{"id":"c6584bdf-0ce5-4543-8dd9-71c5e8bed2c1","arxiv_id":"2412.04981","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"An adaptive constraint-based algorithm that provably recovers context-specific causal skeletons when the context variable is endogenous, under new sufficiency and acyclicity assumptions.","lead":"This paper presents a method for learning causal relationships that change with a context variable, even when that context variable is itself affected by the system. The method adapts the PC algorithm to test some dependencies on all data and others within individual contexts, with a proof that it recovers the true context-specific skeleton under stated assumptions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Single-graph-sufficiency is the load-bearing barrier: the physical-graph claim is not identifiable from the data the algorithm consumes, so the theorem is a conditional guarantee rather than a direct discovery result.","rationale":"The reader's weakest assumption was single-graph-sufficiency, and my stress-test points at the same assumption. I found no internal inconsistency in the proof of Theorem 1; the Markov, faithfulness, and cycle arguments are detailed and appear coherent. The load-bearing concern is instead epistemic: the central claim that the algorithm recovers G^phys_{R=r} depends on an equality of graphs that is not accessible from the observational distribution P(V,R). The concrete two-model construction shows that two SCMs with identical observational data can have different G^phys_{R=r}, so Algorithm 1 cannot tell them apart. This does not refute the theorem, which explicitly assumes single-graph-sufficiency, but it does mean the paper's causal-physical interpretation rests on an untestable condition. The authors acknowledge this in Sec. 3.3 and Def. C.2, and the reader's CONDITIONAL verdict already captures the resulting limitation. I therefore see no reason to change the verdict, and I agree with the reader's identification of the weakest assumption.","tokens_in":31639,"tokens_out":11179,"duration_ms":273471,"concrete_test":"Generate two SCMs with identical observational distribution P(V,R) as described: one satisfying single-graph-sufficiency (mechanism disabled by R) and one violating it through support restriction (T <= T0 whenever R=0), with the same structural equations for the remaining variables. Run Algorithm 1 in the oracle setting on both. If the recovered context-specific skeletons coincide while the true G^phys_{R=r} graphs differ, then single-graph-sufficiency is confirmed as non-identifiable from the data the algorithm sees. This would demonstrate that the physical interpretation of Algorithm 1's output cannot be settled by any available test and must be imposed by assumption.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Definition C.2 (single-graph-sufficiency) is the assumption that converts Algorithm 1's output from a descriptive skeleton into G^phys_{R=r}. The problem is not just that the assumption is hard to verify; it is not identifiable from the joint distribution P(V,R) that the algorithm actually uses. Example 3.2 can be turned into a pair of SCMs with identical P(V,R): take (i) Y = R*g(T) + eta_Y with g(T)=max(T,T0), and (ii) Y = max(T,T0) + eta_Y with P(T>T0|R=0)=0. In both cases the conditional independence relations are the same, so the oracle output of Algorithm 1 is identical. Yet in (i) G^descr_{R=r}=G^phys_{R=r}, while in (ii) G^descr_{R=r} != G^phys_{R=r} because the missing T->Y link is due to restricted support, not a mechanism change. Thus no conditional independence test on any amount of data can distinguish the support problem from a genuine physical mechanism change. Theorem 1 is internally sound as a conditional statement, but the equality G^descr_{R=r}=G^phys_{R=r} is an input assumption, not something the algorithm discovers. The authors state this limitation honestly, so the concern is about scope and practical interpretability, not about a hole in the proof. Still, it is the single most load-bearing issue for the paper's central causal claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies constraint-based causal discovery of context-specific causal graphs when the context variable R is endogenous. It introduces three graph notions (descriptive, physical, and counterfactual), identifies two failure modes (selection bias and the support problem), and proposes a hierarchy of context-sufficiency assumptions. The main result (Theorem 1) states that, under causal sufficiency, single-graph-sufficiency, R-faithfulness, and strong context-acyclicity with union cycles of length at most 2, an adaptive modification of PC (Algorithm 1) recovers the skeleton of G^descr_{R=r}=G^phys_{R=r}. The paper also presents a simulation study comparing the proposed method with masking, pooling, an intersection baseline, FCI-JCI0, and CD-NOD.","tokens_in":31949,"tokens_out":5938,"duration_ms":62005,"significance":"If the result holds, the paper makes a useful contribution by formalizing why naive masking or pooling can fail for endogenous contexts and by identifying sufficient conditions under which a simple adaptive testing rule is sound. The distinction between descriptive and physical context-specific graphs, and the explicit treatment of the support problem, are valuable and clearly presented. The proof of Theorem 1 is detailed, the assumptions are stated explicitly, and the code is provided. The main caveat, which the authors acknowledge, is that the physical interpretation depends on single-graph-sufficiency, an assumption that is not identifiable from the observable distribution.","major_comments":[{"comment":"The equality G^descr_{R=r}=G^phys_{R=r} in Theorem 1 is an input assumption, not a discovered property. Definition C.2 (single-graph-sufficiency) is not identifiable from P(V,R): as Example 3.2 can be modified to show, two SCMs can have identical P(V,R) but differ in whether a missing T->Y edge is due to an explicit mechanism change or to restricted support. The manuscript says this assumption is 'potentially harder to verify directly,' but it should state more strongly that no oracle independence test can distinguish the two cases. Please add an explicit non-identifiability statement (e.g., a proposition exhibiting two SCMs with the same P(V,R) and different G^phys) and temper the abstract/title wording so that the algorithm is described as recovering descriptive context-specific graphs, with physical interpretation only under an untestable assumption.","section":"§4.2, Theorem 1 and §C, Definition C.2"},{"comment":"The simulation study only generates SCMs that satisfy single-graph-sufficiency by construction: every edited variable is made a child of R, and operations add/remove/flip edges as explicit nonzero coefficients in the structural equations. The study therefore never includes support-problem instances of the kind illustrated in Example 3.2(ii), where G^descr_{R=r} differs from G^phys_{R=r} while P(V,R) is compatible. As a result, the finite-sample experiments do not probe the main limitation of the physical interpretation identified in §3.2. The authors should either add a synthetic support-problem scenario that violates single-graph-sufficiency (e.g., an edge removal due to restricted support rather than an explicit mechanism change) or at least explicitly state why such scenarios are excluded and how the method is expected to behave when they occur.","section":"§5.1 and §E.1"}],"minor_comments":[{"comment":"In line 8, the notation 'S \\ R, R= r' is ambiguous; it should read 'S \\ {R}, R = r' to make clear that R is removed from the conditioning set and then set to the context value.","section":"§4.1, Algorithm 1"},{"comment":"The text refers to 'the left plot' and 'the right plot' of Figure 3, but the figure has two panels; 'left panel' and 'right panel' would be clearer.","section":"§5.2, Figure 3"},{"comment":"The data-generation description says that after each edit an edge from R to the affected child is added; the authors should explicitly note that this makes the generated models satisfy weak context-sufficiency, and in fact single-graph-sufficiency, by construction, since this is important context for interpreting the simulation results.","section":"§E.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically sound within its stated assumptions, and the authors are honest about the main limitation. The stress-test concern about single-graph-sufficiency is real and should be addressed by sharpening the claims: the physical-graph interpretation is conditional on an untestable assumption, and the paper currently does not state this non-identifiability as explicitly as it could. The simulation study's omission of support-problem scenarios is the main empirical gap. I would support publication after a revision that clarifies these points."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe short version: this paper deserves a serious referee. It gives the first SCM-based treatment of context-specific causal graphs when the context indicator is endogenous, and it proves soundness of a simple PC modification (PC-AC) under explicit assumptions. The main theorem is real: oracle PC-AC recovers the skeleton of G^{descr} = G^{phys} under causal sufficiency, single-graph-sufficiency, R-faithfulness, and strong context-acyclicity with cycles of length ≤2. The proof is in the appendix and looks careful.\n\nWhat's genuinely new: the adaptive testing rule (pooled unless R is in the separating set) is a simple but clever modification that avoids both selection bias from masking and the loss of context-specific information from pooling. The paper also clarifies an important distinction that prior work glossed over: descriptive graphs (what you see in a context) vs physical graphs (mechanism changes) vs counterfactual graphs. The support problem—where a context-specific independence arises from restricted support rather than a mechanism change—is real, and the paper handles it honestly.\n\nThe soft spots are real but not fatal. The biggest is single-graph-sufficiency (Definition C.2). As the stress-test note correctly observes, this assumption is not identifiable from P(V,R), the data the algorithm consumes. You can build two SCMs with identical observable distribution where one has G^{descr}=G^{phys} and the other doesn't. So Theorem 1's conclusion G^{descr}_{R=r}=G^{phys}_{R=r} is conditional on an assumption no test can verify. The authors state this plainly, but it does mean the physical interpretation is a modeling bet, not a discovery result. The fallback is weaker and still useful: under weak context-sufficiency, the algorithm remains descriptively sound.\n\nSecond, the simulation study is honest but limited. Without prior knowledge of R's adjacencies, FPR is high; the paper attributes this to the mixed-data CIT problem, and the non-parametric CIT experiment supports that. Practical utility currently depends on a computationally expensive CIT.\n\nThird, the theory handles only cycles of length ≤2; larger cycles are a known failure mode (D.7-D.8). Not a flaw if scoped, but it narrows the claim.\n\nThe citation pattern looks fine; they engage explicitly with the closest prior work (LDAGs, JCI) and delimit from it. Code and data are public, which helps.\n\nWho should read this: anyone working in Earth sciences or other settings where regime indicators are endogenous. The formalization of descriptive vs physical vs counterfactual graphs will be useful even if they never use PC-AC. I'd bring it to reading group.\n\nRecommendation: send it to peer review. The theorem is sound, the limitations are stated, and the contribution is original. It will need heavy revision on practicality, but it deserves referees, not a desk reject.","headline":"Solid theoretical contribution: a sound adaptive PC variant for endogenous context variables, but the physical-graph interpretation rests on an unverifiable identifiability assumption.","tokens_in":32450,"tokens_out":2123,"would_cite":true,"duration_ms":25026,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A modified PC algorithm provably recovers context-specific causal skeletons when the context indicator is endogenous.","keywords":["causal discovery","context-specific independence","endogenous context variables","constraint-based algorithm","structural causal models","selection bias","PC algorithm","regime-dependent causality"],"falsifier":"Generate a system like the right-hand example of Sec. 3.2, where $P(T>T_0|R=0)<\\epsilon$ makes $T\\to Y$ disappear only because support is restricted, and run Algorithm 1 with oracle independence tests together with a direct check of whether $T$ remains a parent of $Y$ in the intervened mechanism $M_{\\mathrm{do}(R=0)}$. If the algorithm removes $T\\to Y$ while the physical graph keeps it, single-graph-sufficiency is violated and the physical interpretation is not valid.","tokens_in":31458,"feed_emoji":"🔬","tokens_out":8339,"duration_ms":84456,"temperature":0.7,"pith_summary":"Context indicators—variables that mark a change in causal mechanisms, like a soil-moisture regime—are usually treated as exogenous in causal discovery, yet in many systems they are influenced by the very variables they condition. The paper shows that the obvious fixes fail: masking data by context introduces selection bias when the context indicator is a collider or descendant, while pooling data loses exactly the context-specific edges one wants. It proposes a small modification of the PC algorithm: test conditional independence on the pooled dataset, except when the context indicator $R$ is in the conditioning set, in which case test within each context $R=r$. Under causal sufficiency, single-graph-sufficiency, $R$-faithfulness, and strong context-acyclicity with minimal union cycles of length at most 2, the oracle version provably recovers the skeleton of the physical context-specific graph. The proof rests on a careful separation of three graph notions—descriptive, physical, and counterfactual—and on a hierarchy of sufficiency assumptions that the paper makes explicit.","feed_headline":"Regime-only causal links found despite endogenous contexts","feed_subtitle":"A single conditioning rule decides when to test within a context, avoiding selection bias and preserving pooled data.","key_machinery":"The central object is the context-specific graph triple ($G^{\\mathrm{descr}}_{R=r}$, $G^{\\mathrm{phys}}_{R=r}$, $G^{\\mathrm{CF}}_{R=r}$) together with the adaptive test-selection rule. The rule determines, for every candidate separating set $S$, whether the independence is tested on the pooled data (if $R\\notin S$) or per context (if $R\\in S$). The load-bearing property is that, under weak context-sufficiency, physical mechanism changes are confined to children of $R$, while selection-bias-induced spurious links are confined to ancestors of $R$; with strong context-acyclicity these two sets do not overlap, so the rule never needs to test both ways on the same edge. The soundness proof uses the counterfactual graph to transfer d-separation in the context-specific DAG to a statement about the observed conditional distribution via counterfactual consistency.","core_discovery":"At the center of the paper is a distinction between three graphs associated with an SCM and a context value $r$: the descriptive graph (mechanisms in the intervened model, evaluated on the observed conditional support), the physical graph (mechanisms in the intervened model, evaluated on the full pooled support), and the counterfactual graph (what would be observed had $R$ been set to $r$ under the same noises). The paper's central claim is that a single adaptive decision rule lets a constraint-based algorithm recover the skeleton of the first two when they coincide. The rule is simple: for a candidate independence $X \\perp\\!\\!\\perp Y \\mid S$, use the pooled test unless $R \\in S$; when $R$ is in the conditioning set, test $X \\perp\\!\\!\\perp Y \\mid S\\setminus\\{R\\}, R=r$ inside each context. Theorem 1 states that in the oracle case, under causal sufficiency, single-graph-sufficiency, $R$-faithfulness, and strong context-acyclicity with minimal union cycles of length at most 2, Algorithm 1 recovers the skeleton of $G^{\\mathrm{descr}}_{R=r}=G^{\\mathrm{phys}}_{R=r}$. The paper also shows that without single-graph-sufficiency the output remains descriptively correct under weaker conditions, but should not be read as a physical mechanism change.","pith_inferences":["Editorial inference: a direct analogue of the decision rule should apply in time-series and FCI-style settings, where the context indicator's role as collider or ancestor determines whether per-context tests are safe; the paper notes the extension but does not develop it.","Editorial inference: the support problem implies that a context-specific independence discovered by any conditional-independence test should be checked against the observational support of the conditioning variables, otherwise a descriptive edge removal may be mistaken for a physical mechanism change.","Editorial inference: in parametric settings such as linear-Gaussian models per context, one could test single-graph-sufficiency directly by checking whether a link that vanishes per context corresponds to a zero coefficient in the intervened mechanism rather than to a restricted support."],"forward_implications":["A user who observes $R$ and satisfies the stated assumptions can replace an expensive masked search or an intersection-graph baseline with a single PC-style run that only occasionally goes per-context.","The adaptive rule can be attached to other constraint-based algorithms, giving a general recipe for adding context-specific information to skeleton discovery under endogenous contexts.","When single-graph-sufficiency fails, the algorithm still returns a graph between the descriptive and physical extremes; the correct interpretation is descriptive, not physical.","The number of context-specific tests concentrates near $R$ and scales roughly linearly in the number of context variables, so the method remains feasible when many regimes are present."],"supporting_citations":[{"why":"Defines the PC algorithm that Algorithm 1 modifies and whose oracle skeleton-search guarantees are reused.","marker":"[5]"},{"why":"Provides the counterfactual consistency result used to convert per-context conditional independence into a statement about the intervened SCM.","marker":"[17]"},{"why":"Gives the standard Markov property connecting d-separation in a context-specific DAG to conditional independence.","marker":"[21]"},{"why":"Supplies sigma-separation and Markov-property foundations for SCMs with cycles, needed for union graphs containing cycles.","marker":"[2]"},{"why":"Defines the JCI pooled-data treatment of context indicators as endogenous variables, the starting point the paper extends.","marker":"[16]"},{"why":"Describes the closest prior constraint-based approach for labeled DAGs, which tests inside each context and which the adaptive rule is designed to improve upon.","marker":"[10]"},{"why":"Introduces LDAGs, the context-specific independence models whose semantics the paper contrasts with its SCM-based context-specific graphs.","marker":"[19]"},{"why":"Provides the robustness argument that under weak context-sufficiency the algorithm remains descriptively complete even when single-graph-sufficiency fails.","marker":"[23]"}],"fun_headline_variants":["Endogenous contexts: one rule for when to condition","One conditioning rule recovers context-specific causal graphs","Causal discovery adapts to endogenous context variables","Pooled tests mislead; context-conditional tests uncover true graph"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is single-graph-sufficiency: within each context the descriptive, physical, and counterfactual graphs must coincide, so that an observed context-specific independence reflects a real mechanism change rather than merely a restricted region of the data.","fun_headline_variants_meta":{"raw":{"variants":["Endogenous contexts: one rule for when to condition","One conditioning rule recovers context-specific causal graphs","Causal discovery adapts to endogenous context variables","Pooled tests mislead; context-conditional tests uncover true graph"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001464,"raw_usage":{"total_tokens":5943,"prompt_tokens":1054,"completion_tokens":4889,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":670,"completion_tokens_details":{"reasoning_tokens":4825}},"tokens_in":670,"tokens_out":4889,"duration_ms":37889,"temperature":1.0,"reasoning_tokens":4825,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:01:08.994330+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Generate a system like the right-hand example of Sec. 3.2, where $P(T>T_0|R=0)<\\epsilon$ makes $T\\to Y$ disappear only because support is restricted, and run Algorithm 1 with oracle independence tests together with a direct check of whether $T$ remains a parent of $Y$ in the intervened mechanism $M_{\\mathrm{do}(R=0)}$. If the algorithm removes $T\\to Y$ while the physical graph keeps it, single-graph-sufficiency is violated and the physical interpretation is not valid.","supporting_citations":[{"cited_title":"Glymour, K","cited_arxiv_id":null,"evidence_quote":"Defines the PC algorithm that Algorithm 1 modifies and whose oracle skeleton-search guarantees are reused."},{"cited_title":"Peters, D","cited_arxiv_id":null,"evidence_quote":"Gives the standard Markov property connecting d-separation in a context-specific DAG to conditional independence."},{"cited_title":"Bongers, P","cited_arxiv_id":null,"evidence_quote":"Supplies sigma-separation and Markov-property foundations for SCMs with cycles, needed for union graphs containing cycles."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the JCI pooled-data treatment of context indicators as endogenous variables, the starting point the paper extends."},{"cited_title":"Hyttinen, J","cited_arxiv_id":null,"evidence_quote":"Describes the closest prior constraint-based approach for labeled DAGs, which tests inside each context and which the adaptive rule is designed to improve upon."},{"cited_title":"Pensar, H","cited_arxiv_id":null,"evidence_quote":"Introduces LDAGs, the context-specific independence models whose semantics the paper contrasts with its SCM-based context-specific graphs."},{"cited_title":"Causal Modeling in Multi-Context Systems: Distinguishing Multiple Context-Specific Causal Graphs which Account for Observational Support","cited_arxiv_id":"2410.20405","evidence_quote":"Provides the robustness argument that under weak context-sufficiency the algorithm remains descriptively complete even when single-graph-sufficiency fails."}],"review_version":1}