Contemporaneous Causal ID Algorithm

• Contemporaneous causal identification is a method for discerning causal effects within the same time layer by leveraging finite-window graphical and algebraic criteria.
• The finite-window theorem establishes that identifiability can be determined using a truncated subgraph, eliminating the necessity to analyze the entire infinite time series.
• Algorithms extend Tian–Shpitser–Pearl methods to detect hedges and compute identifying functionals, addressing challenges posed by latent confounders and dynamic dependencies.

A contemporaneous causal identification algorithm formally decides whether an interventional distribution involving variables at the same time layer (such as $P(Y_t \mid \mathrm{do}(X_t))$) is identifiable from observed time series data and a given causal graph with possible latent confounding. Addressing this problem requires explicit structural assumptions about the temporal dependencies and confounding mechanisms present in the data-generating process. Approaches integrate graphical, algebraic, and statistical tools to characterize, detect, and compute such causal effects in infinite or recurrently structured graphs, as well as in practical finite settings.

1. Graphical Model Setup and Notation

The foundational model for contemporaneous causal identification in time series is a periodic acyclic directed mixed graph (ADMG) on countably infinite vertices partitioned as $X_{i,t}$ for $i=1,\dots,w$ (number of observed variables per time step) and $t\in \mathbb{Z}$ (discrete time). The graph includes:

• Directed edges $X_{i,t} \to X_{j,t'}$ with $t \le t'$, encoding forward-time causation.
• Bidirected edges $X_{i,t} \leftrightarrow X_{j,t'}$, representing influence by latent variables (confounding) between times $|t-t'| \le L$ for some fixed maximum lag $L$.

This periodic structure assumes time-invariance (stationarity) of both observed and hidden mechanisms. Formally, the model can be defined as a collection of time-invariant local Markov kernels and a graph $G$ of width $n=w$ and maximum lag $L$, with layer notation $\text{layer}(X_{i,t}) = t$ and $\mathcal{X}_{*,t} = \{X_{1,t},\dots,X_{w,t}\}$ (Jahn et al., 28 Apr 2025).

A contemporaneous causal effect refers to the change in the distribution of variables $Y_t \subseteq X_{*,t}$ at time $t$ under an intervention $\mathrm{do}(X_t)$, where $X_t \subseteq X_{*,t}$ and $Y_t \cap X_t = \emptyset$.

2. Causal Identification Algorithm: The Finite-Window Theorem

The central problem is to decide, given infinite time-series data, whether $P(Y_t \mid \mathrm{do}(X_t))$ is identifiable from the observed joint distribution and the assumed graph, i.e., whether it can be written without unknown parameters of unmeasured sources.

The procedure is based upon the Tian–Shpitser–Pearl ID-algorithm for ADMGs, which in general searches for combinatorial obstructions (hedges) to identifiability. For contemporaneous effects, key structural results allow reduction to a finite segment of the time series:

• Main finite-window theorem: There exists a computable function $f(n,L) = L \cdot 2^{Ln} (Ln+1)^{2Ln+2}$ such that $P(Y_t|\mathrm{do}(X_t))$ is identifiable in the infinite ADMG if and only if it is identifiable in the induced subgraph on time layers $[t-f(n,L), t]$. No dependence on an unbounded past is required; this provides the first such bound for time series with latent confounders (Jahn et al., 28 Apr 2025).

Algorithmic Steps:

1. Compute $C = f(n,L)$.
2. Form subgraph $G_\mathrm{seg}$ of $G$ on layers $[t-C, t]$.
3. Apply the finite-graph ID-algorithm to $(G_\mathrm{seg}, X_t, Y_t)$.
4. If a hedge is found, report non-identifiability; otherwise, return an explicit identifying functional.

3. Key Graph-Theoretic and Algebraic Concepts

Several structural constructs are central to identification in mixed graphs:

• C-component: Maximal set of nodes connected by bidirected paths.
• Hedge: A graphical witness of non-identifiability: a pair of nested C-forests violating identifiability via the do-calculus (Jahn et al., 28 Apr 2025).
• d-separation: Graphical criterion for conditional independence, with "mutilated" graphs used to justify do-calculus invocations.
• Ancestors and parents: Standard definitions in ADMG analysis; $\operatorname{An}(Z)$ and $\operatorname{Pa}(Z)$.

Identifiability is preserved under restriction to ancestors of $Y$ and after recursive absorption of irrelevant nodes, following the recursive logic of the ID-algorithm. Only C-components rooted at or before time $t$ are relevant for hedge detection.

4. Worked Example and Illustrative Procedure

A canonical example is a bivariate, lag-1 model:

• Variables: $A_t$, $B_t$.
• Edges: $A_t \to B_t$ (contemporaneous), $A_t \leftrightarrow B_{t-1}$ (lagged latent confounding).

To determine if $P(B_t|\mathrm{do}\,A_t)$ is identifiable:

• Examine the two-layer segment $[t-1, t]$.
• The existence of a hedge—bidirected $A_t \leftrightarrow B_{t-1}$ combined with $B_{t-1}\to A_t \to B_t$—implies the effect is not identifiable.
• Removing the confounding edge renders the effect identifiable, with standard back-door adjustment sufficing. This demonstrates both the obstruction role of hedges and practical use of the procedure (Jahn et al., 28 Apr 2025).

5. Extensions: Related Models and Methodologies

Contemporaneous causal identification algorithms have been extended and adapted in several frameworks:

• Dynamic Causal Networks (DCNs): In DCNs, identification leverages the notion of static and dynamic confounders within a folded "unrolled-slice" representation. When confounders are static (span at most as many slices as the maximal direct delay), applying the ID algorithm to a small, finite-slice subgraph suffices. For dynamic confounders (spanning longer temporal ranges), additional matrix multiplications over time-varying transition laws are used, but still require only a finite window for computability. Completeness and soundness are proven by reduction to graphical hedge criteria (Blondel et al., 2016).
• Segregated Graphs in Dependent Data: Sherman and Shpitser's SG-ID algorithm addresses contemporaneous effects in settings with spatial, network, or general dependent data. Here, undirected edges encode non-causal contemporaneous association (e.g., network ties), while bidirected edges capture hidden confounders. The ID problem is resolved by decomposing blocks (undirected) and districts (bidirected) into chain graph and nested Markov factors, and verifying reachability in induced CADMGs; non-reachability indicates a hedge and non-identifiability (Sherman et al., 2019).
• Constraint and Score-Based Algorithms: Recent approaches exploit kernel-based Granger causality and smooth sparsity-inducing penalties within a Gaussian process regression framework to recover both lagged and instantaneous (contemporaneous) edges in nonlinear, nonparametric settings. Such methods iterate over lagged and contemporaneous configurations, orienting adjacencies based on structural rules (e.g., absence of fully-instantaneous colliders), and are empirically validated against state-of-the-art benchmarks (Murphy et al., 14 Jan 2026).

6. Limitations, Theoretical Guarantees, and Computational Aspects

• Limitations: General identifiability is restricted by the presence of hedges. Certain graphical properties, such as periodicity and acyclicity, are essential. The identifiability of contemporaneous effects typically fails in the presence of particular latent confounder structures, and detection depends on hedge-finding, which is computationally tractable only on finite segments as per the window bound (Jahn et al., 28 Apr 2025).
• Theoretical guarantees: Soundness and completeness derive directly from the properties of the ID-algorithm and the structure of the finite-window theorem; identifiability failure in the truncated window certifies global non-identifiability, backed by explicit counterexamples (hedges).
• Computational aspects: The procedure scales polynomially in the width $n$, maximum lag $L$, and the finite window size $f(n,L)$. No computation over the infinite graph is needed. For practical implementation, all operations—including ancestor/parent set computation, C-component decomposition, and hedge detection—are performed on the segment $[t-f(n,L),t]$ (Jahn et al., 28 Apr 2025).

7. Connections to Causal Discovery and Future Directions

Integration of contemporaneous causal identification algorithms with joint structure learning procedures remains an active field. Approaches such as TS-ICD (Rohekar et al., 2023) prioritize detection of long-lagged temporal links before contemporaneous ones to enhance computational efficiency and reduce statistical error rates in the presence of latent confounders. Nonlinear and non-Gaussian models, such as score-based GPs with smooth $\ell_0$ penalties, further generalize these methods beyond linear-Gaussian or finite-state Markovian settings (Murphy et al., 14 Jan 2026). Ongoing challenges include extension to non-stationary processes, settings with feedback, and models with selection bias or more general forms of partial observation.

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References:

• "Causal Identification in Time Series Models" (Jahn et al., 28 Apr 2025)
• "Identification and Estimation of Causal Effects from Dependent Data" (Sherman et al., 2019)
• "Identifiability and Transportability in Dynamic Causal Networks" (Blondel et al., 2016)
• "From Temporal to Contemporaneous Iterative Causal Discovery in the Presence of Latent Confounders" (Rohekar et al., 2023)
• "Constraint- and Score-Based Nonlinear Granger Causality Discovery with Kernels" (Murphy et al., 14 Jan 2026)