CICME: Common & Individual Causal Mechanism Estimation

• CICME is a multi-domain causal discovery framework that distinguishes between common (invariant) and individual (domain-specific) causal mechanisms.
• It uses a three-step process: pooled causal discovery, residual-domain HSIC testing for invariant mechanism detection, and domain-specific refinement via hard freezing or soft regularization.
• Empirical results on manufacturing-inspired linear Gaussian models show that CICME can outperform pooled-only or individual-domain approaches under heterogeneous sensor conditions.

Common and Individual Causal Mechanism Estimation (CICME) is a three-step causal discovery framework for multi-sensor and multi-domain datasets in which some causal mechanisms are shared across domains and other mechanisms are domain-specific. It was introduced in “Causal Mechanism Estimation in Multi-Sensor Systems Across Multiple Domains” (Yu et al., 23 Jul 2025) as a mechanism-centric approach that first performs causal discovery on pooled data, then detects variables with domain-invariant causal mechanisms, and finally uses those stable mechanisms to guide domain-specific estimation. The framework is explicitly built on the principle of causal transfer learning (CTL): if a mechanism is truly stable, residuals from a model fit should be independent of the domain. In the reported experiments on linear Gaussian models under scenarios inspired from a manufacturing process, CICME leverages the benefits of applying causal discovery on the pooled data and repeatedly on data from individual domains, and it even outperforms both baseline methods under certain scenarios (Yu et al., 23 Jul 2025).

1. Problem setting and conceptual basis

CICME is designed for heterogeneous data collected across multiple domains, especially multi-sensor systems in which the data-generating process can vary across domains. The motivating settings include different machine IDs, product batches, sensor configurations, time-of-day, operating regimes, and environmental conditions. In such settings, distribution shift often invalidates the i.i.d. assumption that standard causal discovery methods rely on (Yu et al., 23 Jul 2025).

The framework is organized around a distinction between common causal mechanisms and individual causal mechanisms. Common mechanisms are shared across all domains, whereas individual mechanisms are domain-specific and may change when operating conditions, sensors, or production contexts change. The central motivation is that pooled causal discovery can be data-efficient but may blur domain-specific shifts, while per-domain causal discovery can capture heterogeneity but may be sample-inefficient. CICME combines these two regimes by finding stable mechanisms first and then using them to regularize domain-wise estimation (Yu et al., 23 Jul 2025).

The conceptual basis is CTL. In this formulation, causal mechanisms that are invariant across environments or domains can be identified and transferred. CICME operationalizes this principle by training models on pooled data, testing whether residuals are independent of domain labels, treating such variables as domain-invariant, and then using those stable mechanisms as transferable causal knowledge for domain-specific discovery (Yu et al., 23 Jul 2025).

A related but distinct mechanism-estimation perspective appears in “Extracting Mechanisms from Heterogeneous Effects: An Identification Strategy for Mediation Analysis” (Fu, 2024). There, a common causal mechanism is represented by a constant slope $\beta$ across groups, while an individual or heterogeneous causal mechanism is represented by $\beta_k$. This suggests a broader methodological theme: common and individual mechanisms can be inferred from structured heterogeneity, although the formal setting in (Fu, 2024) is mediation analysis rather than multi-domain DAG learning.

2. Formal model and optimization framework

The paper formulates the problem using a functional causal model (FCM). For variables $X_1,\dots,X_d$,

$X_j = f_j(\mathbf{PA}_j, N_j), \quad j=1,\dots,d,$

where $\mathbf{PA}_j$ are the parents of $X_j$ in a DAG, $N_j$ is an exogenous noise term, and the noises $N_1,\dots,N_d$ are assumed jointly independent. Under additive noise, this becomes

$X_j = f_j(\mathbf{PA}_j) + N_j.$

The framework then builds on continuous optimization-based causal discovery, especially NOTEARS-style methods (Yu et al., 23 Jul 2025).

A weighted adjacency matrix $W\in\mathbb{R}^{d\times d}$ defines the DAG, and acyclicity is enforced by a differentiable constraint

$\beta_k$0

where

$\beta_k$1

The generic continuous optimization problem is

$\beta_k$2

For NOTEARS-MLP, the model is parameterized by MLPs, one per variable, with adjacency extracted from first-layer weights:

$\beta_k$3

The training objective is

$\beta_k$4

CICME extends this optimization engine rather than replacing it (Yu et al., 23 Jul 2025).

The identifiability and modeling assumptions are standard for causal discovery. The paper relies on DAG structure, causal sufficiency, independent noises, and an additive noise model for the experimental section. It references identifiability results for linear non-Gaussian models, nonlinear ANMs, and linear Gaussian models with equal variances. In the experiments, identifiability is ensured by restricting parameters so the relevant edges are recoverable from data (Yu et al., 23 Jul 2025).

3. The three-step CICME procedure

CICME consists of three steps: causal discovery on pooled data, detection of stable variables via residual-domain independence, and causal discovery on each individual domain guided by the stable mechanisms (Yu et al., 23 Jul 2025).

In Step 1, suppose there are $\beta_k$5 domains,

$\beta_k$6

These are pooled into one dataset while keeping a domain index $\beta_k$7:

$\beta_k$8

NOTEARS-MLP is then trained on the pooled data exactly as in standard continuous DAG learning. The purpose of Step 1 is to obtain an initial estimate of the global causal structure, exploit the larger pooled sample size, and provide candidate mechanisms for later invariance testing. This step is essentially NOTEARS-pool (Yu et al., 23 Jul 2025).

In Step 2, CICME computes residuals from the pooled model for each variable $\beta_k$9:

$X_1,\dots,X_d$0

It then tests whether the residuals $X_1,\dots,X_d$1 are independent of the domain index $X_1,\dots,X_d$2 using HSIC, a nonparametric kernel-based independence test. The test statistic is $X_1,\dots,X_d$3, where

$X_1,\dots,X_d$4

The associated p-value is approximated as

$X_1,\dots,X_d$5

where $X_1,\dots,X_d$6 is the CDF of a Gamma distribution. If $X_1,\dots,X_d$7, CICME fails to reject independence and the mechanism for $X_1,\dots,X_d$8 is considered domain-invariant; otherwise, $X_1,\dots,X_d$9 is considered unstable. The variables classified as invariant are collected into $X_j = f_j(\mathbf{PA}_j, N_j), \quad j=1,\dots,d,$0 (Yu et al., 23 Jul 2025).

In Step 3, causal discovery is re-run on each individual domain, but the common mechanisms from Steps 1 and 2 are used to constrain estimation. The paper proposes two variants. In CICME-f, the parameters of stable variables are initialized from the pooled model and then frozen during domain-specific training: $X_j = f_j(\mathbf{PA}_j, N_j), \quad j=1,\dots,d,$1 for every stable variable $X_j = f_j(\mathbf{PA}_j, N_j), \quad j=1,\dots,d,$2, and the gradients of those parameters are set to zero during optimization. In CICME-l, the stable mechanisms are not frozen; instead, the domain-specific optimization includes an extra loss term that penalizes deviation from the pooled or common causal structure (Yu et al., 23 Jul 2025).

For CICME-l, let $X_j = f_j(\mathbf{PA}_j, N_j), \quad j=1,\dots,d,$3 be a mask matrix with

$X_j = f_j(\mathbf{PA}_j, N_j), \quad j=1,\dots,d,$4

Let $X_j = f_j(\mathbf{PA}_j, N_j), \quad j=1,\dots,d,$5 be the adjacency learned on pooled data and $X_j = f_j(\mathbf{PA}_j, N_j), \quad j=1,\dots,d,$6 the adjacency learned on domain $X_j = f_j(\mathbf{PA}_j, N_j), \quad j=1,\dots,d,$7. The common-structure penalty is

$X_j = f_j(\mathbf{PA}_j, N_j), \quad j=1,\dots,d,$8

The full objective in domain $X_j = f_j(\mathbf{PA}_j, N_j), \quad j=1,\dots,d,$9 becomes

$\mathbf{PA}_j$0

where $\mathbf{PA}_j$1 controls how strongly the domain model is encouraged to match the common structure. In the paper’s wording, CICME-f is a hard constraint via freezing, whereas CICME-l is a soft constraint via an MSE-like alignment penalty (Yu et al., 23 Jul 2025).

4. Experimental design and benchmark structure

The empirical study is conducted on synthetic data from a manufacturing leakage test process. The ground-truth causal graph contains the variables $\mathbf{PA}_j$2 (chamber temperature), $\mathbf{PA}_j$3 (air flow rate), $\mathbf{PA}_j$4 (pressure before test), $\mathbf{PA}_j$5 (pressure after some time), and $\mathbf{PA}_j$6 (latent leakage area factor). The causal structure is $\mathbf{PA}_j$7, $\mathbf{PA}_j$8, $\mathbf{PA}_j$9, and $X_j$0. The model is

$X_j$1

The noises $X_j$2 are standard Gaussian unless otherwise stated (Yu et al., 23 Jul 2025).

Four domain-shift scenarios are considered. In E1, $X_j$3 changes across domains, which changes the mechanism $X_j$4, and $X_j$5 is sampled from $X_j$6. In E2, the setup is the same as E1 except that in one domain $X_j$7, so the edge $X_j$8 is removed in that domain. In E3, $X_j$9 and the mean of $N_j$0 changes across domains, so $N_j$1 shifts in distribution while the structural equation remains. In E4, $N_j$2 and $N_j$3 is fixed to a constant in each domain, so $N_j$4 becomes constant within each domain but differs across domains (Yu et al., 23 Jul 2025).

For each scenario, $N_j$5 domains are generated, the sample sizes are $N_j$6, and each experiment is repeated $N_j$7 times. The model uses one hidden layer, $N_j$8 hidden units, sigmoid activation, $N_j$9, and $N_1,\dots,N_d$0 for CICME-l. No hyperparameter tuning is performed (Yu et al., 23 Jul 2025).

The baselines are NOTEARS-pool and NOTEARS-ind. NOTEARS-pool applies NOTEARS-MLP to pooled data only and is equivalent to CICME Step 1. NOTEARS-ind applies NOTEARS-MLP separately in each domain and is equivalent to CICME Step 3 with $N_1,\dots,N_d$1. These baselines are chosen to isolate the value of combining pooled and individual-domain discovery (Yu et al., 23 Jul 2025).

The evaluation metrics include stable variable detection, local structural Hamming distance (LSHD), structural Hamming distance (SHD), and execution time. For stable variable detection, the study counts how often each variable is classified as stable over $N_1,\dots,N_d$2 runs. LSHD evaluates whether the causal parents of stable variables are correctly identified, with LSHD $N_1,\dots,N_d$3 meaning the local parent set is perfectly recovered. SHD compares full learned graphs against ground truth and counts edge additions, deletions, and reversals; lower SHD is better. Execution time is reported to compare computational cost (Yu et al., 23 Jul 2025).

5. Empirical behavior and comparative performance

The reported results show that CICME can reliably identify stable variables when sample size is large enough, especially at $N_1,\dots,N_d$4. The variable whose mechanism changes across domains is usually detected as unstable. With fewer samples, HSIC has lower power and misclassification increases. Stable-variable misclassification can hurt Step 3 because a truly stable mechanism might not be carried forward (Yu et al., 23 Jul 2025).

A notable failure case is E4, where $N_1,\dots,N_d$5 is often incorrectly labeled unstable, even at large sample size. The paper attributes this to convergence issues in the constrained optimization and suggests that pre-finetuning with least squares might help (Yu et al., 23 Jul 2025). This suggests that the stable-mechanism detection stage is not only a statistical bottleneck but also an optimization-sensitive stage.

For full causal structure recovery, the overall pattern is that NOTEARS-ind tends to do better when each individual domain has enough samples, because it captures domain-specific structures, whereas NOTEARS-pool tends to do better when sample size is small, because pooling increases statistical strength. CICME is presented as a method that combines these advantages (Yu et al., 23 Jul 2025).

The paper identifies settings in which CICME outperforms the baselines. In E1 and E2 with moderate sample size ($N_1,\dots,N_d$6), CICME-l can outperform both pooled and individual baselines. In some settings, CICME-l matches the best baseline at $N_1,\dots,N_d$7 and $N_1,\dots,N_d$8. In E4, CICME can outperform NOTEARS-ind because pooling in Step 1 makes the otherwise constant $N_1,\dots,N_d$9 identifiable (Yu et al., 23 Jul 2025).

The paper also reports settings in which CICME does not fully win. In E4 with $X_j = f_j(\mathbf{PA}_j) + N_j.$0, CICME may fail to match NOTEARS-pool because of false instability detection of $X_j = f_j(\mathbf{PA}_j) + N_j.$1. If a stable mechanism is misclassified as unstable, the corresponding causal edge may not be transferred, causing downstream errors. Accordingly, the method works best when the number of samples is sufficient, stable mechanisms are detected accurately, and the pooled structure is informative for later per-domain refinement (Yu et al., 23 Jul 2025).

The two CICME variants have different trade-offs. CICME-l often achieves the best accuracy when enough data are available, because it allows flexibility while still regularizing toward common structure. CICME-f is more computationally efficient and can be nearly as accurate in some low-sample settings. Freezing stable mechanisms reduces training cost and can help estimation by reducing degrees of freedom (Yu et al., 23 Jul 2025).

6. Runtime characteristics, limitations, and broader methodological context

Execution time is reported as a distinct axis of comparison. NOTEARS-pool is fastest because it runs only once. CICME-f has only modest overhead above pooled training and is often much cheaper than repeatedly training per-domain models. CICME-l is slower because of the extra alignment term and optimization overhead. The HSIC test itself is comparatively cheap; most runtime comes from NOTEARS optimization (Yu et al., 23 Jul 2025).

The limitations stated in the paper are specific. All domains must contain the same set of variables; otherwise variables missing in some domains must be dropped. CICME assumes causal sufficiency. Stable-variable detection depends on sample size, and with too few samples HSIC may fail. The HSIC test sometimes misclassifies stable variables as unstable, which hurts Step 3. The current version does not explicitly identify which domains changed, only whether a mechanism is stable across all domains. The study is evaluated on a synthetic linear Gaussian manufacturing-inspired model, and broader empirical validation is needed (Yu et al., 23 Jul 2025).

These limitations delimit the scope of the method. A plausible implication is that CICME is best viewed as a transfer-learning-inspired causal discovery procedure for settings where heterogeneity is structured but not arbitrary, and where invariant mechanisms are expected to recur across all observed domains.

Within a broader mechanism-estimation landscape, (Fu, 2024) provides a complementary viewpoint. That paper decomposes heterogeneous treatment effects as

$X_j = f_j(\mathbf{PA}_j) + N_j.$2

where a constant $X_j = f_j(\mathbf{PA}_j) + N_j.$3 corresponds to a common mechanism and a varying $X_j = f_j(\mathbf{PA}_j) + N_j.$4 corresponds to heterogeneous mechanism strength. The formal objects differ from the DAG-based formulation of CICME, but the conceptual parallel is direct: both approaches use structured heterogeneity to distinguish common from individual mechanisms, and both treat invariant mechanism components as transferable across contexts (Fu, 2024).

7. Significance and interpretation

The main contributions attributed to CICME are introducing a three-step method for learning causal mechanisms from multi-domain data, using residual-domain independence tests to detect variables with invariant causal mechanisms, proposing two domain-specific refinement strategies—hard freezing of stable mechanisms and soft regularization toward pooled mechanisms—and demonstrating on a manufacturing-inspired linear Gaussian benchmark that CICME can outperform both pooled-only and per-domain-only causal discovery under certain conditions (Yu et al., 23 Jul 2025).

The framework is explicitly motivated by heterogeneous sensor systems, especially in manufacturing and other industrial environments where sensor distributions vary across machines, batches, and operating conditions. Its practical logic is to exploit invariances where they exist, preserve transferable causal structure, adapt locally where mechanisms change, and combine pooled and per-domain learning rather than choosing one exclusively (Yu et al., 23 Jul 2025).

A common misconception would be to treat CICME as a generic prediction transfer method. The description in the paper is narrower and more specific: CICME is a causal mechanism transfer procedure based on pooled discovery, residual-domain independence testing, and constrained domain-specific refinement (Yu et al., 23 Jul 2025). Another possible misconception would be to read the method as identifying arbitrary cross-domain change patterns; the current version detects whether a mechanism is stable across all domains, but it does not explicitly identify which domains changed.

In summary, CICME denotes a multi-domain causal discovery strategy in which common and individual causal mechanisms are estimated sequentially rather than separately. The method uses pooled data to increase statistical strength, residual-domain independence to identify invariant mechanisms, and domain-specific refinement to capture remaining heterogeneity. In the reported experiments, this combination often outperforms either pooled or individual discovery alone in heterogeneous multi-domain sensor settings (Yu et al., 23 Jul 2025).