Statistical Realizability Constraint

• Statistical realizability constraint is a set of algebraic and probabilistic conditions ensuring a candidate distribution is consistent with a model's latent structure.
• It involves deriving equality and inequality constraints that define the convex polytope of observable distributions, using methods like Fourier–Motzkin elimination.
• These constraints enable the falsification of incompatible models and guide efficient parameter estimation and model-based dimension reduction.

A statistical realizability constraint is a necessary and/or sufficient condition, expressed in probabilistic or algebraic terms, that determines whether a candidate distribution over observed or constructed quantities could actually arise from a given model, experimental procedure, protocol, or computational setup. Such constraints are central in graphical models, latent variable analysis, quantum information, high-dimensional statistics, causal inference, and verification of physical and computational models. Statistical realizability is typically formulated either as a set of equalities and inequalities that must be satisfied by observable distributions, or as procedural limitations reflecting what is attainable under a model’s specified mechanisms and allowed operations.

1. General Framework for Statistical Realizability

Statistical realizability arises fundamentally when only partial knowledge (e.g., marginals, moments, projections of measures, or empirical statistics) of a putative law is accessible, or when the mapping from model parameters (including hidden or latent variables) to observables is non-invertible. Given a generative or candidate model $\mathcal{M}$ imposing structure via latent variables, functional dependencies, or physical constraints, the realization set for the observables is often a strict subset of the ambient probability space. Formally, statistical realizability is characterized by the existence (or constructibility) of model parameters, hidden variables, or physical mechanisms such that a proposed distribution $P_\mathrm{obs}$ matches the observable implications of $\mathcal{M}$.

In hidden variable graphical models with categorical data, every observable law arises via a finite mixture over deterministic policies (response functions) as

$$P(o) = \sum_r \lambda_r f_r(o), \qquad \lambda_r \ge 0,\;\sum_r \lambda_r = 1,$$

where $f_r(o)$ are deterministic mappings and $\lambda_r$ are mixing weights. The set of realizable distributions $\{P(o)\}$ over the observables is then a convex polytope in the probability simplex, and every observable probability must satisfy all (in general, many) affine constraints—equalities and inequalities—that define this polytope (Sachs et al., 16 Jan 2026).

2. Complete Constraint Derivation and Polyhedral Characterization

Given a finite-dimensional parametric setting with latent variables, the complete set of statistical realizability constraints is determined by the image of the response-function simplex under the observable map. In matrix terms,

$$\boldsymbol{p} = \boldsymbol{B}\boldsymbol{r}, \qquad \boldsymbol{r} \ge 0,\,\mathbf{1}^\top \boldsymbol{r} = 1,$$

where $\boldsymbol{B}$ encodes the deterministic structure. The observable region is then the convex hull of the response patterns, i.e., a polytope. The full constraint set is obtained by converting the vertex representation (V-representation) of this polytope to a half-space (H-representation), which yields all equalities and inequalities. This is accomplished by algorithmic polyhedral conversion—Fourier–Motzkin elimination or the double-description method implemented in cddlib (Sachs et al., 16 Jan 2026).

The practical significance is that each face of the polytope corresponds to a testable constraint on observables; e.g., in the classical binary instrumental variable (IV) model, the instrumental inequalities are necessary and sufficient for a candidate $P(Y,X|Z)$ to be realizable under the model. For any empirical distribution $\boldsymbol{p}$, violation of any face constraint falsifies the model; satisfaction of all constraints guarantees statistical realizability by the Kreĭn–Milman theorem (Sachs et al., 16 Jan 2026).

3. Applications: Testing Realizability and Model Selection

The methodology generalizes well beyond textbook IV to sequential IVs, compound-instrument models, front-door/instrument hybrids, and also multipartite Bell-type settings (where the number of facets—Bell inequalities—can be massive). In all cases, the statistical realizability constraint translates model structure and latent-variable content into explicit, necessary and sufficient algebraic constraints on the observable joint probabilities (Sachs et al., 16 Jan 2026).

A generic statistical realizability test proceeds as:

Step	Description	Output
1. Compute polytope	Derive $\boldsymbol{B}$ for model; compute H-rep $(H_i,b_i,H_e,b_e)$	Matrix constraints $H_i \boldsymbol{p} \le b_i$, $H_e \boldsymbol{p} = b_e$
2. Evaluate data	For candidate/empirical $\boldsymbol{p}$, evaluate all left-hand-sides	Realizability status determined by constraint satisfaction

For any violation, the observable is not compatible with the hidden variable model.

4. Interpretational and Procedural Consequences

This formalism does not merely refine model selection; it provides a sharp operational threshold for what can be observed under hidden variable models. Violating the realizability constraint means that no parameterization consistent with the latent variable structure—regardless of parameter values—can ever produce the given data. This enables sharp falsification of classes of models that would be otherwise indistinguishable via conditional independence or other constraints.

Moreover, the approach establishes a framework for model-based dimension reduction: since the observable region is in general lower-dimensional, parameter estimation and statistical inference can be systematically restricted to the feasible region, enhancing efficiency and protecting against overfitting to spurious or nonphysical patterns.

5. Illustrative Examples

Binary Instrumental Variable Model: $Z,X,Y \in \{0,1\}$ with hidden $U$. The $8$-dimensional vector of joint probabilities $p_{xyz}$ is representable as a convex combination over $16$ possible response functions:

$$\boldsymbol{p}_{xyz} = \boldsymbol{B} \boldsymbol{r}, \quad \boldsymbol{r} \in \Delta_{15}$$

with $\boldsymbol{B} \in \{0,1\}^{8 \times 16}$. The H-representation recovers the instrumental inequalities (e.g., $p_{11,0}+p_{10,1} \le 1$) and normalization equations. These are necessary and sufficient for the observed $P(Y,X|Z)$ to be compatible with the model.

Sequential Instrumental Variable Model: For two successive IV districts, each associated 16-point simplex and induced polytope yields four inequalities analogous to the standard IV, again as facets of the respective polytope.

Generalization: The machinery directly extends to more complex models as long as the c-degree (max number of latent parents) is 1. For more intricate hidden-variable DAGs, the approach must be refined, but the principle persists: realizability equates to membership in a convex polytope defined by the model’s mixture structure.

6. Significance and Broader Impact

Statistical realizability constraints provide the foundational bridge between graphical/causal modeling and empirical testability. They go beyond conditional independence by encoding all functional constraints imposed by latent variable structure. In practical causal discovery, quantum nonlocality detection, and econometrics, realizability constraints yield effective, finite, and algorithmically checkable certificates of model compatibility.

The ability to systematically derive and test these constraints has direct implications for the design and falsification of statistical models, the interpretation of non-identifiable parameters, and the automation of scientific discovery across domains reliant on latent structure (Sachs et al., 16 Jan 2026).