no fitted constants and no parameter is calibrated to the evaluation data; the experiments are benchmarked against synthetic ground-truth graphs and real networks, so the reported reductions in CI tests and runtime are not forced by construction. The only adoption of prior work is the use of RFCI orientation rules from Colombo et al. (2012), which is an external, independently established algorithm used as a building block, not an unverified premise that is doing the load-bearing work. The skeptical concern in the review about Lemma B.3 possibly being applied with mismatched conditioning sets is a potential correctness gap in the k=1 proof, not a circularity: it would make a theorem unproven if valid, but it does not make the theorem's conclusion an input to its own proof. There are no self-citations of the present authors at all, and no uniqueness theorem is invoked from the authors' prior work. Overall, the derivation chain is self-contained relative to standard external results, and no step was found in which a prediction reduces to its own inputs by definition or by fitting.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim depends on the standard causal discovery assumptions (causal sufficiency, Markov, faithfulness), oracle CI tests for the formal theorems, and imported ancestral-set results. The algorithm introduces no free parameters fitted to data and no new ontological entities; the user-chosen maximum order k is a configuration variable, not a fitted quantity.

assumptions (4)
  • domain assumption Causal sufficiency: no unobserved confounders or selection bias
    Stated in Section 2 and used by all theoretical results. Without this assumption, CPDAG discovery and the adjustment criteria used here would need latent-variable methods.
  • domain assumption Markov and faithfulness of the observational distribution to the causal DAG
    Section 2 assumes conditional independence in the distribution is equivalent to d-separation in the true graph. This is used in Lemma B.1 and throughout the soundness proofs.
  • domain assumption Oracle conditional independence tests for the theoretical guarantees
    Theorems 3.1 and 3.2 assume oracle CI tests, as stated in the theorem statements. The experiments switch to Fisher-Z, KCI, and chi-squared tests, which introduce finite-sample errors.
  • standard math Known results on ancestral sets and CPDAG restriction (Lauritzen Proposition 3.22 and Guo et al. Lemma D.1)
    Used in the proof of Lemma 3.1 to argue that a possibly ancestral set preserves the induced CPDAG when restricted.

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Cite this review

Pith. "Pith review of SNAP: Sequential Non-Ancestor Pruning for Targeted Causal Effect Estimation With an Unknown Graph." pith.science (2026). https://pith.science/paper/ZJJFX3P3

@misc{pith2026250207857,
  author       = {Pith},
  title        = {Pith review of: SNAP: Sequential Non-Ancestor Pruning for Targeted Causal Effect Estimation With an Unknown Graph},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZJJFX3P3}},
  note         = {Machine review of arXiv:2502.07857}
}
read the original abstract

Causal discovery can be computationally demanding for large numbers of variables. If we only wish to estimate the causal effects on a small subset of target variables, we might not need to learn the causal graph for all variables, but only a small subgraph that includes the targets and their adjustment sets. In this paper, we focus on identifying causal effects between target variables in a computationally and statistically efficient way. This task combines causal discovery and effect estimation, aligning the discovery objective with the effects to be estimated. We show that definite non-ancestors of the targets are unnecessary to learn causal relations between the targets and to identify efficient adjustments sets. We sequentially identify and prune these definite non-ancestors with our Sequential Non-Ancestor Pruning (SNAP) framework, which can be used either as a preprocessing step to standard causal discovery methods, or as a standalone sound and complete causal discovery algorithm. Our results on synthetic and real data show that both approaches substantially reduce the number of independence tests and the computation time without compromising the quality of causal effect estimations.

Figures

Figures reproduced from arXiv: 2502.07857 by the authors.

Figure 2
Figure 2. Number of CI tests (top row) and computation time (bottom row) over number of nodes with nT = 4, d = 3, dmax = 10 and nD = 1000 data-points for different simulation settings in each column. The shadow area denotes the range of the standard deviation. SNAP (∞) is consistently one of the best methods. performs well with oracle tests, but requires more CI tests and is hence slower with linear Gaussian data. For KCI and… view at source ↗
Figure 3
Figure 3. Dierence in computation time for each Figure 3: Dierence in computation time for each Figure 3: Dierence in computation time for each Figure 3: Dierence in computation time for each () Figure 3: Difference in computation time for each [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Number of d-separation tests performed by SNAP(k) during prefiltering for k = 0, 1, 2, 3 and SNAP(∞) (pink), to which we add the number of tests PC performs after prefiltering (blue) on graphs with |V| = 50, nT = 4, d = 5 and dmax = 10. and 15 show that SNAP achieves mostly comparable, but in some cases slightly higher Adjustment Identifica￾tion Distances (AID) than other methods. AID counts how many times the learn… view at source ↗
Figures from the paper (35 more)
Figure 5
Figure 5. Figure 5: Example of a bidirected edge, based on Figure 2 in [Wienöbst and Liskiewicz, 2020]. Fig. 5a shows [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: An example graph with targets T = {X1, X2} on which SNAP(∞) performs more CI tests than PC. B.6 Rough approximation of the expected number of possible ancestors In practice, the performance of SNAP depends on the number of possible ancestors of targets in the graph. We…
Figure 7
Figure 7. Figure 7: The MAGIC-NIAB network [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]
Figure 8
Figure 8. Figure 8: The Andes network [PITH_FULL_IMAGE:figures/full_fig_p024_8.png]
Figure 9
Figure 9. Figure 9: Number of CI tests and computation time for baseline methods combined with SNAP(0) over number of nodes, with nT = 4, d = 3, dmax = 10 and nD = 1000 data-points. PC MARVEL LDECC* MB-by-MB* FGES SNAP(∞) d-separation tests Fisher-Z tests KCI tests χ 2 tests 50 100 150 20…
Figure 10
Figure 10. Figure 10: Quality of estimation over number of nodes, with [PITH_FULL_IMAGE:figures/full_fig_p026_10.png]
Figure 11
Figure 11. Figure 11: Quality of estimation over number of nodes for baseline methods combined with SNAP [PITH_FULL_IMAGE:figures/full_fig_p027_11.png]
Figure 12
Figure 12. Figure 12: Additional results over number of nodes for identifiable targets, with [PITH_FULL_IMAGE:figures/full_fig_p028_12.png]
Figure 13
Figure 13. Figure 13: Additional results over number of nodes for identifiable targets, with [PITH_FULL_IMAGE:figures/full_fig_p029_13.png]
Figure 14
Figure 14. Figure 14: Adjustment identification distance over number of nodes for identifiable targets, with [PITH_FULL_IMAGE:figures/full_fig_p030_14.png]
Figure 15
Figure 15. Figure 15: Adjustment identification distance over number of nodes for identifiable targets, with [PITH_FULL_IMAGE:figures/full_fig_p031_15.png]
Figure 16
Figure 16. Figure 16: Number of CI tests and computation time over number of targets, with nV = 10 for KCI tests and nV = 200 otherwise, d = 3, dmax = 10 and nD = 1000 data-points. The shadow area denotes the range of the standard deviation. We plot values on a log scale. PC MARVEL LDECC* …
Figure 17
Figure 17. Figure 17: Estimation quality over number of targets, with [PITH_FULL_IMAGE:figures/full_fig_p032_17.png]
Figure 18
Figure 18. Figure 18: Number of CI tests and computation time over number of targets, with nV = 10 for KCI tests and nV = 200 otherwise, d = 3, dmax = 10 and nD = 1000 data-points. We also show baseline methods combined with SNAP(0). We plot values on a log scale. PC MARVEL LDECC* MB-by-MB…
Figure 19
Figure 19. Figure 19: Estimation quality over number of targets, with [PITH_FULL_IMAGE:figures/full_fig_p033_19.png]
Figure 20
Figure 20. Figure 20: Number of CI tests and computation time over number of identifiable targets, with nV = 10 for KCI tests and nV = 200 otherwise, d = 3, dmax = 10 and nD = 1000 data-points. We plot values on a log scale. The shadow area denotes the range of the standard deviation. PC M…
Figure 21
Figure 21. Figure 21: Estimation quality over number of identifiable targets, with [PITH_FULL_IMAGE:figures/full_fig_p034_21.png]
Figure 22
Figure 22. Figure 22: Number of CI tests and computation time over number of identifiable targets, with nV = 10 for KCI tests and nV = 200 otherwise, d = 3, dmax = 10 and nD = 1000 data-points. We also show baseline methods combined with SNAP(0). We plot values on a log scale. PC MARVEL LD…
Figure 23
Figure 23. Figure 23: Estimation quality over number of identifiable targets, with [PITH_FULL_IMAGE:figures/full_fig_p035_23.png]
Figure 24
Figure 24. Figure 24: Adjustment identification distance over number of identifiable targets, with [PITH_FULL_IMAGE:figures/full_fig_p036_24.png]
Figure 25
Figure 25. Figure 25: Adjustment identification distance for baseline methods combined with SNAP [PITH_FULL_IMAGE:figures/full_fig_p037_25.png]
Figure 26
Figure 26. Figure 26: Additional results over expected degree, with [PITH_FULL_IMAGE:figures/full_fig_p038_26.png]
Figure 27
Figure 27. Figure 27: Additional results for baseline methods combined with SNAP [PITH_FULL_IMAGE:figures/full_fig_p039_27.png]
Figure 28
Figure 28. Figure 28: Additional results over number of samples, with [PITH_FULL_IMAGE:figures/full_fig_p040_28.png]
Figure 29
Figure 29. Figure 29: Additional results for baseline methods combined with SNAP [PITH_FULL_IMAGE:figures/full_fig_p041_29.png]
Figure 30
Figure 30. Figure 30: Number of CI tests over number of nodes for SNAP(k) with k = 0, . . . , 2, with nT = 4, d = 3, dmax = 10 and nD = 1000 data-points [PITH_FULL_IMAGE:figures/full_fig_p042_30.png]
Figure 31
Figure 31. Figure 31: Computation time over number of nodes for SNAP( [PITH_FULL_IMAGE:figures/full_fig_p043_31.png]
Figure 32
Figure 32. Figure 32: Structural Hamming Distance (SHD) over number targets for SNAP( [PITH_FULL_IMAGE:figures/full_fig_p044_32.png]
Figure 33
Figure 33. Figure 33: Intervention distance over number targets for SNAP( [PITH_FULL_IMAGE:figures/full_fig_p045_33.png]
Figure 34
Figure 34. Figure 34: Tests per order for d-separation CI tests with nT = 4, nV = 20, d = 3 and dmax = 10. SNAP variants use lower order CI tests than baselines. The best method is indicated in bold. CI tests Time Int. dist. SHD PC 12806.95(±2085.96) 79.27(±24.75) 0.009(±0.007) 22.33(±6.02…
Figure 35
Figure 35. Figure 35: Type I and type II error rates of different CI tests performed by SNAP( [PITH_FULL_IMAGE:figures/full_fig_p047_35.png]
Figure 38
Figure 38. Figure 38: Our results show that SNAP(∞) performs on par with most baselines. For MAGIC-NIAB network, the average number of nodes involved in optimal adjustment sets among target pairs is 21.78. While there is still [PITH_FULL_IMAGE:figures/full_fig_p047_38.png]
Figure 36
Figure 36. Figure 36: Number of extra and missing edges in the output graphs by SNAP( [PITH_FULL_IMAGE:figures/full_fig_p048_36.png]
Figure 37
Figure 37. Figure 37: SHD on the induced subgraph of the true CPDAG over only the nodes included in the optimal adjustment sets for some target pair. Results are shown over number of nodes for identifiable targets, with nT = 4, d = 3, dmax = 10 and nD = 1000 data-points. The shadow area de…
Figure 38
Figure 38. Figure 38: SHD on the induced subgraph of the true CPDAG over only the nodes included in the optimal adjustment sets for some target pair, for the MAGIC-NIAB and Andes networks with nT = 4 identifiable targets and nD = 1000 data-points. The dots denote the mean, while the error …

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Cited by 1 Pith paper

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.