---
title: Strong d-Convex Hulls in Causal Graphs & Geometry
url: https://www.emergentmind.com/topics/strong-d-convex-hulls
type: topic
---

# Strong d-Convex Hulls in Causal Graphs & Geometry

Searching arXiv for recent and foundational papers on strong d-convex hulls and closely related notions.
Strong d-convex hulls arise in at least two technically distinct settings. In causal graphical models, a strong d-convex hull is a graph-theoretic closure of a variable set \(R\) inside a DAG or CPDAG, introduced to characterize the smallest retained subgraph that preserves estimate collapsibility and causal effect estimation under marginalization [2606.08941]. In convex geometry, the closely related literature on strongly convex hulls studies \(K\)-hulls, that is, intersections of all translates of a fixed convex body \(K\) containing a set, written as \(\operatorname{conv}_K X = K \stackrel{*}{-} (K \stackrel{*}{-} X)\) or \(\bh_K(A)=\bigcap_{x:\,A\subset K+x}(K+x)\) [1509.08783], [2102.10009]. The two notions share the language of “strong convexity,” but they are defined on different objects, use different closure operators, and serve different theoretical purposes.

## 1. Graph-theoretic definition in DAGs

In the causal-inference formulation, the ambient object is a DAG \(G=(V,E)\), and strong d-convexity is defined for subsets \(R\subseteq V\). The construction depends on two ingredients: inducing paths and a parent-adjacency condition called linear ordering. For a path \(l_{xy}\) between non-adjacent vertices \(x,y\in R\), \(l_{xy}\) is an inducing path of \(R\) if
\[
R\cap V^o(l_{xy}) \subseteq V^c(l_{xy}) \subseteq an_G(\{x,y\}).
\]
Here \(V^o(l_{xy})\) denotes internal vertices and \(V^c(l_{xy})\) the colliders on the path. The associated inducing structure is
\[
\mathrm{IS}_G(l_{uv}) = l_{uv}\cup \mathcal L_G(l_{uv}),
\]
where \(\mathcal L_G(l_{uv})\) is the set of all directed paths from colliders on \(l_{uv}\) to \(An_G(\{u,v\})\) [2606.08941].

The second ingredient is linear ordering with respect to \(R\). A vertex \(v\) is linearly ordered with respect to \(R\) if any two distinct vertices in \(pa_G(v)\) are adjacent, except when both belong to \(R\). A set \(A\subseteq V\) is linearly ordered with respect to \(R\) if every \(w\in A\) is.

With this notation, the formal definition is:
\[
\text{A subset }R\subseteq V\text{ is strongly d-convex if}
\]
1. there is no inducing path for \(R\) in \(G\); and  
2. \(Ch_G(M)\cap An_G(R)\) is linearly ordered with respect to \(R\), where \(M=V\setminus R\), \(Ch_G(M)=ch_G(M)\cup M\), and \(An_G(R)=an_G(R)\cup R\).

If only condition (i) holds, the set is d-convex. The distinction is structural: d-convexity suffices for CI-collapsibility and model collapsibility, but not for estimate collapsibility. Strong d-convexity is therefore strictly stronger in the sense used by the causal paper [2606.08941].

## 2. Collapsibility, minimality, and hull existence

The causal paper distinguishes three marginalization properties:
\[
I(G)_A = I(G_A),\qquad \mathcal P(G)_A = \mathcal P(G_A),\qquad \hat{\mathcal P}(G_A)=\hat{\mathcal P}(G)_A.
\]
These are, respectively, CI-collapsibility, model collapsibility, and estimate collapsibility. The paper states explicitly that estimate collapsibility implies model collapsibility, but not conversely [2606.08941].

The central characterization is Lemma 3.1. Under the assumption of non-triviality, for a CBN \(\mathcal B=(G,\mathcal P(G))\) and \(R\subseteq V\), the following are equivalent:

1. \(\hat P(G_R)=\hat P(G)_R\),
2. there exists a graph \(G'\) in the Markov equivalence class of \(G\) such that \(M=V\setminus R\) is a terminal set in \(G'\),
3. \(R\) is a strong d-convex set in \(G\) [2606.08941].

This equivalence turns strong d-convexity into the exact graphical criterion for estimate collapsibility. The hull notion then follows from an intersection property: if \(\{H_i\}_{i=1}^n\) is a collection of strong d-convex subsets containing \(R\), then
\[
H=\bigcap_{i=1}^n H_i
\]
is strongly d-convex. Consequently, both the d-convex hull \(ch(R)\) and the strong d-convex hull \(sch(R)\) exist and are unique.

The minimality theorem is the decisive closure result:
\[
\textbf{Theorem 3.}\quad H \text{ is the minimal collapsible set containing }R \text{ if and only if }G_H\text{ is the strong d-convex hull of }R.
\]
Here “minimal collapsible set” means that \(\hat P_{G_H}(x_H)=\hat P(x_H)\), while for every \(H'\) with \(R\subseteq H'\subsetneq H\), the equality fails. Thus the strong d-convex hull is not merely a convenient reduction; it is the exact minimal induced subgraph preserving the MLE-based estimation behavior of the full model [2606.08941].

## 3. Algorithmic construction in DAGs and CPDAGs

The algorithmic construction proceeds in two layers. First, one computes the d-convex hull by absorbing vertices forced by minimal inducing structures. Second, one enforces the linear-ordering condition by adding parent vertices of problematic nodes.

A key graph-theoretic reduction states that if \(r_1,r_2\in R\) are non-adjacent and \(G_H\) is d-convex or strongly d-convex, then the condition
\[
V^o(\mathrm{MIS}_G(l_{r_1r_2}))\subseteq H
\]
is equivalent to requiring that every shortest path \(l_{r_1r_2}\) in
\[
\bigl(G_{An_G(\{r_1,r_2\})}\bigr)^m
\]
satisfy
\[
V^o(l_{r_1r_2})\subseteq H.
\]
This justifies computing shortest paths in a moralized ancestor graph rather than enumerating arbitrary inducing paths [2606.08941].

The three algorithms used in the paper are summarized below.

| Algorithm | Role | Complexity |
|---|---|---|
| CVM(\(G,R\)) | Collect vertices on shortest inducing paths | \(O(k^2 |V|^2)\) time, \(O(|V|^2)\) space |
| ICHA(\(G,R\)) | Compute the d-convex hull by iterating CVM | \(O(k^2 |V|^3)\) time, \(O(|V|^2)\) space |
| ISCHA(\(G,R\)) | Upgrade the d-convex hull to the strong d-convex hull | \(O(k^2 |V|^4)\) time, \(O(|V|^2)\) space |

Here \(k=|R_1|\), with \(R_1=mb_G(M)\cap R\) and \(M=V\setminus R\).

CVM computes shortest paths in moralized ancestor graphs for non-adjacent pairs in \(R\) and returns the vertices on those paths. ICHA iterates CVM until closure. ISCHA alternates an ICHA step with the update
\[
PA \gets \left\{ pa_G(w)\ \middle|\ w\in Ch_G(V\setminus H)\cap An_G(H),\ w \text{ is not linearly ordered}\right\},
\]
then enlarges \(H\) by \(PA\) until no violation remains. The correctness theorem states that the set returned by ISCHA is a strong d-convex hull of \(R\) [2606.08941].

For CPDAGs, the construction is transferred through Markov equivalence rather than by a separate partially directed algorithm. Theorem 4.4 states that if \(G\) is a DAG Markov equivalent to a CPDAG \(\mathcal G\), then for distinct vertices \(X\) and \(Y\notin pa_{\mathcal G}(X)\), a vertex set \(R\supseteq \{X,Y\}\) is causal estimate collapsible in \(\mathcal G\) if and only if it is causal estimate collapsible in \(G\). Operationally, one may choose any consistent DAG \(G\in[\mathcal G]\), compute \(\mathrm{ISCHA}(G,\{X,Y\})\), and use the resulting set for graph reduction in the CPDAG [2606.08941].

## 4. Role in causal effect estimation

The causal use of strong d-convex hulls is not to produce an adjustment set directly, but to define a minimal retained subgraph in which adjustment-based estimation remains valid. For non-adjacent \(X\) and \(Y\notin pa_G(X)\), a DAG \(G\) is causal estimate collapsible onto \(R\) if for every non-empty valid back-door adjustment set \(Z\) in \(G_R\),
\[
\hat P(Y\mid do(X=x)) = \sum_z \hat P_{G_R}(Y\mid X=x,Z=z)\hat P_{G_R}(Z=z).
\]
The preserved estimand is therefore the post-intervention distribution \(P(Y\mid do(X=x))\), and for binary \(X\), the ACE is
\[
ACE = E(Y\mid do(X=1)) - E(Y\mid do(X=0)).
\]

The main validity statement is Theorem 4.1: if \(Z\) is a non-empty back-door adjustment set in \(G_R\), then \(G\) is causal estimate collapsible onto \(R\) if \(R\) is a strong d-convex hull of \(X\) and \(Y\) [2606.08941]. Thus the hull is a sufficient graphical certificate for reduction before causal estimation.

The paper treats the empty-adjustment case separately. If the empty set is a valid back-door adjustment set for \((X,Y)\) in \(G_R\), then the relevant hull must be computed in the manipulated graph \(G_{\bar X}\), obtained by deleting all edges into \(X\). This captures omitted parents of \(X\) that only become relevant after intervention.

The CPDAG implementation is “Subgraph IDA.” The procedure is: obtain a consistent DAG via Meek’s rules; compute
\[
R_{\text{init}} \leftarrow \mathrm{ISCHA}(G,\{X,Y\});
\]
for each DAG in the equivalence class, set \(R\leftarrow R_{\text{init}}\), let
\[
Z\leftarrow pa_{G_R}(X),
\]
and, if \(Z=\emptyset\), construct \(G_{\bar X}\), recompute \(R\leftarrow \mathrm{ISCHA}(G_{\bar X},\{X,Y\})\), and reset \(Z\leftarrow pa_{G_R}(X)\). Estimation then uses \(P(Y\mid X)\) when \(Z=\emptyset\), and otherwise the adjustment formula
\[
P(Y\mid do(X=x)) = \sum_z P(Y\mid X=x,Z=z)P(Z=z).
\]
The hull is therefore a preprocessing reduction layered on top of IDA, not a replacement for adjustment theory [2606.08941].

## 5. Worked examples, empirical behavior, and limitations

A representative example uses the DAG on \(\{A,B,C,D,X,Y\}\) with edges
\[
X\to D\to Y,\quad A\to X,\quad A\to B,\quad B\to X,\quad B\to Y,\quad C\to Y.
\]
For the target set \(R=\{X,Y\}\), the paper states that there are \(3\) inducing paths between \(X\) and \(Y\). A naive union of vertices on inducing structures gives \(\{X,A,B,D,Y\}\), which is d-convex but not minimal. The d-convex hull is \(\{X,B,D,Y\}\). The strong d-convex hull is larger, because \(Y\) violates the linear-ordering condition with
\[
pa_G(Y)=\{B,C,D\},
\]
so \(C\) must be added, yielding
\[
\{X,B,C,D,Y\}
\]
as the final strong d-convex hull [2606.08941].

The same paper gives a larger reduction example on the 56-node Hailfinder Bayesian network for the target pair \(\{\text{AreaMoDryAir},\ \text{MountainFcst}\}\). ISCHA reduces the graph from 56 variables to a 16-variable subgraph. The initial AIP is
\[
\{\text{AreaMoDryAir},\ \text{CldShadeOth},\ \text{InsInMt},\ \text{MountainFcst}\},
\]
this set is already d-convex, and further linear-ordering checks add variables such as \(\text{CombVerMo}\), \(\text{SubjVertMo}\), and \(\text{AreaMeso\_ALS}\) [2606.08941].

The empirical results reported are consistent with the theoretical interpretation of the hull as a minimal safe retained subgraph. In benchmark probabilistic reasoning experiments, KL divergence is essentially zero (\(\approx 10^{-18}\)), node reduction is up to \(96\%\), and local inference is faster. In random-graph CPDAG experiments, recall and precision are essentially \(1\) across nearly all settings, and speedups increase with graph size, reaching around \(18\times\) for 100-node random graphs. On real-world Bayesian networks including Sachs, Insurance, Alarm, Hepar2, Pathfinder, and Munin1, precision remains \(1.0\), recall remains high (\(\ge 0.86\)), and speedup reaches \(30.37\times\) on Munin1 [2606.08941].

The paper also states several scope restrictions. The framework currently applies to non-adjacent treatment-outcome pairs, does not handle latent variables, and focuses on back-door-type adjustment rather than other adjustment strategies. It also emphasizes that collapsibility may fail if the retained set is only d-convex but not strongly d-convex.

## 6. Relation to strongly convex hulls in geometry and to other nearby notions

A separate line of work studies strong convexity relative to a fixed convex body \(K\subset\mathbb R^d\). In that setting, a set is \(K\)-strongly convex if it is an intersection of translates of \(K\),
\[
C = K \stackrel{*}{-} T = \bigcap_{t\in T}(K-t),
\]
and the strongly convex hull is
\[
\operatorname{conv}_K X = K \stackrel{*}{-} (K \stackrel{*}{-} X).
\]
This hull is defined only for sets \(X\) contained in a translate of \(K\). When \(K\) is generating, the strong-convexity Carathéodory theorem gives a \(d+1\) bound, and colorful as well as very colorful Carathéodory theorems hold with \(d+1\) color classes. The same paper shows that in \(\mathbb R^3\), for arbitrary \(K\), the Carathéodory number can be arbitrarily large or infinite, so the generating-set assumption is not merely technical [1509.08783].

The random-geometry literature develops this \(K\)-hull viewpoint further. For a compact set \(A\subset\mathbb R^d\),
\[
\bh_K(A):=\bigcap_{x\in \mathbb R^d:\,A\subset K+x} (K+x),
\]
and a set \(Q\) is \(K\)-strongly convex if \(Q=\bh_K(Q)\). For i.i.d. uniform samples \(\Xi_n\subset K\), one studies
\[
Q_n=\bh_K(\Xi_n),\qquad X_n=K\ominus \Xi_n=\bigcap_{i=1}^n (K-\xi_i).
\]
The paper on facial structure proves that \(nX_n\) converges in distribution to the zero cell \(Z\) of a Poisson hyperplane tessellation, while \(n^{-1}X_n^o\) converges to \(Z^o\); under strict convexity and regularity, the generalized \(f\)-vector of \(Q_n\) converges in distribution, without normalization, to the ordinary \(f\)-vector of \(Z^o\), and all moments converge [2102.10009]. A later paper introduces \(m\)-point peelings and recursive convex hull peelings for families of compact convex sets, together with the dual wrapping operations for intersections, and applies them to \(K\)-hulls generated by random samples; under strict convexity and regularity of \(K\), the corresponding peelings and wrappings of the rescaled random objects converge in distribution to Poisson limits [2606.30617].

Several nearby literatures use similar language for different objects. The planar lattice-based discrete hull
\[
C^0 = \mathrm{ConvexHull}(C \cap \mathbb Z^2)
\]
is an ordinary Euclidean convex hull applied after filtering to lattice points, and the source explicitly states that it is not about strong \(d\)-convex hulls in the usual digital-convexity sense [2601.00392]. The finite-element literature proves a strong discrete convex hull property for discretely \(p\)-harmonic functions on acute triangulations, where an interior extreme point of \(\operatorname{conv\,hull}(U(\Omega))\) forces constancy [1302.0112]. The computational-geometry literature on \(\mathcal D\)-convex hulls defines planar directional hulls by excluding empty translated halfplanes and wedges induced by a direction set \(\mathcal D\), and the paper explicitly does not define a notion called strong \(d\)-convex hull [1111.5340].

Taken together, these sources delimit the current usage. In causal graphical models, the strong d-convex hull is the unique minimal induced subgraph guaranteeing estimate collapsibility and supporting reduced-graph causal estimation [2606.08941]. In convex geometry, strongly convex hulls are \(K\)-hulls generated by intersections of translates of a convex body, with their own Carathéodory theory, facial structure, and random-sample asymptotics [1509.08783], [2102.10009], [2606.30617]. The terminological overlap is substantial, but the operators, ambient spaces, and intended applications are not the same.

Source: https://www.emergentmind.com/topics/strong-d-convex-hulls