Quasiprobabilistic Models in Bayesian Networks

• Quasiprobabilistic models are formalisms that maintain Bayesian network factorization while relaxing standard positivity constraints in latent or response functions.
• They employ both signed-kernel and algebraic weak-product approaches to capture observable correlations in complex networks with latent variables.
• By isolating structural causal constraints from positivity conditions, these models offer new perspectives on classical, quantum, and non-signaling correlation boundaries.

Searching arXiv for the cited papers and closely related context. Quasiprobabilistic models for Bayesian networks are formalisms that preserve the factorization structure of a directed acyclic graph while relaxing the standard probabilistic calculus in one of two ways. In one line of work, latent-variable priors or local response functions are allowed to be normalized real-valued functions that need not be nonnegative, yielding a quasi set $\mathcal{W}(\mathcal{G})$ of observable correlations that outer-approximates both the classical and the quantum sets (Becsi et al., 22 Jun 2026). In another line, probabilities in Boolean Bayesian networks are represented by algebraic quasi-probabilities built from complement and an idempotent weak product, yielding an exact symbolic characterization of noisy AND-OR-NOT networks (Schubert, 2012). These approaches are related by a common objective: to isolate the structural content of Bayesian-network factorization from the positivity constraints that ordinarily define probability distributions.

1. Classical factorization and the motivation for quasiprobabilistic relaxations

A Bayesian network on a directed acyclic graph $\mathcal{G}$ with observed nodes $V$ and latent nodes $L$ specifies observable correlations by marginalizing a factorized joint model over the latent sector. In the reduced-graph formulation, a distribution $p$ over observed variables $X_V$ lies in the classical set $\mathcal{C}(\mathcal{G})$ when there exist finite latent sample spaces and local conditional distributions such that

$$p(x_V)=\sum_{x_L}\prod_{v\in V} p(x_v\mid x_{pa(v)})\prod_{l\in L} p(x_l).$$

This factorization is straightforward to state, but the induced observable model is difficult to characterize once latent nodes are present: beyond conditional-independence constraints, it is governed by generally hard algebraic equality constraints, including Verma constraints, and inequality constraints (Becsi et al., 22 Jun 2026).

This difficulty motivates quasiprobabilistic relaxations. The central idea is to retain normalization, chain rule, and factorization algebraically, while discarding positivity in selected internal components of the model. The observable distribution itself remains an ordinary nonnegative distribution. The resulting framework separates structural constraints inherited from the graph from additional restrictions imposed by positivity.

A distinct but historically earlier motivation appears in Schubert’s algebraic treatment of Boolean Bayesian networks. There, quasi-probabilities are introduced as symbolic expressions that encode the recursive semantics of noisy AND, noisy OR, and noisy NOT nodes. Rather than relaxing positivity of distributions over hidden variables, that framework replaces ordinary multiplication by an idempotent operator tailored to prevent double counting of shared ancestors (Schubert, 2012). The two usages share terminology but operate at different levels: one enlarges the causal model class by allowing signed internal factors, and the other supplies an exact symbolic algebra for standard Bayesian-network probabilities.

2. Signed quasidistributions and quasi factorization on directed acyclic graphs

In the causal-network formulation, a quasiprobability distribution over a finite set $\Omega$ is a real-valued function $p:\Omega\to\mathbb{R}$ satisfying normalization, $\mathcal{G}$0, without any requirement of nonnegativity. Conditional quasidistributions use the usual quotient $\mathcal{G}$1 whenever $\mathcal{G}$2. For a DAG $\mathcal{G}$3, a quasiprobabilistic model consists of kernels $\mathcal{G}$4 and priors $\mathcal{G}$5 that may take negative values but satisfy

$\mathcal{G}$6

The observable distribution is then defined by

$\mathcal{G}$7

with the additional requirement that the resulting $\mathcal{G}$8 be entrywise nonnegative (Becsi et al., 22 Jun 2026).

The 2026 formulation isolates three equivalent realizations of the same observable family. In the quasilatents variant, latent kernels may be negative while observed kernels remain stochastic. In the quasiresponses variant, latent kernels remain nonnegative while observed local response functions may be negative. In the quasinoise variant, each observed node $\mathcal{G}$9 is assigned an auxiliary independent noise variable $V$0; the response $V$1 is deterministic, ordinary latent variables remain nonnegative, and the quasiprobabilistic freedom is moved into the marginals of the noise variables. Theorem 1 establishes that these constructions generate the same set of observed distributions,

$V$2

so the location of negativity is not identifiable at the level of observed correlations (Becsi et al., 22 Jun 2026).

A frequent misunderstanding is that quasiprobabilistic realizability permits negative observed probabilities. It does not. Negativity is confined to latent priors, local kernels, or auxiliary noise distributions; the observable marginal must still be a valid probability distribution. The relaxation is therefore internal to the factorization, not external to the empirical predictions.

3. Position of the quasi set among classical, quantum, and nested-Markov models

For any DAG $V$3, the relevant correlation classes satisfy the inclusion chain

$V$4

where $V$5 is the classical set, $V$6 the quantum set, $V$7 the generalized-probabilistic-theory set, $V$8 the nested Markov model, and $V$9 the ordinary Markov model (Becsi et al., 22 Jun 2026). The quasi set $L$0 is an outer approximation to the classical and quantum sets because it preserves factorization while discarding positivity on latent or response factors. At the same time, Proposition 1 proves that

$L$1

since the symbolic factorizations and fixing operations defining the nested Markov model do not depend on positivity (Becsi et al., 22 Jun 2026).

This leads to the central conjecture of the 2026 work: $L$2 If this conjecture holds, quasiprobabilities would provide a constructive parameterization of the equality content of latent-variable Bayesian networks. The resulting interpretation is precise: nested-Markov constraints would be the full equality skeleton of the DAG, while any further restrictions on classical or quantum realizability would arise from positivity.

Bell scenarios provide the canonical example. In the bipartite Bell network with inputs $L$3, outputs $L$4, and latent variable $L$5, the classical factorization

$L$6

implies Bell inequalities such as CHSH, for example

$L$7

When either the hidden-variable distribution or the local response functions are allowed to be negative, the quasi model reproduces exactly the non-signalling correlations,

$L$8

including PR-box correlations. In this setting, $L$9, which strictly contains the quantum set $p$0 (Becsi et al., 22 Jun 2026). The Bell case is therefore the prototype for the broader claim that quasiprobabilities remove inequality constraints while retaining structural equalities.

4. Tree-structured correlation scenarios and the tensor-network construction

A correlation scenario is a reduced DAG in which observed nodes have no children and latent nodes are exogenous with at least two observed children. From such a DAG one forms the undirected observed-adjacency graph $p$1 on the observed nodes $p$2, connecting two observed vertices when they share a latent parent. The scenario is tree-structured when $p$3 is a tree. In all correlation scenarios, Proposition 2 states that the nested Markov model and ordinary Markov model coincide,

$p$4

and $p$5 consists exactly of those distributions for which any two sets of observed variables with no common latent ancestor are independent (Becsi et al., 22 Jun 2026).

Theorem 2 proves the main structural result: $p$6 for every tree-structured correlation scenario. In such networks, quasiprobabilistic models reproduce all non-signalling or Markov correlations compatible with the separation structure of the DAG, and there are no further inequality constraints. This extends the Bell fact $p$7 from a single latent bipartite setting to a broad class of multi-latent, multi-observed networks (Becsi et al., 22 Jun 2026).

The proof proceeds through tensor-network decompositions. If $p$8 is regarded as a tensor on the tree $p$9, standard tree tensor-network theory yields a decomposition

$X_V$0

Minimal decompositions exist and are unique up to gauge transformations. A key Nonzero Sums Lemma shows that gauge freedom can be used so that, for every observed node $X_V$1 and every assignment of internal indices, the sum over $X_V$2 is nonzero. One then defines normalized local responses

$X_V$3

factorizes the remaining scalar terms along the edges into functions $X_V$4, and sets the latent prior on each edge $X_V$5 proportional to $X_V$6. This converts the tensor-network representation into a quasiprobabilistic Markov factorization (Becsi et al., 22 Jun 2026).

The “4-on-line” example illustrates the construction concretely. For observed nodes $X_V$7 and latent variables $X_V$8, a tensor decomposition of the form

$X_V$9

maps directly to latent priors and normalized local response kernels. The example exemplifies the theorem’s general content: every distribution in the path-graph Markov model belongs to $\mathcal{C}(\mathcal{G})$0, and conversely (Becsi et al., 22 Jun 2026).

5. Cyclic scenarios, strict gaps, and expressivity beyond generalized probabilistic theories

Tree structure is sufficient for $\mathcal{C}(\mathcal{G})$1, but not obviously necessary. The Triangle scenario, in which three observed nodes $\mathcal{C}(\mathcal{G})$2 are pairwise connected by three latent variables, is the leading cyclic case discussed in detail. The 2026 paper shows that the perfect-correlation distribution

$\mathcal{C}(\mathcal{G})$3

belongs to $\mathcal{C}(\mathcal{G})$4 for the Triangle graph, even though this correlation is known not to lie in the generalized-probabilistic-theory model $\mathcal{C}(\mathcal{G})$5 for that graph (Becsi et al., 22 Jun 2026).

Two explicit quasiprobabilistic realizations are provided. One uses quasilatents: the three pairwise latent variables $\mathcal{C}(\mathcal{G})$6 are assigned specified signed priors, while the observed responses of $\mathcal{C}(\mathcal{G})$7 are deterministic. The other uses quasiresponses: the latent variables are unbiased and positive, while each observed node employs the signed response kernel

$\mathcal{C}(\mathcal{G})$8

with $\mathcal{C}(\mathcal{G})$9 and $$p(x_V)=\sum_{x_L}\prod_{v\in V} p(x_v\mid x_{pa(v)})\prod_{l\in L} p(x_l).$$0 (Becsi et al., 22 Jun 2026).

By extending the Triangle latents to carry many independent shared coin flips, the quasiprobabilistic Triangle can simulate a Common Ancestor scenario and hence generate any distribution on three observed nodes. This establishes a strict expressivity gap: in the Triangle case, $$p(x_V)=\sum_{x_L}\prod_{v\in V} p(x_v\mid x_{pa(v)})\prod_{l\in L} p(x_l).$$1 strictly contains $$p(x_V)=\sum_{x_L}\prod_{v\in V} p(x_v\mid x_{pa(v)})\prod_{l\in L} p(x_l).$$2. It also sharpens the broader point that the quasi set is not a reformulation of quantum or GPT compatibility. Rather, it is a larger algebraic envelope that keeps normalization and factorization but discards positivity.

The status of more general cyclic networks remains open under the conjecture $$p(x_V)=\sum_{x_L}\prod_{v\in V} p(x_v\mid x_{pa(v)})\prod_{l\in L} p(x_l).$$3. The tree theorem and the Triangle construction together provide evidence in favor of that conjecture, but they do not yet resolve it for arbitrary DAGs (Becsi et al., 22 Jun 2026).

6. Algebraic quasi-probabilities in Boolean noisy AND-OR-NOT networks

Schubert’s quasi-probabilities form a separate but complementary theory. Here the basic objects are algebraic expressions built from a set $$p(x_V)=\sum_{x_L}\prod_{v\in V} p(x_v\mid x_{pa(v)})\prod_{l\in L} p(x_l).$$4 of elementary probability symbols, together with complement $$p(x_V)=\sum_{x_L}\prod_{v\in V} p(x_v\mid x_{pa(v)})\prod_{l\in L} p(x_l).$$5 and a weak product operator $$p(x_V)=\sum_{x_L}\prod_{v\in V} p(x_v\mid x_{pa(v)})\prod_{l\in L} p(x_l).$$6. Atomic quasi-probabilities are $$p(x_V)=\sum_{x_L}\prod_{v\in V} p(x_v\mid x_{pa(v)})\prod_{l\in L} p(x_l).$$7, $$p(x_V)=\sum_{x_L}\prod_{v\in V} p(x_v\mid x_{pa(v)})\prod_{l\in L} p(x_l).$$8, or products of distinct elementary probabilities. Equivalence of quasi-probabilities is defined by a reduction system whose central rule is idempotent product reduction: repeated elementary factors collapse to a single occurrence. This yields Lemma 2,

$$p(x_V)=\sum_{x_L}\prod_{v\in V} p(x_v\mid x_{pa(v)})\prod_{l\in L} p(x_l).$$9

which is the algebraic mechanism that prevents double counting of shared ancestral parameters (Schubert, 2012).

The framework applies to Boolean Bayesian networks after conversion to noisy AND-OR-NOT form. A noisy AND node $\Omega$0 with parents $\Omega$1 and joint reliability $\Omega$2 satisfies

$\Omega$3

A noisy OR node $\Omega$4 with parents $\Omega$5 and link labels $\Omega$6 satisfies

$\Omega$7

A noisy NOT node $\Omega$8 with parent $\Omega$9 and inhibitory link label $p:\Omega\to\mathbb{R}$0 satisfies

$p:\Omega\to\mathbb{R}$1

For a set $p:\Omega\to\mathbb{R}$2, the joint quasi-probability is simply

$p:\Omega\to\mathbb{R}$3

These recursive definitions mirror standard Bayesian-network semantics while encoding shared ancestry through the weak product (Schubert, 2012).

Three lemmas organize the calculus. Lemma 1 gives an inclusion-exclusion-style expansion of $p:\Omega\to\mathbb{R}$4. Lemma 2 establishes idempotency. Lemma 3, the decoupling rule,

$p:\Omega\to\mathbb{R}$5

is especially important computationally because it eliminates repeated occurrences of a shared factor $p:\Omega\to\mathbb{R}$6 across multiple terms (Schubert, 2012).

The central correctness theorem states that for any nonempty set of nodes $p:\Omega\to\mathbb{R}$7 in a noisy AND-OR-NOT Bayesian network,

$p:\Omega\to\mathbb{R}$8

Thus quasi-probabilities do not approximate BN probabilities in this setting; they are an exact symbolic representation of them. The same calculus yields formulas for conditioning and mixed truth/falsehood patterns. For example, if $p:\Omega\to\mathbb{R}$9 and $\mathcal{G}$00 are sets of possibly negated nodes, then

$\mathcal{G}$01

with division performed after sufficient elimination of $\mathcal{G}$02 (Schubert, 2012).

The weak-product formalism is accompanied by practical simplification rules—book-keeping, resolution, and decoupling—and by a square-wave pulse-train representation in which elementary probability symbols are represented by pulse trains whose areas equal their numeric values. Pointwise multiplication of pulse trains corresponds to weak product, and complementation corresponds to $\mathcal{G}$03 (Schubert, 2012). The pulse-train construction is presented as a computational representation rather than as a redefinition of Bayesian-network semantics.

7. Computational consequences and conceptual interpretation

For signed-kernel quasiprobabilistic models, membership in $\mathcal{G}$04 is, by definition, a linear-feasibility problem in the unknown kernels $\mathcal{G}$05 and $\mathcal{G}$06, subject to normalization and to the linear constraints imposed by the observable marginal formula. Without cardinality bounds on latent variables, the problem is underdetermined, and a concrete certificate consists of a particular quasifactorization (Becsi et al., 22 Jun 2026). In tree-structured correlation scenarios, however, the proof of Theorem 2 is constructive: one computes a minimal tree tensor-network decomposition, uses gauge transformations to enforce the nonzero-sum property, extracts the edgewise factors $\mathcal{G}$07, and thereby constructs finite latent priors and local responses. Although tensor rank is NP-hard for general tensors, in trees the minimal bond dimensions are governed by matrix ranks of flattenings and are computable in polynomial time; numerical conditioning and gauge-fixing still require care (Becsi et al., 22 Jun 2026).

For Schubert’s algebraic quasi-probabilities, the computational emphasis lies elsewhere. The calculus often permits substantial symbolic simplification before numeric substitution, especially in polytrees where weak products can frequently be replaced immediately by ordinary products because no elementary symbol appears in more than one factor. The rules of book-keeping, resolution, and decoupling are designed to prevent combinatorial blow-up. Nevertheless, worst-case complexity remains NP-hard, and the square-wave pulse-train representation may require exponentially long pulse trains to achieve a desired accuracy (Schubert, 2012).

Conceptually, the two traditions converge on a common interpretation of what is “quasi” in a Bayesian network. In the signed-kernel approach, quasiprobabilities remove inequality constraints and expose the equality structure of latent-variable models. In the Boolean algebraic approach, quasi-probabilities provide the “natural” algebra of Bayesian networks and make explicit how shared causal ancestry is handled without double counting. The first enlarges the admissible internal models while preserving the observable distribution; the second re-expresses exact BN probabilities in a symbolic calculus.

If the conjecture $\mathcal{G}$08 is correct for all DAGs, a precise causal interpretation follows: all inequality constraints on classical marginal models, beyond the nonnegativity of the observed distribution itself, would arise solely from positivity in the latent or response sector (Becsi et al., 22 Jun 2026). Under that reading, quasiprobabilistic models would furnish a constructive bridge between causal graphical structure, tensor-network parameterizations, and the hierarchy of classical, quantum, generalized-probabilistic, and non-signalling correlations.