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Cyclic functional causal models beyond unique solvability with a graph separation theorem

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Cyclic causal models get a sound and complete graph-separation rule, p-separation, for finite-valued variables.

desk verdict A serious and likely-correct contribution to cyclic causal modeling, but the completeness proof of the main p-separation theorem is informal in the general case and needs a full derivation before the result is fully proven. read the letter →

arxiv 2502.04171 v2 pith:AYJQIBII submitted 2025-02-06 math.ST quant-phstat.MLstat.TH

Carla Ferradini , Victor Gitton , V. Vilasini This is my paper · ORCID
classification math.STquant-phstat.MLstat.TH MSC 05C9062H0568T37
keywords functionalcausalmodelscyclicgraphsp-separationd-separationpost-selectedteleportationMarkovfactorizationaverageuniquesolvabilityloops
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 extends functional causal models (fCMs) from acyclic to cyclic graphs with finite-valued variables, including models that are not uniquely solvable and therefore previously had no well-defined probability distribution. It introduces a post-selection-based probability rule that assigns a unique distribution to every consistent cyclic fCM and reduces to the standard rule when solutions are unique. On top of that it defines a graph-separation property, p-separation, and proves it sound and complete: a p-separation between vertex sets holds exactly when every consistent model on the graph must satisfy the corresponding conditional independence. In acyclic graphs p-separation collapses to the classical d-separation theorem. If correct, this gives the first graph-theoretic criterion that fully captures conditional independences for non-uniquely solvable cyclic causal models over finite variables.

What carries the argument

The central object is the classical post-selected teleportation protocol. A teleportation protocol (Definition 3) is a function $f(a,b,c)$ and priors $P_B,P_C$ such that post-selecting on $f=1$ copies any distribution $p_A$ onto $C$; the canonical 'uniform prior' protocol uses $f(a,c)=\delta_{a,c}$ and $P_C$ uniform, with success probability $1/|X_A|$. Definition 7 turns any cyclic graph $G$ into a family of acyclic teleportation graphs $G_{tp}(G)$ by replacing selected vertices $v$ with a pre-selection copy $R_v$ and a post-selection vertex $T_v$, and Definition 9 builds acyclic fCMs on these graphs. Post-selecting $T_v=1$ and conditioning yields the probability rule $\Pr(x)_G = \frac{\sum_u \prod_v p_v(u_v)\delta_{x_v,f_v(x_{Pa(v)},u_v)}}{\sum_y \sum_u \prod_v p_v(u_v)\delta_{y_v,f_v(y_{Pa(v)},u_v)}}$, which is independent of the choice of $G_{tp}$ and of the teleportation implementation. p-separation is then defined as: there exists $G_{tp}\in G_{tp}(G)$ with $(V_1 \perp_d V_2 | V_3 \cup V_{post})_{G_{tp}}$. This machinery transfers d-separation arguments from acyclic to cyclic models by making the hidden collider effect of cycles explicit.

What would settle it

Exhibit any consistent finite-cardinality cyclic fCM and a triple $V_1,V_2,V_3$ where p-separation holds but the distribution given by the paper's Eq. (33) violates the promised conditional independence, or where p-connection holds but every fCM on $G$ satisfies that independence. A concrete place to look is a continuous-variable model on the graph of Eq. (66) that realizes the sigma-separation correlation; if such a model can be approximated by finite-cardinality models while preserving the dependence, the soundness claim would be at risk.

Watch

Extended reading notes

Core claim

The paper's central theorem (Theorem 20) states: for any directed graph $G$ and three disjoint non-empty vertex sets $V_1,V_2,V_3$, $(V_1 \perp_p V_2 | V_3)_G$ holds if and only if every consistent finite-cardinality functional causal model on $G$ satisfies $(X_1 \perp\!\!\perp X_2 | X_3)_P$. p-separation is defined by passing to an acyclic teleportation graph $G_{tp}$ obtained by splitting vertices into pre- and post-selection copies, and asking for d-separation of $V_1$ from $V_2$ given $V_3$ together with all post-selection vertices. The same construction yields the probability rule, which counts all solutions of the functional equations and renormalizes, so inconsistent models (zero solutions on average) are exactly those with zero post-selection success probability. The paper also identifies averagely uniquely solvable fCMs --- those with average number of solutions equal to one --- as exactly the class whose probabilities admit the Markov factorization.

Load-bearing premise

The whole construction assumes variables with finitely many values, because the probability rule post-selects on exact variable values; and the completeness proof for the general d-connection case is argued through examples rather than a full derivation.

Editorial extensions

If this is right

  • Every consistent finite-cardinality cyclic fCM gets a unique probability distribution, resolving the ambiguity in non-uniquely solvable models.
  • p-separation gives sound and complete conditional-independence semantics for all such models, and reduces to d-separation for acyclic graphs.
  • Averagely uniquely solvable models are exactly the Markovian ones; uniquely solvable models are a strict subset of them.
  • The post-selection success probability is proportional to the average number of solutions, connecting consistency to solution counting.
  • Because p-separation is sound and complete, it could support causal discovery and causal-compatibility tests for cyclic finite-valued causal structures.

Reading between the lines

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

  • Editorial: The probability rule effectively weights each consistent solution equally; if this is accepted as a resolution principle, non-uniquely solvable cyclic models become predictable without extra modeling assumptions.
  • Editorial: The same post-selection construction suggests finite cardinality is not a technical convenience: for continuous variables the post-selection event has probability zero, so p-separation may have no direct continuous analogue.
  • Editorial: Because d- and p-separation agree on the graph of Eq. (66) while sigma-separation does not, the paper implies that every finite-cardinality model on that graph is fine-tuned relative to sigma-separation; a full taxonomy of separation properties on cyclic graphs is a natural next step.
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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 / 5 minor

Summary. The paper develops a probability rule for finite-cardinality functional causal models (fCMs) on arbitrary directed graphs, including cyclic and non-uniquely solvable models, by mapping a cyclic model to a family of acyclic 'classical teleportation graphs' with post-selection. It defines a new graph-separation property, p-separation, and claims in Theorem 20 that p-separation is sound and complete for all consistent finite-cardinality fCMs, recovering classical d-separation on DAGs. The paper also introduces 'average unique solvability' and proves, in Corollary 29, that this property characterizes when the probability rule yields a Markov factorization.

Significance. If Theorem 20 is fully established, this would be the first sound and complete graph-separation property for non-uniquely solvable cyclic functional causal models over finite variables, a genuine open problem in the area. The probability rule (Eq. (33)) is explicit, simple, and recovers the standard rule for uniquely solvable models, and the solvability results are clean and well-motivated. The connection to classical post-selected teleportation is conceptually interesting and links the work to quantum causal models. The main limitation is the finite-cardinality restriction, which the authors acknowledge, and the present incompleteness of the general-case completeness proof for Theorem 20, which is load-bearing for the universality claim.

major comments (3)
  1. [Appendix B, 'Proof for the general case' (after Eq. (116))] The completeness half of Theorem 20 is not proven for the general case. After reducing the problem to the three d-connections of Eq. (109), the proof treats three special path shapes (Eqs. (111)-(113)) and then asserts that the argument 'generalizes in a straightforward manner.' This is load-bearing because Theorem 20 claims universality over all directed graphs and all p-connection patterns. A complete proof must either provide an inductive or case-based argument showing that arbitrary unblocked paths among v1, v2 and the post-selection vertices reduce to the structure of Eq. (114), or explicitly construct the required dependence for the general path geometry. The current text leaves the general case at the level of plausibility rather than proof.
  2. [Appendix B, Definition 31 and the following paragraph] The constructed model assigns x_T = XOR of parents to post-selection vertices and then post-selects on t_T = 0, but Definition 9 requires the post-selection vertex to implement a classical post-selected teleportation protocol from Definition 3, whose success event is t_T = 1 and whose function f satisfies Lemma 5. The proof must explicitly define f'_T = 1 - (x_v xor x_R), or otherwise relabel the success event, and verify that the resulting model on Gtp belongs to the family of classical teleportation functional models used to define Pr_G in Definition 12. As written, the construction does not obviously fall inside the framework whose probability rule is being invoked.
  3. [Appendix B, paragraph around Eq. (115)] The step from conditional dependence in the acyclic model on Gtp to conditional dependence in Pr_G is asserted rather than derived. Pr_G is obtained by conditioning on the post-selection event and marginalizing over pre-selection and other auxiliary variables, and the proof does not explicitly show that the equality x_T xor x_v1 = x_T' xor x_v2, established under x_C = 0, survives this marginalization for arbitrary choices of the remaining vertices in Gtp. This is likely fixable, but it is part of the same completeness gap and should be made explicit.
minor comments (5)
  1. [Proposition 13 (statement)] The statement refers to 'the probability rule in definition 2' but it should refer to Definition 12, since the proposition gives an alternative expression for the cyclic-model probability rule.
  2. [Section 4.5, first paragraph] There is a typo: 'Finally, we we describe the relations' should read 'Finally, we describe the relations.'
  3. [References] The entries [LMGP+11a] and [LMGP+11b] appear to be identical duplicate references to the same paper; this should be cleaned up.
  4. [Appendix B, Eq. (115) and surrounding text] Notation such as 'T = 0' is used ambiguously: it should be clear whether one conditions on the event {T = 0} or on the random variable T as a conditioning variable, especially because Definition 17 uses conditioning on random variables.
  5. [Appendix A, proof of Proposition 11] The parenthetical 'if the equality holds in this case, the general result follows from corollary 14' creates an apparent forward-reference to a corollary that is proved later; the special-case calculation appears to establish the general implementation case directly for shared split vertices, so please clarify or reorder the argument to avoid an apparent circularity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 20 is established from the paper's own definitions using the classical d-separation theorem and a constructive completeness argument; companion-paper citations are not load-bearing.

full rationale

p-separation is a purely graph-theoretic predicate (Definition 19) built from d-separation in acyclic teleportation graphs, while the cyclic probability rule (Definition 12) is defined independently by conditioning on teleportation post-selection outcomes. The soundness half of Theorem 20 follows by applying the acyclic d-separation theorem to a chosen Gtp and then using Proposition 11 and Definition 12; it does not assume the target independence. The completeness half is a constructive reduction: a d-connected teleportation graph is reduced to a subgraph, an XOR model (Definition 31) is built there, and the paper argues that post-selecting the collider outcomes induces a functional model on the original cyclic graph. Whatever gaps exist are proof-gaps, such as the 'general case' being argued through examples with the statement that it 'generalizes in a straightforward manner' and the implicit relabeling of the teleportation success event from t=0 to t=1. These are not reductions of the theorem to its inputs. The only self-citations, mainly [FGV25], are used for the quantum counterpart and for the equivalence of two p-separation formulations; the classical soundness and completeness proof in Appendix B does not rest on those citations. No fitted parameter is renamed as a prediction, and no known result is repackaged as a new one, so the derivation chain is self-contained.

Assumptions & free parameters 0 free parameters · 4 assumptions · 3 invented entities

The central claim rests on standard probability/graph theory, the finite-cardinality domain restriction, the consistency requirement, and a newly introduced teleportation protocol. There are no fitted parameters. The invented concepts (p-separation, average unique solvability, the teleportation protocol) are formal and are justified by the theorems proven in the paper.

assumptions (4)
  • standard math Standard probability theory and the d-separation theorem for acyclic graphs (Theorem 18) are used as background.
    Invoked in the soundness proof of Theorem 20 to transfer conditional independence from the acyclic teleportation graph to the cyclic model.
  • domain assumption All random variables take values in finite sets.
    The probability rule and p-separation are defined only for finite-cardinality variables; the post-selection event has positive probability only in this case. The paper explicitly excludes continuous variables (Section 6).
  • domain assumption Functional models are required to be consistent, i.e., the post-selection success probability ps > 0.
    Inconsistent models (no solutions for any error assignment) are excluded from the framework; their probabilities are undefined (Definition 12).
  • ad hoc to paper A classical post-selected teleportation protocol (Definition 3) exists and satisfies Lemma 5.
    This new construction is the basis for the mapping from cyclic to acyclic models. The uniform prior protocol (Definition 6) provides a concrete instance, so the assumption is justified, but it is introduced specifically for this framework.
invented entities (3)
  • p-separation independent evidence
    purpose: A graph-separation property that characterizes conditional independences in all consistent finite-cardinality cyclic fCMs and reduces to d-separation for acyclic graphs.
    The soundness and completeness theorem provides testable constraints: for any graph, the property predicts which conditional independences must hold in every model in the class.
  • Classical post-selected teleportation protocol
    purpose: A construction used to map cyclic functional causal models to acyclic ones with post-selection, inspired by quantum teleportation.
    This is a mathematical tool internal to the framework; it has no observable handle outside the paper, but it is proven to exist and to yield protocol-independent probabilities.
  • Average unique solvability independent evidence
    purpose: A property of functional causal models (average number of solutions equals 1) that is shown to be equivalent to Markov factorization.
    Corollary 29 gives a precise equivalence: an fCM is probabilistically Markovian if and only if it is averagely uniquely solvable, which can be checked on any model.

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Cite this review

Pith. "Pith review of Cyclic functional causal models beyond unique solvability with a graph separation theorem." pith.science (2026). https://pith.science/paper/AYJQIBII

@misc{pith2026250204171,
  author       = {Pith},
  title        = {Pith review of: Cyclic functional causal models beyond unique solvability with a graph separation theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AYJQIBII}},
  note         = {Machine review of arXiv:2502.04171}
}
read the original abstract

Functional causal models (fCMs) specify functional dependencies between random variables associated to the vertices of a graph. In directed acyclic graphs (DAGs), fCMs are well-understood: a unique probability distribution on the random variables can be easily specified, and a crucial graph-separation result called the d-separation theorem allows one to characterize conditional independences between the variables. However, fCMs on cyclic graphs pose challenges due to the absence of a systematic way to assign a unique probability distribution to the fCM's variables, the failure of the d-separation theorem, and lack of a generalization of this theorem that is applicable to all consistent cyclic fCMs. In this work, we develop a causal modeling framework applicable to all cyclic fCMs involving finite-cardinality variables, except inconsistent ones admitting no solutions. Our probability rule assigns a unique distribution even to non-uniquely solvable cyclic fCMs and reduces to the known rule for uniquely solvable fCMs. We identify a class of fCMs, called averagely uniquely solvable, that we show to be the largest class where the probabilities admit a Markov factorization. Furthermore, we introduce a new graph-separation property, p-separation, and prove this to be sound and complete for all consistent finite-cardinality cyclic fCMs while recovering the d-separation theorem for DAGs. These results are obtained by considering classical post-selected teleportation protocols inspired by analogous protocols in quantum information theory. We discuss further avenues for exploration, linking in particular problems in cyclic fCMs and in quantum causality.

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Pith tools

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