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Bounding Classical and Quantum Correlations in Bayesian Networks with Quasiprobabilities

T0 review · 2 major / 2 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read Quasiprobabilities on unobserved nodes generate all non-signaling correlations in a broad class of Bayesian networks.

desk verdict The generalization of quasiprobability models to broader networks holds up via the tensor-network link, but the conjecture that this recovers the nested Markov model is stated without proof. read the letter →

arxiv 2606.23372 v1 pith:3LIQEELT submitted 2026-06-22 quant-ph

Paul Becsi , Matty J. Hoban This is my paper · ORCID
classification quant-ph
keywords quasiprobabilitiesBayesiannetworksnon-signalingcorrelationscausalinferenceBellnonlocalitynestedMarkovmodeltensor
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 examines quasiprobabilistic models of Bayesian networks in which distributions over hidden nodes obey normalization but may take negative values. It establishes that for many network structures these models produce every non-signaling distribution over the observed nodes, extending earlier results known only for Bell scenarios. The construction is linked to tensor-network factorizations of the joint distribution. This leads the authors to conjecture that the resulting set of observable marginals coincides with the nested Markov model that encodes the causal constraints of the network.

What carries the argument

The quasi set formed by allowing quasiprobabilities (normalized but possibly negative) solely on unobserved nodes while retaining ordinary probabilities on observed nodes.

What would settle it

A non-signaling correlation over observed nodes in one of the considered networks that cannot be realized by any assignment of quasiprobabilities to the hidden nodes.

Watch

Extended reading notes

Core claim

Quasiprobabilistic models for Bayesian networks, obtained by replacing probability distributions on unobserved nodes with quasiprobabilities that respect only normalization, yield a set of observable correlations (the quasi set) that contains all non-signaling distributions for a broad class of networks; the same construction is conjectured to recover exactly the nested Markov model associated with the network.

Load-bearing premise

That quasiprobabilities placed only on hidden nodes produce observable marginals that match or outer-bound the quantum set for the networks studied.

Editorial extensions

If this is right

  • The quasi set supplies an outer approximation to the quantum correlations permitted by any given network.
  • Tensor-network decompositions become a practical tool for describing the allowed correlations.
  • The conjecture equates the quasi set with the nested Markov model, thereby characterizing all non-signaling distributions compatible with the causal structure.

Reading between the lines

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

  • The tensor-network representation may be reusable for analyzing other factorization problems in classical and quantum probability.
  • If the conjecture holds, linear programming over quasiprobabilities would compute the exact non-signaling bound for those networks without enumerating quantum strategies.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The paper introduces quasiprobabilistic models for Bayesian networks in which probability distributions on unobserved nodes are replaced by quasiprobabilities obeying normalization but not positivity. It shows that these models recover all non-signalling observable marginals for a broad class of networks, via an explicit connection to tensor-network decompositions. This generalization is used to motivate the conjecture that the resulting 'quasi set' exactly coincides with the nested Markov model.

Significance. If the conjecture holds, the quasiprobabilistic construction would supply a concrete outer approximation (or exact characterization) to the set of quantum correlations compatible with a given causal structure, extending the known recovery of the non-signalling set in Bell scenarios. The tensor-network link may be of independent interest for relating causal models to tensor decompositions. The approach is parameter-free once the network is fixed and directly falsifiable against explicit distributions.

major comments (2)
  1. [Abstract] Abstract and conjecture statement: the identification of the quasi set with the nested Markov model is presented only as a conjecture motivated by the generalization; no explicit construction is supplied showing that every distribution in the nested Markov model arises from some quasiprobability assignment on the hidden nodes (or vice versa).
  2. [Main text (generalization)] Generalization claim: while the tensor-network argument is said to establish recovery of all non-signalling marginals for a broad class of networks, the manuscript does not delineate the precise class of networks for which the result holds nor provide worked examples of networks and explicit quasiprobability assignments that realize the non-signalling set.
minor comments (2)
  1. [Introduction] The term 'quasi set' is used repeatedly before a formal definition appears; an early equation or boxed definition would improve readability.
  2. Notation for quasiprobability distributions on unobserved nodes should be distinguished typographically from ordinary probability distributions to avoid confusion with the classical case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We appreciate the positive assessment of the work's significance and address the major comments point by point below, with plans for revision where appropriate.

read point-by-point responses

  1. Referee: [Abstract] Abstract and conjecture statement: the identification of the quasi set with the nested Markov model is presented only as a conjecture motivated by the generalization; no explicit construction is supplied showing that every distribution in the nested Markov model arises from some quasiprobability assignment on the hidden nodes (or vice versa).

    Authors: We agree that the equivalence between the quasi set and the nested Markov model is presented strictly as a conjecture, motivated by the tensor-network generalization result but without a full bidirectional explicit construction. The manuscript establishes one direction (that quasiprobabilistic models recover all non-signalling marginals) for a broad class via the tensor-network link, which provides the motivation for the conjecture. We will revise the abstract and main text to state the conjectural status more explicitly and add a short discussion of the structural features (e.g., the ability of quasiprobabilities to match arbitrary marginals while respecting normalization) that support expecting the converse to hold. A complete proof remains an open question beyond the present scope. revision: partial

  2. Referee: [Main text (generalization)] Generalization claim: while the tensor-network argument is said to establish recovery of all non-signalling marginals for a broad class of networks, the manuscript does not delineate the precise class of networks for which the result holds nor provide worked examples of networks and explicit quasiprobability assignments that realize the non-signalling set.

    Authors: The referee is correct that the precise class of networks is described only as 'broad' without formal delineation and that no explicit worked examples with quasiprobability assignments are provided. In the revised manuscript we will define the class more rigorously as the set of Bayesian networks whose observed marginals admit a tensor-network decomposition allowing quasiprobability assignments on hidden nodes to reproduce any non-signalling distribution. We will also add at least two concrete examples: the Bell scenario (recovering the full non-signalling polytope) and a simple three-node chain with one hidden variable, including explicit quasiprobability values and the resulting observable marginals. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; quasiprobability models and tensor-network generalization are defined independently

full rationale

The paper defines the quasi set by replacing unobserved-node distributions with quasiprobabilities obeying only normalization (no positivity), a construction stated independently of any quantum or nested-Markov target. The generalization to a broad class of networks is shown via an explicit tensor-network connection that recovers all non-signalling marginals; this step does not reduce to a self-definition, fitted parameter renamed as prediction, or self-citation chain. The further claim that the quasi set recovers the nested Markov model is presented only as a conjecture motivated by the generalization, not as a proven identity. No load-bearing self-citations, ansatzes smuggled via citation, or renamings of known results appear in the derivation chain. The central results therefore remain self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Based on abstract only; the central construction rests on the domain assumption that quasiprobabilities suffice for hidden nodes.

assumptions (1)
  • domain assumption Quasiprobabilities on unobserved nodes (normalization preserved, positivity dropped) can be used to generate observable correlations in Bayesian networks
    This is the defining step that creates the quasi set and enables the generalization claim.

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Cite this review

Pith. "Pith review of Bounding Classical and Quantum Correlations in Bayesian Networks with Quasiprobabilities." pith.science (2026). https://pith.science/paper/3LIQEELT

@misc{pith2026260623372,
  author       = {Pith},
  title        = {Pith review of: Bounding Classical and Quantum Correlations in Bayesian Networks with Quasiprobabilities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3LIQEELT}},
  note         = {Machine review of arXiv:2606.23372}
}
read the original abstract

Bell's theorem reveals that quantum theory is in tension with classical causal reasoning and, in particular, the notion of local causality. This is now understood as a particular example of non-classicality in the study of correlations in (Bayesian) networks with both unobserved and observed nodes: the correlations are probability distributions over the observed nodes. There is a great deal of work aiming to understand the bounds on quantum and classical correlations in such networks and one approach is to consider outer approximations to the former. Along these lines, we consider quasiprobabilistic models for Bayesian networks, which can be seen as classical models but the probability distributions involving unobserved nodes are "replaced" with quasiprobabilities that respect normalisation but not positivity. We denote the set of correlations resulting from these models as the quasi set. Such models have a history in the study of Bell-type non-classicality where it has been shown that they can produce all non-signalling correlations. We show a generalisation of this result for a broad class of networks, which motivates a conjecture that the quasi set recovers the so called nested Markov model. Our work utilises a connection to tensor network decompositions, which may be of independent interest.

Figures

Figures reproduced from arXiv: 2606.23372 by the authors.

Figure 1
Figure 1. The bipartite Bell Scenario. the set of distributions consistent with it?". Of course, one needs to define mathematically what consistency means in this context. In the simplest setting where all variables are observed, consistency is given by the three equivalent Markov properties: the factorisation criterion, the local Markov property, and the global Markov property. If latent variables are present, classically on… view at source ↗
Figure 2
Figure 2. Introducing noise variables, illustrated on the Triangle Scenario. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. The correspondence between the 4-on-line Scenario and its underlying undirected graph. Edges become latent variables with the same domain size as the corresponding index. the 4-on-line Scenario. For some finite sets A, B, C, D we consider the order-4 real-valued tensors T(a, b, c, d), where a ∈ A, b ∈ B, c ∈ C, d ∈ D. We say such a tensor T decomposes with respect to T and bond dimensions (I, J, K) ∈ R 3 if there ex… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Graph Comn. Set everything else in p ∗ new to behave the same as in p ∗ . Then p ∗ new also achieves p(xV ): p(xV ) = X xL Y v∈V ∪L p ∗ (xv|xpaG(v) ) = X xL\{l} Y l ′∈L\{l} p ∗ (xl ′) Y v∈V \ch(l) p ∗ (xv|xpaG(v) ) X xl p ∗ (xl) Y v∈ch(l) p ∗ (xv|xpaG(v) ) = X xL\{l} Y…
Figure 5
Figure 5. Figure 5: The Triangle Scenario. By summing this equation over xV we obtain 1 = X {1≤ie≤re}e∈E Y e∈E ψe(ie) = Y e∈E X 1≤ie≤re ψe(ie). Therefore, p(xV ) = X {1≤ie≤re}e∈E Y v∈V p  xv|{xle = ie}e∈inc(v)  · Y e∈E ψe(ie) P 1≤ie≤re ψe(ie) . (27) Finally, we go back to the original c…
Figure 6
Figure 6. Figure 6: The Triangle Scenario after extending the latents with a copy of the perfect correlation [PITH_FULL_IMAGE:figures/full_fig_p044_6.png]
Figure 7
Figure 7. Figure 7: The Common Ancestor Scenario. References 1. Sangmin Byeon, W.L.: Directed acyclic graphs for clinical research: a tutorial. Journal of Minimally Invasive Surgery 26(3), 97–107 (2023). doi: 10.7602/jmis.2023.26.3.97 2. García-Franco, J.D., Díez, F.J., Carrasco, M.Á.: Pr…

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