Context-Specific Independencies

• Context-specific independencies are conditional independence relationships that hold only within specific contexts, allowing localized and precise modeling.
• They are integrated into diverse models such as log-linear frameworks, Bayesian networks, Markov networks, and staged trees to improve model parsimony and inference efficiency.
• Their use in causal inference and meta-learning addresses challenges like non-smooth parameterization and redundant constraints while enhancing effect identification.

Context-specific independencies (CSIs) refer to conditional independence relationships that hold only within selective subsets—or “contexts”—of the outcome or conditioning variable space, as opposed to globally for all assignments of the conditioning set. CSIs have emerged as a powerful conceptual and technical extension to standard conditional independence, enabling the modeling and exploitation of localized, heterogeneous, or structurally asymmetric dependence phenomena in statistical models, graphical models, and causal inference frameworks. The paper and formalization of CSIs span a broad range of paradigms including log-linear and graphical models, Bayesian and Markov networks, staged tree models, causal inference, probabilistic programming, statistical relational learning, and algorithmic meta-learning.

1. Formal Definitions and Core Constructions

A context-specific independency is a conditional independence statement that holds only for a subset of configurations of the conditioning variables, rather than universally. For discrete variables, a typical CSI asserts that for variables $X$, $Y$ and a conditioning set $Z$, $X \perp Y \mid Z = z^*$ for specific $z^*$. In graphical and log-linear frameworks, this is expressed by imposing independence constraints only within particular strata or subspaces:

• In log-linear and staged tree models, a CSI may take the form $X_A \perp X_B \mid X_S, X_C = x_C$, meaning $X_A$ and $X_B$ are conditionally independent given $X_S$ only when $X_C = x_C$.
• For continuous variables, context-set specific independence (CSSI) generalizes this by requiring that $Y \perp X_{A^c} \mid X_A, \mathcal{C}$, that is, the conditional distribution of $Y$ given $X_A$ is invariant across all $X_{A^c}$, within a measurable context set $\mathcal{C}$ in the parent space (Hwang et al., 12 May 2024).

CSIs are naturally represented within staged tree models as vertices (contexts) sharing the same stage group, within log-linear models via non-vanishing linear combinations of parameters selectively imposed in contexts, and in causal models via labeled edges or regime-dependent function graphs (Colombi et al., 2012, Nyman et al., 2013, Alexandr et al., 2022, Rabel et al., 27 Oct 2024).

2. Context-specific Independencies in Log-linear and Graphical Models

In classical graphical models, conditional independence is encoded globally in the absence of edges or via hierarchical vanishing of log-linear parameters. However, repeated imposition of the same interaction in multiple marginal distributions leads to non-smooth, singular parameterizations. To address such issues, (Colombi et al., 2012) introduces a marginal log-linear parameterization that leverages a mixed parameter set—anchoring certain interactions to a reference configuration and allowing for selective omission (replacement) of redundant constraints. The approach enables the formulation of models where smoothness is restored by "removing" problematic constraints in higher-order marginals, but only for corresponding context-specific configurations. The intended global conditional independence "becomes context-specific," i.e., the independence holds except in configurations activating the omitted higher-order interactions.

Mathematically, for a marginal set $M$, subset $I \subset M$, and context $x_0$, the marginal log-linear interaction is: $$\eta_{I;M}(x_I \mid x_{M\setminus I}) = \sum_{J \subset I} (-1)^{|I\setminus J|} \log p_M(x_J, 0_{I\setminus J}, x_{M\setminus I})$$ The selection and omission of redundant parameters follow combinatorial rules ensuring one-to-one and differentiable mappings, as checked via the rank and spectral radius conditions involving key derivative matrices (e.g., the Jacobian $J = A^{-1}B$ and replacement matrices $Q_{I,H|R}$). This machinery provides a robust framework for parameterizing and reconstructing context-specific models.

Stratified graphical models (SGMs) (Nyman et al., 2013) and context-specific graphical log-linear models (Nyman et al., 2014) extend standard graph-based factorizations by labeling edges with contexts or "strata," leading to parameter restriction equations of the form

$$\sum_{B \subset A, \{\delta, \gamma\} \subset B} \phi_B = 0,$$

valid only within the given context, enabling non-hierarchical yet interpretable parameterizations that can be fit via cyclical projection algorithms.

3. CSIs in Bayesian Networks, Markov Networks, and Staged Trees

In Bayesian networks (BNs), standard d-separation conveys only global conditional independence. CSIs, however, are encoded at the level of conditional probability tables (CPTs) or, more efficiently, via tree-structured CPTs ("CPT-trees") (Boutilier et al., 2013). In such representations, branching structure reflects the context—branches not involving a variable $Y$ in the presence of certain assignments imply context-specific independence between $Y$ and the child variable. The formal criterion is: $\text{If %%%%22%%%% does not appear on any path of the CPT-tree consistent with context %%%%23%%%%, then } X \perp Y \mid c.$ Exploitation of these structures in inference algorithms, such as context-specific likelihood weighting (CS-LW) (Kumar et al., 2021) and first-order context-specific likelihood weighting (FO-CS-LW) (Kumar et al., 2022), leads to variance reduction and scaling benefits by targeting only requisite subspaces for sampling.

In undirected Markov networks, CSIs cannot be encoded directly in the undirected graph. The approach in (Edera et al., 2013) circumvents this by using log-linear models where each feature corresponds to a context-specific assignment. CSIs are represented as factorization of feature sets active under a context $x_W$, governed by a context-specific Hammersley-Clifford decomposition.

Staged tree models (Alexandr et al., 2022, Leonelli et al., 28 May 2024) generalize these ideas: tree vertices are colored ("staged") to indicate that the same conditional distribution applies across multiple paths, thus capturing context-specific independence not possible with fixed DAGs. The equivalence between staged tree models and Bayesian network factorizations is formally established, but only for symmetric staging; additional asymmetric (context-specific) dependencies are captured by refining the staging via clustering.

4. Advanced CSI Notions and Axiomatizations

Beyond classical CSIs, several advanced classes have been identified:

• Contextual Weak Independence (CWI) (Wong et al., 2013), a generalization that allows independence in equivalence classes of the domain, with axiomatizations encompassing reflexivity, transport, augmentation, and "weaken" and transitivity rules that clarify the interplay with standard CI.
• Regular and canonical CSSI decompositions for continuous variables (Hwang et al., 12 May 2024), establishing uniqueness of the minimal local parent set under convexity assumptions.
• Stratified chain graphical models (SCGM) for ordinal or categorical data (Nicolussi et al., 2017), which employ labeled arcs in chain graphs to encode CSIs and derive context-conditional Markov properties.
• Decomposable context-specific models (Alexandr et al., 2022) that generalize decomposable graphical model theory via collections of perfect DAGs, toric algebraic characterizations, and Markov bases derived from saturated CSI statements.

These generalizations are crucial for formalizing and reasoning about partial local independence and for enabling refined inferences in large-scale or multi-context systems.

5. CSIs in Causal Modeling and Identification

In causal inference, CSIs serve as essential information for effect identification when global CI is insufficient. LDAGs (labeled DAGs) (Tikka et al., 2020), $lm$-SCMs (Aguas et al., 18 Jun 2025), and context-enriched causal graph objects (Rabel et al., 27 Oct 2024) model CSIs via edge labels or regime-specific mechanisms. For instance, when latent confounding impedes identification by standard do-calculus, the presence of a CSI may sever the confounding path in a context $A = 0$, yielding an identifying formula such as: $$P(Y\,|\,do(X)) = P(Y\,|\,A=0,X)P(A=0) + P(Y\,|\,A=1)P(A=1)$$ (Tikka et al., 2020).

Moreover, in multi-context systems (Rabel et al., 27 Oct 2024), the causal graph objects—such as descriptive, physical, observable, and union graphs—jointly encode both the functional mechanisms and observational support, so that CSIs can emerge either from support limitations or from bona fide mechanism changes. Identifiability results demonstrate that, under strong context-faithfulness, the observed union graph is the union over context-specific graphs, and that context-specific independencies can be used to explain context-dependent anomalies or shifts.

For settings where missing data can trigger mechanism shifts, as in $lm$-SCMs (Aguas et al., 18 Jun 2025), labeled edges capture the activation of alternate assignment functions when a parent variable becomes unobserved. This directly induces CSIs, which are crucial for correctly recovering either hypothetical (FATE) or natural (NATE) causal effects under real-world patterns of missingness.

6. Practical Inference, Learning, and Applications

CSIs substantially affect statistical modeling, inference, and learning:

• Sampling algorithms such as CS-LW (Kumar et al., 2021) and FO-CS-LW (Kumar et al., 2022) exploit CSIs by partitioning variables into assigned/unassigned sets, using contextual assignments and Rao–Blackwellization to reduce variance.
• Algorithmic inference and learning benefit from CSIs by yielding more compact, accurate models and improving learning efficiency, as demonstrated in synthetic and real datasets for Markov network structure learning (Edera et al., 2013), staged tree classifier accuracy (Leonelli et al., 28 May 2024), and stratified graphical model recovery (Nyman et al., 2013).
• CSIs enable the design of modular and granular probabilistic models with commutative coarsening/refinement (nest/unnest) operations (Wong et al., 2013), and contribute directly to meta-learning architectures that mimic context-gated human cognitive adaptation (Dubey et al., 2020).
• Causal inference with CSI extends effect identification in presence of context-dependent mechanism shifts or missingness-specific adaptation (Aguas et al., 18 Jun 2025) and supports transfer learning, generalization, anomaly detection, and regime-aware explanation in multi-context and time-varying systems (Rabel et al., 27 Oct 2024).

A summary of encoding and exploitation methods is presented:

Model Family	CSI Representation	Main Exploitation Mechanism
Log-linear / SGM	Strata/context edge labels	Mixed parameterization, projections
Bayesian Networks	CPT-trees/rule-based CPDs	Tree-structured inference, arc deletion
Markov Networks	Log-linear feature sets	Factorization by context, CSPC algo
Staged Trees	Vertex staging, minimal DAGs	Agglomerative staging refinement
Probabilistic Logic	Distributional clauses	Contextual assignments, lifted sampling
Causal Models	Labeled edges / regime graphs	CSI-calculus, context-specific graphs

7. Challenges, Limitations, and Future Directions

While CSIs considerably increase the expressiveness and parsimony of probabilistic and causal models, the complexity of their identification and exploitation is substantial:

• Determining the optimal contextual decomposition is often NP-hard (Jamshidi et al., 2023, Tikka et al., 2020), as inference and validity verification must be performed across exponentially many contexts or context-induced subgraphs.
• The presence of redundant or repeated constraints in log-linear models can induce singularities that must be carefully resolved by principled omission/replacement schemes (Colombi et al., 2012).
• The interpretability of non-hierarchical parameterizations and context-specific chain decompositions continues to motivate research into more transparent and theoretically grounded frameworks (Nyman et al., 2014, Nicolussi et al., 2017).
• In causal inference, distinguishing between loss of dependence from mechanism changes versus observational support limitations requires refined graph objects and identification criteria (Rabel et al., 27 Oct 2024).
• Development of scalable, automated search procedures (using neural methods or advanced algebraic geometry) is an area of active research (Hwang et al., 12 May 2024, Alexandr et al., 2022).

Potential further work includes the integration of CSI concepts with time-series, handling hidden confounding, refinement of regime-aware intervention graphs, and unification across Bayesian, frequentist, and logic-based modeling traditions. The ongoing extension of the theoretical toolkit for CSIs is expected to yield both deeper foundational understanding and practical gains in statistical, machine learning, and causal modeling domains.