Estimate Collapsibility in Causal Graphs

• Estimate collapsibility is the property that marginalizing a subset in a causal graph preserves the maximum-likelihood estimator for causal effects.
• It employs the strong d-convex hull to characterize minimal collapsible sets, ensuring that reduced subgraphs yield identical estimators as full models.
• The framework supports practical algorithms that speed up causal estimation while maintaining exact alignment with global estimation methods.

Estimate collapsibility is a notion of estimator invariance under marginalization in causal Bayesian networks and completed partially directed acyclic graphs (CPDAGs). For a causal Bayesian network $\mathcal{B}=(G,\mathcal{P}(G))$, it is defined by the equality $$\widehat{\mathcal{P}(G_A)}=\widehat{\mathcal{P}(G)_A}$$, meaning that the maximum-likelihood estimate obtained from the marginal model induced by a subgraph equals the marginal of the maximum-likelihood estimate from the full model. In the CPDAG setting, the concept is used to reduce a graph around a treatment–outcome pair while preserving the causal effect estimator before and after marginalization; the central structural object is the strong d-convex hull, which characterizes the minimal collapsible set (Deng et al., 8 Jun 2026).

1. Formal setting and scope

The 2026 formulation places estimate collapsibility within three collapsibility notions for causal Bayesian networks: CI-collapsibility, where the independence model is preserved after marginalization; model collapsibility, where the distribution family is preserved after marginalization; and estimate collapsibility, where the maximum-likelihood estimators are preserved after marginalization (Deng et al., 8 Jun 2026). This separation is central because the target is not merely preservation of conditional independences or model class, but preservation of the estimator actually used for causal effect computation.

The ambient graphical setting is that of DAGs and CPDAGs. A CPDAG $\mathcal{G}$ represents a Markov equivalence class of DAGs with the same independence model,

$I(G_1)=I(G_2).$

For a causal Bayesian network, the factorization is

$$P(x_V;\theta)=\prod_{v\in V} P(x_v\mid pa_G(x_v);\theta_v).$$

The paper studies how to identify, for a target pair $\{X,Y\}$, a smallest induced subgraph on which estimation can be performed without changing the resulting estimator (Deng et al., 8 Jun 2026).

This formulation makes estimate collapsibility a graph-reduction principle for causal estimation. A plausible implication is that it serves as a local-computation analogue of classical collapsibility in graphical modeling, but specialized to estimator preservation rather than to parameter or independence preservation.

2. Causal estimate collapsibility

For general estimation in a causal Bayesian network $\mathcal{B}=(G,\mathcal{P}(G))$, estimate collapsibility onto a subset $A$ is defined by

$$\widehat{\mathcal{P}(G_A)} = \widehat{\mathcal{P}(G)_A}.$$

This states that fitting the reduced graph $G_A$ and then estimating is equivalent to fitting the full graph and marginalizing afterward (Deng et al., 8 Jun 2026).

For causal effect estimation in a DAG, the paper defines a stronger causal version. Let $$\widehat{\mathcal{P}(G_A)}=\widehat{\mathcal{P}(G)_A}$$0 and $$\widehat{\mathcal{P}(G_A)}=\widehat{\mathcal{P}(G)_A}$$1 be non-adjacent, and let $$\widehat{\mathcal{P}(G_A)}=\widehat{\mathcal{P}(G)_A}$$2 contain $$\widehat{\mathcal{P}(G_A)}=\widehat{\mathcal{P}(G)_A}$$3. Then $$\widehat{\mathcal{P}(G_A)}=\widehat{\mathcal{P}(G)_A}$$4 is causal estimate collapsible onto $$\widehat{\mathcal{P}(G_A)}=\widehat{\mathcal{P}(G)_A}$$5 if, for every non-empty valid back-door adjustment set $$\widehat{\mathcal{P}(G_A)}=\widehat{\mathcal{P}(G)_A}$$6 in $$\widehat{\mathcal{P}(G_A)}=\widehat{\mathcal{P}(G)_A}$$7,

$$\widehat{\mathcal{P}(G_A)}=\widehat{\mathcal{P}(G)_A}$$8

The paper also gives the analogous empty-adjustment case: if $$\widehat{\mathcal{P}(G_A)}=\widehat{\mathcal{P}(G)_A}$$9 is valid in $\mathcal{G}$0, then causal estimate collapsibility holds when $\mathcal{G}$1 is the strong d-convex hull of $\mathcal{G}$2 and $\mathcal{G}$3 in $\mathcal{G}$4 (Deng et al., 8 Jun 2026).

The causal estimand is the average causal effect

$\mathcal{G}$5

and the back-door formula used in the reduced graph is

$\mathcal{G}$6

The framework therefore preserves not only local likelihood-based estimation but also the causal intervention estimate built from valid adjustment in the reduced graph (Deng et al., 8 Jun 2026).

A key point is that estimate collapsibility is defined at the level of the estimator. This distinguishes it from criteria that preserve only structural or probabilistic properties of the graph.

3. Strong d-convexity and minimal collapsible sets

The main structural result is that minimal collapsible sets are exactly strong d-convex hulls. A subset $\mathcal{G}$7 is strongly d-convex if:

1. No inducing path for $\mathcal{G}$8 exists in $\mathcal{G}$9.
2. $I(G_1)=I(G_2).$0 is linearly ordered with respect to $I(G_1)=I(G_2).$1, where $I(G_1)=I(G_2).$2,

$I(G_1)=I(G_2).$3

The paper defines the linearly ordered condition as follows: for any node in that set, any two distinct parents are adjacent unless both parents lie in $I(G_1)=I(G_2).$4 (Deng et al., 8 Jun 2026).

For $I(G_1)=I(G_2).$5, an induced subgraph $I(G_1)=I(G_2).$6 is the strong d-convex hull of $I(G_1)=I(G_2).$7 if it is the smallest strongly d-convex subgraph containing $I(G_1)=I(G_2).$8. The central theorem states: $I(G_1)=I(G_2).$9 This provides the exact graph-theoretic characterization of estimate collapsibility (Deng et al., 8 Jun 2026).

Several auxiliary results organize this characterization. The paper gives the equivalence

$$P(x_V;\theta)=\prod_{v\in V} P(x_v\mid pa_G(x_v);\theta_v).$$0

thereby linking estimator preservation, Markov-equivalent orientations, and strong d-convexity. It also states that for a DAG $$P(x_V;\theta)=\prod_{v\in V} P(x_v\mid pa_G(x_v);\theta_v).$$1 and $$P(x_V;\theta)=\prod_{v\in V} P(x_v\mid pa_G(x_v);\theta_v).$$2, the following are equivalent:

1. $$P(x_V;\theta)=\prod_{v\in V} P(x_v\mid pa_G(x_v);\theta_v).$$3 is d-convex.
2. $$P(x_V;\theta)=\prod_{v\in V} P(x_v\mid pa_G(x_v);\theta_v).$$4, where $$P(x_V;\theta)=\prod_{v\in V} P(x_v\mid pa_G(x_v);\theta_v).$$5 is the set of marginalization forbidden pairs.
3. There are no inducing structures of $$P(x_V;\theta)=\prod_{v\in V} P(x_v\mid pa_G(x_v);\theta_v).$$6 in $$P(x_V;\theta)=\prod_{v\in V} P(x_v\mid pa_G(x_v);\theta_v).$$7.

In addition, strong d-convexity is hereditary inside a strongly d-convex induced subgraph, intersections of strongly d-convex supersets remain strongly d-convex, and both the d-convex hull $$P(x_V;\theta)=\prod_{v\in V} P(x_v\mid pa_G(x_v);\theta_v).$$8 and strong d-convex hull $$P(x_V;\theta)=\prod_{v\in V} P(x_v\mid pa_G(x_v);\theta_v).$$9 exist and are unique (Deng et al., 8 Jun 2026).

These results make the reduced estimation domain canonical rather than heuristic: once $\{X,Y\}$0 is fixed, its minimal collapsible completion is uniquely determined.

4. Algorithms for identifying collapsible sets

The paper gives three algorithms for computing the relevant hulls in DAGs, later generalized to CPDAGs (Deng et al., 8 Jun 2026).

Algorithm	Output	Complexity
CVM($\{X,Y\}$1)	Vertices on shortest inducing paths between non-adjacent vertices in $\{X,Y\}$2	$\{X,Y\}$3
ICHA($\{X,Y\}$4)	A d-convex hull $\{X,Y\}$5	$\{X,Y\}$6
ISCHA($\{X,Y\}$7)	Strong d-convex hull $\{X,Y\}$8	$\{X,Y\}$9

CVM($\mathcal{B}=(G,\mathcal{P}(G))$0) starts with $\mathcal{B}=(G,\mathcal{P}(G))$1, computes

$\mathcal{B}=(G,\mathcal{P}(G))$2

and, for each non-adjacent pair $\mathcal{B}=(G,\mathcal{P}(G))$3 in $\mathcal{B}=(G,\mathcal{P}(G))$4, forms the moralized ancestral graph

$\mathcal{B}=(G,\mathcal{P}(G))$5

restricts it to $\mathcal{B}=(G,\mathcal{P}(G))$6, and collects the vertices on shortest paths. The resulting set records vertices on minimal inducing structures (Deng et al., 8 Jun 2026).

ICHA($\mathcal{B}=(G,\mathcal{P}(G))$7) iterates CVM: it initializes $\mathcal{B}=(G,\mathcal{P}(G))$8, repeatedly updates

$\mathcal{B}=(G,\mathcal{P}(G))$9

and stops when no new vertices are added. This yields the d-convex hull. ISCHA($A$0) then strengthens ICHA by enforcing the linear-ordering condition. After updating $A$1 by ICHA, it computes

$A$2

adds those parents to $A$3, and repeats until $A$4 (Deng et al., 8 Jun 2026).

The algorithmic significance is twofold. First, the minimal collapsible set is computable directly from graph structure. Second, the procedure is constructive enough to be embedded into causal effect estimation workflows rather than used as a purely structural diagnostic.

5. Reduction of CPDAG-based causal estimation

The strong d-convex hull is integrated with IDA, the framework “Intervention calculus when the DAG is Absent,” to form a reduced-graph estimation procedure for CPDAGs (Deng et al., 8 Jun 2026). The modified method proceeds as follows:

1. Obtain an arbitrary DAG $A$5 using Meek’s rules.
2. Compute

$A$6

1. For each DAG $A$7 consistent with $A$8, set $A$9 and $$\widehat{\mathcal{P}(G_A)} = \widehat{\mathcal{P}(G)_A}.$$0.
2. If $$\widehat{\mathcal{P}(G_A)} = \widehat{\mathcal{P}(G)_A}.$$1, build $$\widehat{\mathcal{P}(G_A)} = \widehat{\mathcal{P}(G)_A}.$$2, recompute $$\widehat{\mathcal{P}(G_A)} = \widehat{\mathcal{P}(G)_A}.$$3, and update $$\widehat{\mathcal{P}(G_A)} = \widehat{\mathcal{P}(G)_A}.$$4.
3. If $$\widehat{\mathcal{P}(G_A)} = \widehat{\mathcal{P}(G)_A}.$$5, estimate using $$\widehat{\mathcal{P}(G_A)} = \widehat{\mathcal{P}(G)_A}.$$6; otherwise estimate by back-door adjustment:

$$\widehat{\mathcal{P}(G_A)} = \widehat{\mathcal{P}(G)_A}.$$7

1. Return the multiset of possible causal effects.

This procedure reduces the size of the graph on which IDA is run, the number of equivalent DAGs that must be enumerated, and the number of variables entering adjustment sets. The paper states that the local estimator is exactly identical to the global IDA estimator when the strong d-convex hull is used (Deng et al., 8 Jun 2026).

A further invariance theorem extends the construction from DAGs to CPDAGs: if $$\widehat{\mathcal{P}(G_A)} = \widehat{\mathcal{P}(G)_A}.$$8 is any DAG in the Markov equivalence class of a CPDAG $$\widehat{\mathcal{P}(G_A)} = \widehat{\mathcal{P}(G)_A}.$$9, then for any $G_A$0,

$G_A$1

This result is essential because it shows that collapsibility is not an artifact of one representative DAG but a property stable across the entire equivalence class (Deng et al., 8 Jun 2026).

In effect, estimate collapsibility turns a global equivalence-class problem into a local induced-subgraph problem without changing the causal answer returned by IDA.

6. Relation to other collapsibility frameworks

Estimate collapsibility belongs to a broader family of collapsibility notions, but it is distinguished by its estimator-preservation target. In hierarchical log-linear graphical models, for example, collapsibility onto a target set $G_A$2 is defined by

$G_A$3

and a graphical model is collapsible onto $G_A$4 if and only if $G_A$5 contains every minimal separator between every pair of non-adjacent vertices in $G_A$6; equivalently, $G_A$7 contains at least one minimal $G_A$8-separator for every non-adjacent pair $G_A$9 (Heng et al., 10 Oct 2025). This separator characterization is closely related in spirit to the strong d-convex hull characterization, but the objects preserved are marginal model estimates rather than causal effect estimators in CPDAGs.

In the counterfactual causal framework, collapsibility is defined for effect measures rather than graph-based estimators. The causal risk difference is collapsible over arbitrary baseline covariates with weights $$\widehat{\mathcal{P}(G_A)}=\widehat{\mathcal{P}(G)_A}$$00, the causal risk ratio $$\widehat{\mathcal{P}(G_A)}=\widehat{\mathcal{P}(G)_A}$$01 is collapsible with weights $$\widehat{\mathcal{P}(G_A)}=\widehat{\mathcal{P}(G)_A}$$02, and no general weights exist for the odds ratio (Huitfeldt et al., 2016). A geometric reformulation using Rothman diagrams states that a measure of association is collapsible if and only if all its contour lines are straight; in that framework, risk difference and risk ratio are collapsible, while odds ratio and cumulative hazard ratio are noncollapsible except at the null line (Kenah, 30 Jun 2025). These are properties of effect measures, not of reduced graphical estimators.

For categorical contingency tables, marginal log-linear models define collapsibility by equality of parameters across margins,

$$\widehat{\mathcal{P}(G_A)}=\widehat{\mathcal{P}(G)_A}$$03

with strict collapsibility additionally requiring that all interactions involving removed variables vanish (Ghosh et al., 2017). A further generalization is average collapsibility, under which the conditional average of an association measure equals the marginal association measure; this is weaker than ordinary collapsibility and is developed for the expectation dependence function, the mixed derivative of interaction, and the log-expectation dependence measure (Vellaisamy, 2011).

These neighboring frameworks clarify the specificity of estimate collapsibility. It is not primarily a statement about effect-measure algebra, contingency-table parameter equality, or separator structure alone. It is a statement that marginalization over graph vertices leaves a designated likelihood-based causal estimator unchanged.

7. Empirical behavior and significance

The empirical evaluation attached to the strong d-convex hull framework emphasizes exactness together with substantial reduction in computational burden (Deng et al., 8 Jun 2026).

Domain	Setting	Reported outcome
Probabilistic reasoning	Hailfinder, Win95pts, Pathfinder, Munin1, Andes	56 nodes reduced to about 14; 76 to 8.77; 109 to 4.68; 186 to 22.35; 223 to 68.83
Probabilistic reasoning	Same benchmarks	KL divergence on the $$\widehat{\mathcal{P}(G_A)}=\widehat{\mathcal{P}(G)_A}$$04 scale; node reduction up to 96%
Synthetic causal estimation	Random DAGs with $$\widehat{\mathcal{P}(G_A)}=\widehat{\mathcal{P}(G)_A}$$05 to $$\widehat{\mathcal{P}(G_A)}=\widehat{\mathcal{P}(G)_A}$$06 nodes and densities $$\widehat{\mathcal{P}(G_A)}=\widehat{\mathcal{P}(G)_A}$$07 to $$\widehat{\mathcal{P}(G_A)}=\widehat{\mathcal{P}(G)_A}$$08	Recall near 1; Precision is 1; speedup up to about 18×
Real-world causal estimation	Sachs, Insurance, Alarm, Hepar2, Pathfinder, Munin1	Precision stays at 1.0; Recall typically $$\widehat{\mathcal{P}(G_A)}=\widehat{\mathcal{P}(G)_A}$$09; speedup 1.13× to 30.37×

The paper interprets the slight recall drop in real-world causal experiments as finite-sample noise rather than a theoretical failure (Deng et al., 8 Jun 2026). That interpretation is consistent with the exact structural theorems: the reduced estimator is guaranteed to match the full estimator when the strong d-convex hull criterion is satisfied.

The broader significance of estimate collapsibility is that it provides a precise notion of causally sufficient locality in CPDAGs. Instead of treating graph reduction as an approximation, it identifies the smallest induced subgraph that preserves maximum-likelihood estimation and back-door-based causal estimation exactly. This makes the concept relevant both for theoretical characterizations of estimator invariance and for practical acceleration of equivalence-class causal effect estimation.