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Identifiability from Stationary Covariance in Linear SDEs with Reciprocal Interactions and Latent Variables
Ana Leticia Garcez Vicente, Saber Salehkaleybar
TL;DR
Stationary-covariance identification for linear SDEs has been studied mainly in fully observed systems without reciprocal interactions, while latent variables complicate target-effect recovery. The paper develops graphical rank conditions for reciprocal interactions and an instrumental-variable approach for target direct effects under partial observability. It establishes identifiability conditions for both settings, with instrumental validity and rank assumptions in the latent-variable result.
Problem
Existing stationary-SDE identifiability results largely concern fully observed systems without directed 2-cycles, and latent confounding complicates recovery of direct effects.
Source: §1 · p. 1Method
Results
The paper establishes graphical conditions for generic identifiability with directed 2-cycles and a covariance-based instrumental-variable theorem for identifying target coefficients under its assumptions.
Source: §1 · p. 2, §5.1 · p. 7Takeaways & Limitations
Graphical structure and suitable instruments can support causal-effect identification from stationary covariance even when reciprocal interactions or latent variables are present.
Source: §1 · p. 2The instrumental-variable results assume root-node instruments and marginal independence between hidden parents of the target and the instruments.
Source: §5.1 · p. 6
Abstract
from arXiv · showhide
Continuous-time stochastic differential equations (SDEs) provide a natural framework for modeling interacting dynamical systems. A central causal question is whether causal effects among observed variables can be identified from observational data drawn from the stationary distribution, given the underlying causal graph. For linear SDEs, existing identifiability results are largely restricted to fully observed systems whose causal graphs contain no directed 2-cycles. We first establish new graphical conditions for the generic identifiability of causal effects in systems that may contain directed 2-cycles, thereby allowing for reciprocal interactions. Next, we address the previously unexplored setting of latent variables by developing an instrumental-variable framework for identifying a target direct causal effect under partial observability.
1 INTRODUCTION
Stationary covariance can support causal identification in linear SDEs even when reciprocal interactions or latent variables complicate recovery. The paper develops graphical rank conditions for full drift identification and an instrumental-variable framework for target effects under partial observability.
1 INTRODUCTION
- Directed 2-cycles are the principal obstruction to drift identifiability, and whether added graphical structure resolves their ambiguity remains open.Earlier global-identifiability results covered graphs without directed 2-cycles.Source: §1 · p. 1
- Latent variables can confound observed variables, preventing direct effects from being recovered through standard adjustment.Comparable latent-variable results for stationary SDEs are less established than in structural causal models.Source: §1 · pp. 1–2
- The paper uses a shared rank-based perspective to identify the complete drift under full observability and target coefficients under partial observability.Both settings use graphical structure to certify rank conditions for covariance systems induced by the stationary Lyapunov equation.Source: §1 · p. 2
- New graphical conditions guarantee generic identifiability in continuous Lyapunov models containing directed 2-cycles.The paper also characterizes identifiable four-node motifs outside those conditions and shows that identifiable subgraphs can be localized within larger non-identifiable systems.Source: §1 · p. 2
- An instrumental-variable framework identifies direct drift coefficients under latent confounding using observed covariance.The paper gives graphical conditions for instrumental covariance rank, including a bipartite matching characterization for direct instruments and a sufficient vertex-disjoint-path criterion when intermediate variables are present.Source: §1 · p. 2
2 RELATED WORK
Prior stationary-SDE work established identifiability results mainly for fully observed systems, while related causal-inference methods often address acyclic structural models or use different observations. This paper extends graphical identifiability analysis to reciprocal interactions and target effects under latent confounding.
2 RELATED WORK
- Earlier work developed causal models for stochastic dynamical systems and graphical continuous Lyapunov models for stationary observations.The drift sparsity pattern defines the directed graph, and stationary covariance summarizes equilibrium dependence across the system.Source: §2 · p. 2
- Earlier stationary-SDE results established global drift identifiability for simple graphs, while non-simple identifiable examples lacked a general graphical characterization.This paper complements that work with conditions for non-simple graphs and local structures that repair ambiguity from reciprocal interactions.Source: §2 · p. 2
- Causal methods for latent variables include adjustment and do-calculus, but are mainly developed for acyclic structural causal models.Structural-equation approaches also use graphical path structures to certify rank conditions, whereas this paper uses stationary covariance constrained by the continuous Lyapunov equation.Source: §2 · p. 2
- Stationary-SDE research has primarily considered full drift recovery under full observation, while other work addresses sign identification or integrated covariance from trajectories.This paper instead identifies a target drift coefficient under latent confounding from observed stationary covariance.Source: §2 · pp. 2–3
3 PRELIMINARIES
The paper models continuous-time interactions with stable linear SDEs whose stationary Gaussian covariance satisfies a Lyapunov equation. It formulates full and partial observability identification problems in terms of drift graphs, covariance, and rank criteria.
3.1 Stochastic differential equations
- A linear SDE has a drift matrix encoding direct effects and a diffusion matrix encoding stochastic perturbations; stability yields a stationary Gaussian covariance.The diffusion matrix is assumed known, and the stationary covariance satisfies the Lyapunov equation.Source: §3.1 · p. 3
3.1.1 Graphical Representation
- The drift graph represents nonzero drift coefficients, permits directed cycles including reciprocal 2-cycles, and partitions variables into observed and latent sets.In the stationary Gaussian setting, the covariance is determined by the stationary covariance matrix; trek connections characterize possible nonzero covariance entries.Source: §3.1.1 · p. 3
3.1.2 Problem Formulation
- The graphical continuous Lyapunov model restricts stable drift matrices to a graph and poses full-drift and target-effect recovery as identification problems.Global and generic identifiability distinguish uniqueness across all admissible matrices from uniqueness outside a proper algebraic exceptional set.Source: §3.1.2 · p. 4
- For full observability, generic identifiability is certified by a full-rank condition on H(Σ) restricted to non-edge rows.The criterion is equivalent to exhibiting a nonzero square minor of the required size.Source: §3.1.2 · p. 4
Proofs
- The proofs are placed in the appendix, and Figure 1 depicts generically identifiable three-node motifs.The supplied proof-section passages also label child and parent extensions.Source: Proofs · p. 4, Figure 1 · p. 4
4 GENERIC IDENTIFIABILITY UNDER FULL OBSERVABILITY
Under full observability, the paper gives graphical conditions for generic drift identifiability despite directed 2-cycles, then treats identifiable four-node motifs and recoverable induced subgraphs. The results show that added structure matters when it supplies independent covariance constraints.
4.1 Identifiability Beyond Simple Structures
- Generic identifiability holds when every directed 2-cycle is disjoint from other 2-cycles and has a distinct adjacent witness outside every 2-cycle.The proof specializes the drift locally around each cycle and combines nonsingular local and mixed minors to certify the rank condition.Source: §4.1 · pp. 4–5
4.2 Four-Node Non-Simple Identifiable Motifs
- Some four-node motifs remain generically identifiable even when they violate Theorem 1’s sufficient condition.Adjacency alone does not determine whether a 2-cycle’s rank deficiency is repaired; the covariance constraints induced by the graph matter.Source: §4.2 · p. 5
4.2.1 Extensions of deficient three-node motifs
- Adding one vertex to a deficient three-node motif can restore generic identifiability when the new adjacency has specified parent, child, or reciprocal-edge forms.Exceptions depend on the common-child and intersecting-2-cycle configurations; the extension must supply a new independent covariance constraint.Source: §4.2.1 · p. 5
4.3 Subgraph Identifiability
- An induced subgraph’s drift block is identifiable from its own covariance when the induced model is identifiable and no outside vertex is a parent of a vertex in the subgraph.Under that condition, the corresponding Lyapunov equation involves only the subgraph drift block, its observed covariance, and its known diffusion block—even if the full drift is not identifiable.Source: §4.3 · p. 5
5 CAUSAL EFFECT IDENTIFICATION UNDER LATENT CONFOUNDING
With latent variables, the section shifts from recovering the full drift to identifying a target direct coefficient from observed stationary covariance. It uses instrumental variables and graphical rank conditions, while noting that the path-based condition is sufficient but not necessary for full-rank recovery.
5 CAUSAL EFFECT IDENTIFICATION UNDER LATENT CONFOUNDING
- If every latent parent of y is conditionally independent of x given the other observed variables, the effective observed coefficient for x→y equals the structural coefficient.Without this separation, latent-parent terms can bias the effective coefficient; exact cancellation is nongeneric.Source: §5 · p. 6
5.1 Instrumental Variables
- When hidden parents are uncorrelated with root instruments and the instrumental covariance matrix has full row rank, the target parent coefficients are uniquely identifiable from observational covariance.The instruments must have known contributions and no direct edge into y.Source: §5.1 · pp. 6–7
5.2.1 Direct Instruments
- Without intermediate vertices, generic full rank is characterized by a bipartite matching covering all parents of y except one, plus two instruments sharing that parent with distinct self-drifts.This exact characterization follows from Hall’s theorem.Source: §5.2.1 · p. 7, §5.2 · p. 7
5.2.2 Intermediate Vertices
- With intermediate vertices, vertex-disjoint instrument-to-parent paths, plus two distinct paths to one shared parent with different instrument self-drifts, suffice for generic full rank.The path condition is sufficient, not necessary; the observed subgraph need not be acyclic or simple, and latent parents may be arbitrarily numerous if uncorrelated with instruments.Source: §5.2.2 · pp. 7–8
5.2.3 Bottleneck Investigation
- Accurate recovery is possible even when the bottleneck has too few intermediate nodes for the required vertex-disjoint paths, so Theorem 5’s condition is not necessary.Larger parent sets require a higher bottleneck ratio before estimation error decreases substantially.Source: §5.2.3 · p. 8
6 RESTRICTIVENESS OF THEOREM 1
Theorem 1 covers most numerically full-rank graphs in the sparsest regime, but its coverage falls sharply as graph density increases.
6 RESTRICTIVENESS OF THEOREM 1
- Figure 3 plots target-drift coefficient recovery error as a function of bottleneck ratio |Q|/|P′|, with curves for different parent counts and latent-parent count |U| = 5.Source: Figure 3 · p. 8
- 94.4–97.8% of numerically full-kernel-rank graphs are certified by Theorem 1 at edge probability 0.05; coverage falls to 6.4% for |V| = 12 at edge probability 0.30.The decline is attributed to denser graphs having more overlapping 2-cycles that violate the theorem’s vertex-disjoint structure, despite full kernel rank.Source: §6 · p. 8
7 CONCLUSION AND FUTURE WORK
The paper develops rank-based identifiability results for reciprocal linear SDEs under full and partial observability. It leaves finite-sample properties and broader modeling assumptions for future work.
7 CONCLUSION AND FUTURE WORK
- The paper studies identifiability from stationary covariance in non-simple linear SDEs, allowing directed 2-cycles in both fully and partially observed settings.It establishes graphical conditions for generic identifiability, an instrumental-variable framework for direct drift coefficients under latent confounding, and graphical conditions for instrumental covariance rank.Source: §7 · p. 8
- Finite-sample consistency, convergence rates, and robustness to covariance estimation error remain future-work questions.The conclusion also points to broader reciprocal structures and relaxing diffusion-matrix assumptions.Source: §7 · p. 8
A.1 Proof of Theorem 1
The proof constructs a parameter specialization whose covariance yields a full-rank kernel minor while retaining every directed 2-cycle. This establishes generic identifiability under the theorem’s graphical assumptions.
A.1 Proof of Theorem 1
- Each directed 2-cycle is grouped with a distinct witness into a three-vertex set, and the remaining vertices form singleton blocks.The sets partition the vertices; both cycle edges and the unique directed edge between each cycle endpoint and its witness are retained with positive coefficients.Source: §A.1 · p. 10
- Choosing pq < ab makes each three-vertex block Hurwitz, and the positive diagonal diffusion gives a unique positive-definite stationary covariance.The specialization disconnects distinct neighborhoods, so the covariance is block diagonal; sufficiently small positive off-diagonal coefficients make each covariance block strictly diagonally dominant.Source: §A.1 · p. 10
- For each cycle-with-witness neighborhood, three covariance columns admit a nonsingular 3 × 3 minor while both cycle-edge coefficients remain nonzero.The proof treats both orientations of the witness edge.Source: §A.1 · p. 11
- For covariance columns spanning distinct blocks, selected non-edge rows form strictly diagonally dominant, hence nonsingular, mixed-block minors.Each mixed column receives a distinct available non-edge row; the row’s diagonal entry is a positive variance and the block construction controls its off-diagonal entries.Source: §A.1 · p. 12
- Combining the local and mixed blocks gives a nonsingular selected minor, so the kernel has full rank and the model is generically identifiable.After row and column permutations, the selected matrix is block diagonal with the local and mixed minors as its blocks.Source: §A.1 · p. 12
A.2 Proof of Proposition 1
The appendix supplies rank certificates for additional four-node motifs and derives identities used in the latent-variable identification results.
A.2 Proof of Proposition 1
- For the four-node extensions, a nonzero selected-minor determinant certifies generic identifiability; the reverse-only common-child orientation is excluded by its collider structure.The same certificate also covers an optional reverse edge when its coefficient is set to zero; the reverse-only case forces covariances Σ24 and Σ34 to zero, leaving a kernel column zero.Source: §A.2 · p. 13
A.3 Proof of Lemma 2
- The observed covariance block satisfies a Lyapunov equation with effective drift MO = MOO + MOLΓL|O.Substituting the observed–latent covariance relation into the observed block yields the equation for MO and ΣOO.Source: §A.3 · p. 14
A.4 Proof of Proposition 3
- The (y,z) Lyapunov entry expands into contributions from observed and latent parents of y and parents of z.With diagonal diffusion and z ≠ y, the diffusion term in that covariance entry is zero.Source: §A.4 · p. 14
A.5 Proof of Lemma 3
- When z has no parents except itself, bz = mzzΣyz, and mzz = −czz/(2Σzz) is identifiable from the diagonal diffusion parameter and stationary variance.The proof concludes that bz is identifiable under the stated diagonal-diffusion parametrization.Source: §A.5 · p. 14
A.6 Proof of Theorem 4
Theorem 4 and Theorem 5 prove generic full-rank conditions by constructing parameter choices that make the relevant covariance minors nonzero. The four-node examples likewise emphasize that extensions repair rank deficiency only when they add independent covariance information.
A.6 Proof of Theorem 4
- A matching covering the observed parents, plus two instruments adjacent to one shared parent, yields the rank augmentation required for Theorem 4.The proof constructs a stable parameter choice with a nonzero covariance minor; unequal instrument self-drifts are not an extra generic condition.Source: §A.6 · p. 15
A.7 Proof of Theorem 5
- For Theorem 5, vertex-disjoint instrument-to-parent paths and two paths to a shared parent produce a nonzero covariance minor, giving full row rank generically.The construction retains the selected paths and parent-to-target edges, then chooses sufficiently negative diagonal drift coefficients to ensure stability.Source: §A.7 · p. 16
B FURTHER DETAILS ON THE FOUR-NODE EXTENSIONS
- Adding a vertex alone does not repair a deficient three-node motif; the extension must change trek and covariance structure to add an independent rank constraint.The examples distinguish useful extensions from those that merely reproduce existing constraints.Source: B FURTHER DETAILS ON THE FOUR-NODE… · pp. 16–17
B.1 Identifiable Four-Node Extensions
Certain four-node extensions of deficient motifs restore generic identifiability by adding independent covariance constraints. These motifs can also contribute to identifiability when embedded in larger graphs, subject to block-structure conditions.
B.1 Identifiable Four-Node Extensions
- Figures 4a–4d show identifiable extensions whose added vertex changes covariance structure and recovers the missing rank contribution.The displayed red arrows specify required or optional edges, while blue adjacencies may have either orientation and at least one indicated connection must be present.Source: §B.1 · p. 17, Figure 4 · p. 17
- Coupling two vertex-disjoint directed 2-cycles with directed edges can make the combined four-node graph generically identifiable, even when each cycle alone is deficient.One connecting edge may be removed while identifiability remains, provided the remaining connection preserves the treks needed for full column rank.Source: §B.1 · p. 17
B.2 Embedding the Motifs in Larger Graphs
- A collection of vertex-disjoint identifiable four-node motifs can be embedded in a larger generically identifiable graph when the remaining subgraph is identifiable and cross-block 2-cycles are absent.The block-rank construction then gives generic identifiability of the complete graph.Source: §B.2 · p. 17
B.3 Failed Extensions
Four-node extensions fail when they do not add covariance information that repairs rank deficiency. The supplied trek-graph criterion links separation to conditional independence, while the computational details describe population-level numerical evaluations of identifiability and recovery.
B.3 Failed Extensions
- A four-node extension can remain non-identifiable when the added vertex has no trek to either vertex of the directed 2-cycle, leaving its covariances with both equal to zero.The added rows then cannot repair the missing rank contribution; duplicated common-child or common-parent attachments likewise repeat existing constraints.Source: §B.3 · p. 18
C TREK GRAPHS AND PROPOSITION 2
- In the trek graph, separation of latent h and observed x by L\{h} implies X_h is conditionally independent of X_x given the other observed variables.The stated condition eliminates the marginalization term associated with a latent parent of y.Source: C TREK GRAPHS AND PROPOSITION 2 · pp. 18–19
D.1.1 Graph Generation
- Graph-generation experiments vary 2-cycle-containing graph structures, while model specification treats edge coefficients and diagonal drift entries as free and uses diagonal positive diffusion.Sampled drift matrices are retained only when stable; covariance matrices are obtained by solving the Lyapunov system, and inconsistent or non-positive-definite cases are discarded.Source: §D.1.1 · p. 19, §D.1.2 · p. 20, §D.1.3 · p. 20
D.1.5 Numerical Identifiability Test
- The numerical identifiability test optimizes each free drift coefficient in both directions under the Lyapunov constraints, classifying it as identifiable when its feasible range collapses.Each configuration and parameter is tested over 1000 random trials, and the reported quantity is the proportion classified as identifiable.Source: D COMPUTATIONAL DETAILS · p. 19, §D.1.5 · p. 20
D.2.1 Graph Configurations
- The IV experiments compare a sufficient full-rank configuration with a bottleneck configuration whose bottleneck ratio is 1/3.The graph structure is fixed within each configuration while numerical model parameters are resampled.Source: §D.2.1 · p. 21
D.2.3 Recovery Procedure
- For the IV recovery experiments, only observed covariance is supplied; target coefficients and the target self-drift are estimated jointly, with accuracy measured by mean relative ℓ2 error over 100 repetitions.The stationary covariance is computed directly from the continuous Lyapunov equation, with no trajectories or finite observational samples generated.Source: §D.2.3 · p. 21, §D.2.4 · p. 21, §D.2.2 · p. 21
D.2.4 Evaluation Metric
- The recovery metric is the relative ℓ2 error between estimated and true target drift-coefficient vectors, averaged across independent repetitions.The reported mean is computed over 100 repetitions.Source: §D.2.4 · p. 21
D.3 Restrictiveness of Theorem 1
Theorem 1 covers most numerically full-kernel-rank graphs in sparse regimes, but its sufficient condition becomes increasingly conservative as graph density rises.
D.3 Restrictiveness of Theorem 1
- 94.4%–97.8% coverage across graph sizes at p = 0.05 falls to 6.4% for |V| = 12 at p = 0.30, as Theorem 1 becomes more conservative on denser graphs.Figure 7 reports the share of randomly generated non-simple graphs with numerical full kernel rank that the theorem certifies, by vertex count and edge probability.Source: Figure 7 · p. 22, §D.3 · p. 22