Hidden-Variable Models in Theory and Application

• Hidden-variable models are theoretical frameworks that incorporate latent variables to explain observed statistical dependencies in fields like quantum foundations, graphical modeling, and network science.
• They face challenges in identifiability as multiple parameterizations can yield the same observable statistics, with techniques such as SVD used to analyze these equivalences.
• Applications range from testing quantum nonlocality through Bell-type scenarios to optimizing latent state estimation in network models using methods like EM and sum-of-squares relaxations.

Hidden-variable models are theoretical frameworks in which observed statistical dependencies among variables are explained by introducing additional, unobserved (‘hidden’ or ‘latent’) variables. These models are employed across quantum foundations, probabilistic graphical modeling, and modern network science to account for complex dependencies and emergent phenomena. The explicit mathematical formalism and operational role of hidden variables varies by domain, but always involves a mapping from a joint distribution over both observed and hidden components to the observed statistics alone.

1. Formal Definition and General Structure

A hidden-variable model posits that the observed set of variables $X$ depends probabilistically or deterministically on further variables $\lambda$ (classically called hidden or latent), which themselves follow some distribution. In probabilistic terms, an observed distribution $p_X(x)$ is represented as

$$p_X(x) = \int_\Lambda p_X(x|\lambda) \rho(\lambda) d\lambda$$

where $\rho(\lambda)$ is the hidden-variable density, and $p_X(x|\lambda)$ encodes how $\lambda$ influences the observed outcome.

In directed graphical models (DAGs), hidden variables correspond to unobserved nodes, and the full joint distribution factors as

$$p(x_{V \cup H}) = \prod_{v \in V \cup H} p(x_v | x_{\mathrm{pa}(v)})$$

where $V$ is the set of observed nodes, $H$ is hidden, and $\mathrm{pa}(v)$ denotes parents of $v$ in the DAG. Marginalization over $H$ yields the observable distribution. Models with hidden variables can induce constraints (e.g. Verma constraints) not visible to ordinary conditional-independence tests.

2. Identifiability and Covariance Structure

Hidden-variable models are generically underidentified: multiple distinct assignments of parameters $(A, B, \sigma^2, \Sigma^\epsilon)$ can yield identical marginal statistics. For instance, in the family of one-dimensional Gaussian DAGs relating observed blocks $X \in \mathbb{R}^p$, $Y \in \mathbb{R}^q$ coupled by a hidden $L \sim N(0,\sigma_L^2)$,

$X = A L + \epsilon_X,\quad Y = B L + \epsilon_Y$

the cross-covariance $\Sigma_{XY} = A B^T \sigma_L^2$ is rank-one (Wegelin et al., 2013). The singular value decomposition (SVD) $\Sigma_{XY} = u d v^T$ allows reconstruction of all parameterizations with

$A = a u, \quad B = \frac{d}{\sigma_L^2 a} v$

for $a$ in a feasible interval determined by the observed block covariances and positive semidefiniteness of the error blocks. Thus, the model is underidentified except for degenerate cases.

Alternative parametrizations with two correlated hidden variables per block (“paired-latent” models) yield the same family of cross-covariances, and under unrestricted block error structure, all variants (single latent, paired latent, asymmetric) are covariance-equivalent.

3. Quantum Foundations: Nonlocality, Measurement-Independence, and Outcome-Independence

In quantum mechanics, hidden-variable models have been central in debates around nonlocality and contextuality. The canonical Bell scenario considers bipartite measurements with settings $a, b$ and outcomes $\sigma, \tau$, and postulates a decomposition

$$P_{\sigma, \tau}(a, b) = \int d\lambda\, \mu(\lambda) P_{\sigma, \tau}(a, b, \lambda)$$

with $\mu(\lambda)$ the (potentially hidden) distribution over hidden variables.

Crucial constraints are:

• Measurement Independence (Uncorrelated Choice / ‘Free Will’): $\mu(\lambda|a, b) = \mu(\lambda)$.
• Setting-Independence (`No-Signaling’): The single-site marginal depends only on the local setting and $\lambda$, not the remote setting.
• Outcome-Independence: The joint conditional probability factorizes, $$P_{\sigma,\tau}(a, b, \lambda) = M_\sigma(a, \lambda) M_\tau(b, \lambda)$$.

Bell's theorem is predicated on all three; violation of even one allows quantum correlations to be simulated. Modern hidden-variable constructions achieving perfect quantum statistics for the spin singlet always violate outcome-independence, yet retain measurement independence and no-signaling. The general form is

$$P_{\sigma, \tau}(a, b) = \int d\lambda\, \frac{1}{4}[1 + \sigma \tau(-a \cdot b + C(\lambda, a, b))]$$

with $C(\lambda, a, b)$ capturing deviations which vanish in marginal or mean (Lorenzo, 2013, Lorenzo, 2011).

4. Dimensionality, Dynamics, and Ontological Compression

A central question is the minimal dimensionality required of the hidden-variable space to reproduce quantum statistics. For pure $N$-dimensional quantum systems, the quantum manifold has dimension $2N-2$; Markovian hidden-variable dynamics (short-memory) cannot reduce this (Montina, 2010). However, by abandoning Markovian updating, Montina exhibited one-dimensional non-Markov hidden-variable models for a qubit, compressing the ontic space below the quantum manifold (Montina, 2010). Generalization to higher-$N$ demands careful balancing between regularity, coverage of quantum states, and the structure of conditional probabilities.

Recent developments challenge the possibility of continuous, deterministic dynamics for hidden variables faithfully tracking time-evolving quantum systems, even when snapshots remain Bell-local. Dimensional no-go theorems show that for sufficiently large multipartite systems, no smooth microscopic LHV dynamics exists whose group action matches quantum evolution (Selzam et al., 18 Dec 2025).

5. Learning Hidden Variables: Cardinality, Structure, and State-Aggregation

Estimation of hidden-variable cardinalities is a critical methodological concern in probabilistic graphical modeling. Given incomplete data, Elidan & Friedman established a score-driven agglomerative clustering algorithm wherein candidate hidden states are greedily merged to maximize a decomposable score (e.g. BDe, MDL), efficiently exploring latent-state cardinality and structure (Elidan et al., 2013). For multiple interacting hiddens, a coordinate-descent round-robin extension jointly optimizes state cardinalities.

Approach	Cardinality Selection	Model Complexity	Performance
Agglomerative score	Greedy merging	Efficient	Strong
EM exhaustive	Full search	Costly	Variable
FindHidden	Semi-clique discovery	Integrated	Improved

The agglomerative approach dramatically reduces computational overhead compared to exhaustive EM and improves generalization.

6. Network Models: Hidden Variables for Structure, Fluctuations, and Dynamics

In network science, hidden-variable models assign each node a latent fitness $h_i$, with connection probability $p(h_i, h_j)$. This can reproduce arbitrary degree distributions (including scale-free power laws) (Balogh et al., 2019, Ostilli, 2014) and encode rich structural properties:

• Degree distribution scaling: Proper choice of hidden-variable cut-off $h_\max \sim N^\lambda$ with $\lambda \geq 1$ is necessary to reproduce all power-law moments; failure leads to mis-scaling of clustering and motif counts.
• Motif fluctuations: Density of motifs exhibits strong non-self-averaging in the random-$h$ ensemble, with giant fluctuations unless the degree exponent lies outside specific intervals.
• Dynamic models: Extending static hidden-variable models to temporal versions incorporates stochastic jumps of $h_i$ and link-resampling. Two parameters $(\alpha, \beta)$ control hidden-variable and link dynamics, yielding coexistence of frozen, quasi-static, and fully random regimes (Hartle et al., 2021). In the quasi-static regime, snapshots statistically match static hidden-variable draws, but in generic settings, structural persistence, suppressed heterogeneity, and out-of-equilibrium phenomena emerge.

7. Foundational Themes: Logical Structure, Constraints, and Falsifiability

Hidden-variable models are deeply connected to formal properties of dependence, independence, and context in quantum mechanics. Logics of dependence and independence enable uniform definitions of locality, outcome-independence, and contextuality across both probabilistic and relational frameworks (Albert et al., 2021). Notably:

• Bell's and Kochen-Specker no-go theorems are re-expressed as logical impossibility statements, clarifying the essential role of assumptions.
• Contextuality and measurement-independence alone are empirically vacuous: only their conjunction (context-irrelevance) produces falsifiable constraints, e.g., Bell’s inequalities (Dzhafarov, 2023).

Furthermore, any violation of Bell-CHSH can be attributed precisely to trade-offs among measurement-dependence, hidden-variable cardinality (‘hidden information’), and output factorization (Takakura et al., 2022). These trade-offs underlie the epidemic of non-classical correlations in quantum and classical models.

8. Applications and Methodological Extensions

Hidden-variable methodologies extend to algorithmic testing of model adequacy, for example, using polynomial sum-of-squares relaxations to construct ‘hidden-variable tests’—linear inequalities (Bell-type and generalized) whose violation rules out latent-variable explanations (Steeg et al., 2011). This framework generalizes beyond quantum mechanics to causal inference and network modeling.

Expectation-Maximization (EM) and its modern online and distributed extensions are fundamentally driven by divergence-based combinations of singleton hidden-variable models, enabling scalable parameter estimation for mixtures, HMMs, and Kalman filters (Amid et al., 2019).

• Gaussian latent variable DAGs: (Wegelin et al., 2013)
• Bell-scenario admissible hidden-variable models: (Lorenzo, 2013, Lorenzo, 2011, Lorenzo, 2011)
• Dimensionality, dynamics, and ontological compression: (Montina, 2010, Selzam et al., 18 Dec 2025, Montina, 2010)
• Learning hidden-variable cardinality: (Elidan et al., 2013)
• Network hidden-variable models: (Balogh et al., 2019, Ostilli, 2014, Hartle et al., 2021)
• Logical structure and contextuality: (Albert et al., 2021, Dzhafarov, 2023)
• Trade-offs and constraints: (Takakura et al., 2022, Ghirardi et al., 2012)
• SOS relaxations, tests: (Steeg et al., 2011)
• EM and divergence-based updates: (Amid et al., 2019)

Hidden-variable models thus constitute a foundational and versatile paradigm for explaining, analyzing, and testing structured dependencies in probabilistic, quantum, and network systems. Their interplay with identifiability, computational methods, and fundamental logical constraints continues to shape the landscape of statistical inference and physical theory.