{"id":"27bd955d-cf87-4db6-85dd-60bbaaf7d0c7","arxiv_id":"2607.23439","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The orthocomplement of the tangent space of a general CI Markov model is the direct sum of the known single-constraint orthocomplements, yielding the full class of influence functions.","lead":"The paper gives closed-form expressions for the orthogonal complement of the tangent space of any Markov model defined by conditional independences, including undirected, chain, and ADMG models. This unlocks the full class of influence functions needed for efficient root-n estimation in those models, which previously lacked such a characterization outside DAGs.","discovery_kind":"new_method","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"The proof of the load-bearing inclusion ∩Ti ⊆ T uses a submodel construction that provably leaves the model at O(ε²); the main theorem is likely true (and repairable) for finite-state models, but it is not established as written.","rationale":"The reader identified the correct load-bearing assumption (T = ∩Ti, established via pε = p0(1+εf)) but mischaracterized the issue as \"usual regularity that such submodels remain inside the model for small ε.\" No choice of ε fixes the problem: the multiplicative submodel violates the CI at second order whenever f has components in both the X|Z and Y|Z subspaces, as the explicit binary counterexample shows. So the proof of the paper's central theorem contains a genuine, checkable gap, not merely an unstated regularity condition. Three considerations keep this from being a REJECT-level concern. First, the claim itself is very likely correct in the paper's core setting: for finite-state variables and positive p0, each single-CI model is a log-linear (hierarchical) model defined by vanishing interaction parameters, intersections of such models are again log-linear and smooth on the simplex interior, and for log-linear models the tangent space of an intersection equals the intersection of tangent spaces — so a correct proof should be obtainable by standard arguments, even though the paper does not give one. Second, all downstream results (Lemma 1, the DAG consistency check between Eqs. 10 and 30, the worked IF classes, and Theorem 4's variance improvement, which only needs Π(φ|Ti⊥) ∈ T⊥ — a fact that follows from the correct inclusion T ⊆ ∩Ti) are either independently correct or depend only on the uncontested direction of the inclusion. Third, the paper is unusually candid about what it does not deliver (no projection operator, no EIF, no Verma constraints). Nonetheless, a main theorem whose displayed proof contains a step that is false as stated cannot stand at ACCEPT/HIGH without repair; the fix is likely routine for discrete models but the general (continuous/mixed) case additionally needs a closedness argument for ⊕Ti⊥. CONDITIONAL: accept once the inclusion ∩Ti ⊆ T is given a valid proof (e.g., via log-linear parametrization) and the closedness assertion is qualified to finite-dimensional settings.","tokens_in":46477,"tokens_out":6477,"duration_ms":2036719,"concrete_test":"Two-part check. (1) Settle the proof gap: binary X,Y, p0=1/4, f=(−1)^x+(−1)^y ∈ Ti; with pε=p0(1+εf) compute pε(0,0) − pε(0)pε(0) = (1+2ε)/4 − (1+ε)²/4 = −ε²/4 ≠ 0 for all ε≠0. This confirms the construction in Theorem 3's proof does not stay in the model. (2) Settle whether the theorem nonetheless holds: in the binary Bell model (Fig. 1d) at uniform p0, take f ∈ T1∩T2 with components coupling both constraints, e.g. f=(−1)^{a+c}+(−1)^{b+d}; either exhibit an explicit curve pε ∈ P1∩P2 with score f (e.g., via the log-linear parametrization with the AB, AD, ABD, BC, ABC interactions set to zero) or show none exists. Existence for all such f confirms ∩Ti⊆T and rescues the main theorem; non-existence refutes Eq. 27.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorem 3 rests entirely on T = T1∩...∩TK; the nontrivial direction is ∩Ti ⊆ T. The Appendix proof takes f ∈ ∩Ti, sets pε = p0(1+εf), and asserts: \"Following the proof of Lemma 1, the i-th constraint must hold in pε(v).\" That is not what Lemma 1's proof shows, and the assertion is false. Lemma 1 verified the constraint only for f drawn from each orthogonal subspace separately; for a general sum, the multiplicative perturbation violates the CI at second order. Concretely: single constraint X⊥Y (Z,W empty), p0 uniform on {0,1}², f(x,y)=(−1)^x+(−1)^y. Then f ∈ Ti since Π(f|Ti⊥)=E[f|xy]−E[f|x]−E[f|y]+E[f]=0. But pε(x,y)=(1+ε(−1)^x+ε(−1)^y)/4 while pε(x)pε(y)=(1+ε(−1)^x)(1+ε(−1)^y)/4, differing by ε²(−1)^{x+y}/4 ≠ 0 for every ε≠0 — no range of ε keeps this submodel in the model. The reader's weakest_assumption flags this exact step but describes it as \"usual regularity\"; it is not regularity — the displayed construction fails outright. The direction is realizable by a different curve (factor-wise product), so K=1 survives, but for K≥2 constraints no valid construction is given, and realizing directions obeying all constraints simultaneously is close to the implicitization problem the paper itself calls open (§3.2). If ∩Ti strictly contained T anywhere, then T⊥ would be strictly larger than ⊕Ti⊥ and the closed-form characterization (Eq. 27) — the paper's central deliverable — would miss part of the true orthocomplement. The claim is very likely true for finite-state p0 with full support: each CI is then equivalent to the vanishing of a set of log-linear interaction parameters, so the intersection is a hierarchical log-linear model, smooth on the simplex interior, with tangent space exactly ∩Ti. But that argument (or any correct one) is absent. Secondary issue: in infinite-dimensional H the claim that ⊕Ti⊥ \"is exactly the direct sum\" because each Ti⊥ is closed is invalid — sums of closed subspaces need not be closed — so for continuous variables Eq. 27 may only describe a","agreement_with_reader":"partial"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The paper studies semiparametric Markov models — statistical models defined solely by a list of conditional independence constraints Xi ⊥ Yi | Zi, i = 1,...,K — with emphasis on graphical models not equivalent to any DAG model (undirected graphs, chain graphs, ADMGs under the ordinary Markov property). The main result (Theorem 3) characterizes the orthocomplement of the model's tangent space as the direct sum of the orthocomplements of the K single-constraint models, each element written in closed form via the projection Π(h|T⊥i) = E[h|xi,yi,zi] − E[h|xi,zi] − E[h|yi,zi] + E[h|zi]. The argument views the model as an intersection of single-independence DAG models, identifies the tangent space with the intersection of the individual tangent spaces (T = ∩Ti), and invokes the Hilbert-space identity (∩Ti)⊥ = closure(ΣT⊥i) in the spirit of Van Der Laan & Robins (2003, Lemma 1.7). Applications include the class of influence functions for conditional-mean targets in the graphs of Figure 1 (§5.1) and an iterative efficiency-improvement scheme based on sequential projections onto the T⊥i (Theorem 4). The authors correctly note that the projection operator onto T⊥, and hence the efficient influence function, remains open.","tokens_in":30720,"tokens_out":15160,"duration_ms":843358,"significance":"If the main result holds, this is a useful and cleanly formulated contribution: the first closed-form characterization of the orthocomplement of the tangent space for ordinary Markov models associated with undirected graphs, chain graphs, and ADMGs, a class for which no explicit likelihood factorization is known and for which the DAG technique provably fails. The payoff is concrete: the full linear variety of influence functions for any pathwise-differentiable target (Eq. 28), an implementable variance-reduction iteration (Theorem 4) requiring only single-constraint projections in closed form, and instructive worked examples — notably the Bell scenario, where the one-step improvement of the saturated-model IF recovers the familiar AIPW form (Appendix D, Example 2). The paper is also commendably honest about what it does not deliver: the projection onto T⊥, the efficient influence function, and the implicitization of the likelihood remain open, and Example 1 in Appendix C explicitly flags that the natural candidate operator is not evidently a projection. These are genuine strengths. However, the characterization is parameter-free only if the identification T = ∩Ti is correct, and as","major_comments":[{"comment":"The nontrivial inclusion T1∩...∩TK ⊆ T is not established. The proof takes f ∈ ∩Ti, sets pε = p0(1+εf), and asserts: 'Following the proof of Lemma 1, the i-th constraint must hold in pε(v).' This is not what Lemma 1's proof shows. Lemma 1 (Appendix B) verifies the constraint only for f drawn from each of the four factor subspaces TW|XYZ, TX|Z, TY|Z, TZ separately, where the perturbation factorizes through the conditional densities; for a general element of Ti (a sum of such pieces) the multiplicative perturbation breaks the constraint at O(ε²). Concrete counterexample: single constraint X⊥Y with Z,W empty, p0 uniform on {0,1}², f(x,y) = (−1)^x + (−1)^y. Then Π(f|T⊥) = E[f|x,y] − E[f|x] − E[f|y] + E[f] = 0, so f ∈ T (and indeed f is realizable by the factor-wise curve pε(x)pε(y) with pε(x) = (1+ε(−1)^x)/2). But the paper's curve gives pε(x,y) = (1+ε(−1)^x+ε(−1)^y)/4 while pε(x)pε(y) = (1+","section":"Appendix C, proof of Theorem 3, second paragraph"},{"comment":"The proofs assert that the sum T⊥1 + ... + T⊥K is closed and equals the direct sum ⊕i T⊥i ('Since both T⊥1 and T⊥2 are closed, T⊥1 ⊕ T⊥2 = overline{T⊥1 ⊕ T⊥2}'). A sum of two closed subspaces of a Hilbert space need not be closed; the correct general statement is (T1 ∩ T2)⊥ = closure(T⊥1 + T⊥2). The equality used is fine for finite state spaces (all subspaces finite-dimensional) but this assumption is never stated. Relatedly, the 'direct sum' terminology is misleading: the T⊥i need not be linearly independent. For instance, with the constraint list {X⊥Y, X⊥Y|Z} one has T⊥1 ⊆ T⊥2, and for the undirected square the two constraints both remove the same highest-order log-linear interaction, so the summands overlap. Equation (27) as a Minkowski sum is the correct object; the text should say 'sum', state the state-space assumption under which the sum is closed, and note the representation hi i","section":"§4.1 Lemma 2 and Appendix C, closedness of the sum of orthocomplements"},{"comment":"The manuscript never specifies whether variables are discrete or general, whether p0 is required to have full support, or what class of scores is admitted. The construction pε = p0(1+εf) with 'δ small enough so that pε ≥ 0' (Lemma 8, Lemma 1, and Theorem 3 proofs) requires f bounded; for unbounded L²(p0) scores the standard truncation-plus-density argument is needed and should be written down once. Similarly, the reduction of pathwise differentiability (Eq. 4) to multiplicative submodels is flagged 'under regularity conditions' but the conditions are never stated, and conditional expectations E[·|z] are used at points where p0(z) may be small. These are routine in the semiparametric literature but must be collected into an explicit assumption set, especially since the paper's main claim is about general state spaces where, per Major Comment 2, the topological step also needs care.","section":"Global: regularity framework for all submodel constructions"}],"minor_comments":[{"comment":"The main text repeatedly refers to 'Equation 81' (e.g., §4.2 after Theorem 3, and §5.1 'according to Equation 81'), which is the supplementary-material numbering of Eq. (27). Please unify the numbering or add cross-reference notes.","section":"§4.2 and §5.1, equation numbering"},{"comment":"The iteration φm = φm−1 − Π(φm−1|T⊥im) with cyclic im is exactly the method of alternating (cyclic) projections onto the subspaces Ti. By von Neumann's theorem (K=2) and Halperin's extension (K≥2), φm converges to Π(φ0|T), i.e., to the efficient influence function, whenever T = ∩Ti and the relevant sums are closed. Citing this literature would answer, or at least sharply frame, the open question raised about the M→∞ limit, and would connect Theorem 4 to known convergence-rate results.","section":"§5.2, Theorem 4"},{"comment":"The computation of ⟨(h−Γ(h)), Γ(g)⟩ is left inconclusive ('It is not evident that term2 = term1'). Using the Bell constraints at p0 (A⊥B,D and B⊥A,C), several cross terms factorize (e.g., E[E[h|A]E[g|B,D]] = E[E[h|A]·E[E[g|B,D]|A]]-type simplifications); it should be possible to either exhibit h, g with nonzero inner product (settling that Γ is not the projection) or prove equality. An inconclusive displayed computation should not be left in the supplement.","section":"Appendix C, Example 1"},{"comment":"In §5.1 the projection operators are subscripted Πa, Πb, Πc, then Πd for the bidirected square model P(e) of Figure 1e; the Bell scenario P(d) is treated in §4.1 instead. The lettering mismatch (Πd attached to model (e)) is confusing; consider subscripting by the model superscripts (a)–(e) or by constraint index.","section":"§5.1, notation for projection operators"},{"comment":"Typos and small items: 'irrelevance,and' (§1); 'referred to asgraphical Markov models' (§1); 'important subsclass' (§2); 'BDs' in the local-Markov bullet list versus 'BGs' elsewhere (§2.1); 'syntatically' (§4.2); 'auxillary' (§5.1); 'The second one is is an iterative method' (§6); the equality sign in 'Its orthogonal complement is the direct sum' vs. sum (see Major Comment 2). Please also verify the numbering of the Van Der Laan & Robins (2003) lemma cited as Lemma 1.7.","section":"Typos and references"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is an accepted UAI 2026 conference paper; for a statistics journal the regularity framework (state space, positivity, boundedness, closure arguments) will need to be made explicit in any case. The reader report accompanying this submission scored soundness 8/10 with an accept verdict; I find that too generous given that the central inclusion in Theorem 3 is asserted via a construction that provably fails (see Major Comment 1), though I agree the result is very likely correct for finite-state models and repairable. The citation pattern is heavy on the second author's own work but that seems appropriate to the topic rather than self-promotional. The attribution to Van Der Laan & Robins (2003) \"Lemma 1.7\" should be checked against the published numbering."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is that this paper finally writes down an explicit orthocomplement for ordinary Markov models (UG, CG, ADMG) that are not DAG-equivalent, so you can immediately write the full class of influence functions once you have one. That is the real deliverable, and it is useful.\n\nWhat they do well is straightforward. They take the classical single-CI formula (Lemma 1) and the Van Der Laan–Robins intersection fact, apply them to the local Markov property of the usual graph classes, and give concrete IF classes for the conditional mean on the Bell scenario, undirected square, chain graph, and bidirected square. Theorem 4’s iterative variance-reduction scheme is a practical bonus. The limitations (no projection operator, no EIF, no Verma constraints) are stated honestly. The math is elementary Hilbert-space geometry; the citations are appropriate and not padded.\n\nThe soft spot is real and sits at the center. Theorem 3 rests on T = ∩ Ti. The Appendix constructs the submodel pε = p0(1 + εf) for f in every Ti and claims the constraints survive because “Lemma 1” already checked them. That only works for the orthogonal pieces separately. For a general sum the multiplicative perturbation violates the CIs at second order; a simple binary counter-example (uniform on {0,1}², f = (-1)^x + (-1)^y) shows pε leaves the model for every ε ≠ 0. So the displayed proof does not establish the inclusion. The claim is almost certainly true for finite-state full-support distributions (they become hierarchical log-linear models whose tangent space is exactly the intersection), and a correct curve can be built factor-wise, but that argument is missing. In infinite dimensions the algebraic direct-sum claim also needs a closure caveat. These are fixable, not fatal, but they are not “usual regularity.”\n\nThe paper is for people who already do semi-parametric efficiency on graphs and who have been stuck once they leave DAGs. It deserves a serious referee who will demand the repaired tangent-space argument (or an explicit finite-state restriction). I would cite the IF expressions and the improvement scheme once the proof is cleaned; I would bring the corrected version to reading group.","headline":"Useful closed-form IF classes for non-DAG Markov models, but the load-bearing proof that T equals the intersection of the single-CI tangent spaces is broken as written.","tokens_in":33015,"tokens_out":556,"would_cite":true,"duration_ms":11455,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G05","62H22","62F12"],"pacs":[],"model":"grok-4.5","headline":"The orthocomplement of the tangent space for any Markov model is the direct sum of the orthocomplements of its single-constraint pieces, each given by a simple conditional-expectation formula.","keywords":["semiparametric efficiency","influence functions","tangent space","Markov models","conditional independence","graphical models","ADMGs","chain graphs"],"falsifier":"Exhibit a concrete Markov model and a mean-zero square-integrable function that lies in every single-constraint tangent space yet fails to be a score of any parametric submodel that simultaneously satisfies all the model’s conditional independences.","tokens_in":32644,"feed_emoji":"📊","tokens_out":821,"duration_ms":13571,"temperature":0.7,"pith_summary":"Markov models are defined only by conditional independence restrictions. Efficient estimation of a finite-dimensional parameter inside such a model requires the class of all influence functions, which is one influence function plus the orthocomplement of the model’s tangent space. For DAG models that orthocomplement was already known; for undirected graphs, chain graphs and ordinary ADMG models it was not. The paper shows that any Markov model is the intersection of single-constraint models, so its tangent-space orthocomplement is simply the direct sum of the already-known single-constraint orthocomplements. Each piece is an explicit four-term conditional expectation, giving a closed-form description of every influence function. The same construction yields an iterative projection that produces a sequence of influence functions with non-increasing variance, improving any initial regular asymptotically linear estimator.","feed_headline":"Closed-form influence functions for every Markov model","feed_subtitle":"Orthocomplement of the tangent space is just the sum of the single-constraint pieces","key_machinery":"Theorem 3: the identification T⊥ = ⊕ i T⊥i together with the explicit four-term projection formula for each single-constraint orthocomplement. This identity converts the geometric fact that the model is an intersection into a concrete recipe for every influence function.","core_discovery":"For a Markov model defined by K conditional independences Xi ⊥ Yi | Zi, the orthocomplement of its tangent space equals the direct sum of the K single-constraint orthocomplements. Every element of that orthocomplement is therefore written in closed form as the sum over i of the operators Π(hi | T⊥i) = E[h|xi,yi,zi] - E[h|xi,zi] - E[h|yi,zi] + E[h|zi].","pith_inferences":["The missing projection onto the full orthocomplement is now the only remaining obstacle to efficient influence functions; successive single-constraint projections may converge to it under additional regularity.","The same direct-sum geometry should extend immediately to models that also impose Verma constraints once the orthocomplement of a single Verma constraint is characterized.","Software that already computes single-constraint influence functions can be reused, without new algebraic factorization, to produce the full class for undirected and mixed-graph models."],"forward_implications":["Every influence function for any smooth target in a UG, CG or ordinary ADMG model is now obtained by adding an arbitrary sum of the four-term operators to one known influence function.","An iterative sequence of single-constraint projections produces influence functions of non-increasing variance, yielding more efficient RAL estimators from any initial one.","The same orthocomplement formula applies to non-graphical Markov models defined by arbitrary lists of conditional independences.","Once a projection onto the full orthocomplement is found, the efficient influence function itself becomes available for these models."],"fun_headline_variants":["Markov orthocomplement is the sum of single-CI pieces","Closed-form tangent orthocomplement for any Markov model","Orthocomplement equals direct sum of K single-constraint spaces","Influence functions via summed CI orthocomplement operators","General Markov tangent orthocomplement in closed form"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"That the tangent space of the full intersection model is exactly the intersection of the individual closed tangent spaces, so that the orthocomplement is their direct sum.","fun_headline_variants_meta":{"raw":{"variants":["Markov orthocomplement is the sum of single-CI pieces","Closed-form tangent orthocomplement for any Markov model","Orthocomplement equals direct sum of K single-constraint spaces","Influence functions via summed CI orthocomplement operators","General Markov tangent orthocomplement in closed form"]},"model":"grok-4.5","effort":"low","cost_usd":0.004756,"raw_usage":{"total_tokens":1365,"prompt_tokens":804,"num_sources_used":0,"completion_tokens":62,"cost_in_usd_ticks":47564000,"prompt_tokens_details":{"text_tokens":804,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":499,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":804,"tokens_out":62,"duration_ms":7569,"temperature":1.0,"reasoning_tokens":499,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T22:03:19.095807+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a concrete Markov model and a mean-zero square-integrable function that lies in every single-constraint tangent space yet fails to be a score of any parametric submodel that simultaneously satisfies all the model’s conditional independences.","supporting_citations":[],"review_version":1}