Title: DSCS: Fast CPDAG-Based Verification of Collapsible Submodels in High-Dimensional Bayesian Networks
Venue: NeurIPS 2025
Accepted: yes
Keywords: directed acyclic graph

Bayesian network

Markov equivalence class

estimation-collapsibility

completed partially directed acyclic graph
Abstract: Bayesian networks (BNs), represented by directed acyclic graphs (DAGs), provide a principled framework for modeling complex dependencies among random variables. As data dimensionality increases into the tens of thousands, fitting and marginalizing a full BN becomes computationally prohibitive—particularly when inference is only needed for a small subset of variables. Estimation-collapsibility addresses this challenge by ensuring that directly fitting a submodel, obtained by ignoring non-essential variables, still yields exact inference on target variables. However, current DAG-based criterion for checking estimation-collapsibility is computationally intensive, involving exhaustive vertex searches and iterative removals. Additionally, practical applications typically identify the underlying DAG only up to its Markov equivalence class, represented by a completed partially directed acyclic graph (CPDAG). To bridge this gap, we introduce sequential $c$-simplicial sets—a novel graphical characterization of estimation-collapsibility applicable directly to CPDAGs. We further propose DSCS, a computationally efficient algorithm for verifying estimation-collapsibility within CPDAG framework that scales effectively to high-dimensional BNs. Extensive numerical experiments demonstrate the practicality, scalability, and efficiency of our proposed approach.

DSCS: Fast CPDAG-Based Verification of Collapsible Submodels in High-Dimensional Bayesian Networks

Abstract

Bayesian networks (BNs), represented by directed acyclic graphs (DAGs), provide a principled framework for modeling complex dependencies among random variables. As data dimensionality increases into the tens of thousands, fitting and marginalizing a full BN becomes computationally prohibitive-particularly when inference is only needed for a small subset of variables. Estimation-collapsibility addresses this challenge by ensuring that directly fitting a submodel, obtained by ignoring non-essential variables, still yields exact inference on target variables. However, current DAG-based criterion for checking estimation-collapsibility is computationally intensive, involving exhaustive vertex searches and iterative removals. Additionally, practical applications typically identify the underlying DAG only up to its Markov equivalence class, represented by a completed partially directed acyclic graph (CPDAG). To bridge this gap, we introduce sequential c -simplicial sets-a novel graphical characterization of estimation-collapsibility directly applicable to CPDAGs. We further propose DSCS, a computationally efficient algorithm for verifying estimation-collapsibility within CPDAG framework that scales effectively to high-dimensional BNs. Extensive numerical experiments demonstrate the practicality, scalability, and efficiency of our proposed approach.

1 Introduction

Bayesian networks (BNs), represented by directed acyclic graphs (DAGs), offer a powerful and transparent framework to model complex dependency structures among random variables. By representing each variable as a vertex and each direct influence as a directed edge, BNs allow practitioners to read off causal or correlational relationships from their DAGs and to leverage powerful algorithms (such as message-passing) to compute marginals and posteriors efficiently. As a result, BNs have become indispensable tools across numerous domains, including econometrics [1], engineering [2-4], bioinformatics [5, 6], causal inference [7-9], and machine learning [10-12]. Recently, hybrid architectures that integrate DAG structure with deep neural networks have further extended BNs' expressive power while preserving their causal semantics [e.g. 13-15].

Despite these strengths, the explosive increase of data dimensionality has highlighted a critical bottleneck: constructing and marginalizing a full BN becomes infeasible when thousands-or even tens of thousands-of variables are involved, yet in many applications one cares about inference on only a small subset of variables. For instance, the Cancer Network Galaxy (TCNG) database hosts 768 networks encompassing over 20,000 genes and more than 16 million interactions. Typically, researchers only focus on disease-related or hub genes (genes with many interactions), which usually constitute a significantly smaller subset of the entire network. To draw inference and probability query for the target variables, it is both computationally expensive and wasteful to fit the full model and then marginalize to obtain a target submodel.

DSCS

(

)

sequential

-

simplicial  set

sequential

-

removable set

[28]

simplicial  set

[23,25]

Algorithm complexity

when

is a DAG

when

is a UG

Algorithm complexity

CPDAG

Algorithm complexity

Our work

In such scenarios, we would like to 'collapse' away irrelevant graph vertices (variables) and directly obtain an exact estimate of the submodel. This desideratum is captured by the notion of estimationcollapsibility , which ensures a submodel's direct estimation and inference aligns perfectly with that from the full-model, yielding drastic savings in data collection, computation, and robustness to unobserved variables [16-20]. However, neglecting proper validation of the submodel's collapsibility risks misleading or contradictory conclusions, exemplified by phenomena such as the Yule-Simpson paradox [21, 22]. Hence, identifying the precise conditions under which collapsibility holds is pivotal for trustworthy statistical inference with submodels.

Most existing works addressing collapsibility focused on undirected graphical models [e.g., 23-26], while the literature for Bayesian networks (or DAG models) remains notably limited [27, 28]. Despite its theoretical and practical significance, estimation-collapsibility in Bayesian networks remains as substantial challenges that are far from fully resolved. In particular, existing DAG-based criterion, sequential c -removability [28], can be computationally intensive to verify. It requires multiple laborious rounds of vertex examination, repeatedly identifying and removing c -removable vertices until none remain. The complexity can escalate up to O ( | V | 4 ) , making the algorithm impractical for a high-dimensional Bayesian network with a large cardinality | V | for its vertex set V .

Moreover, existing works [27, 28] assume the DAG structure is known, which is rarely true in practical scenarios. Typically, the underlying DAG must be inferred from observational data using causal discovery methods [29-31, 8, 32-36]. Lacking strong assumptions like linearity [33] or additive noise [31, 37], causal discovery process usually yields a Markov equivalence class, which is succinctly represented by a completed partially directed acyclic graph (CPDAG) [29, 38-40, 9, 4144]. Consequently, a pressing open problem emerges:

How can we efficiently verify estimation-collapsibility directly on the CPDAG for a high-dimensional Bayesian network?

To address this challenge, the current work presents the following novel contributions:

In Section 3.1, we introduce a novel concept, sequential c -simplicial sets , and find it as a necessary-and-sufficient criterion for estimation-collapsibility based on CPDAG.

As shown in Figure 1, our framework recovers and unifies prior collapsibility results for DAGs and undirected graphs (UGs). When a CPDAG G ∗ only has directed edges, a sequential c -simplicial set reduces to a sequential c -removable set of [28]. When G ∗ only contains undirected edges, our sequential c -simplicial criterion is equivalent to the simplicial criterion for undirected graphs [23, 25].

In Section 3.2, we develop the Detecting Sequential C -simplicial Sets (DSCS) algorithm, for rapid verification of estimation-collapsibility. It achieves a complexity order O ( | V | + | E ∗ | ) , where | V | and | E ∗ | denote the number of vertices and edges in a CPDAG G ∗ , respectively.

Extensive experiments have been conducted in Section 4 to demonstrate our method's practical effectiveness and computational efficiency.

Lastly, we emphasize that while our method addresses typical scenarios where the underlying DAG is unknown, the DSCS Algorithm also offers significant advantages when the DAG is explicitly available. Indeed, verifying estimation-collapsibility is substantially simpler with our CPDAG-based criterion. Given a known DAG ⃗ G , converting it into its CPDAG representation G ∗ and applying our DSCS Algorithm enables efficient verification at significantly reduced computational cost.

2 Preliminaries and Related Works

We begin by providing a review of graphical model concepts in Section 2.1. More terminologies and notations can be found in Appendix A. The notion of estimation-collapsibility for Bayesian networks is formally defined in Section 2.2. In Section 2.3, we review the existing criteria for checking estimation-collapsibility, as summarized in the right panel of Figure 1.

2.1 Graphical terminologies

A graph G = ( V, E ) is defined by a set V of nodes (or vertices), and a set E consisting of directed and (or) undirected edges. We use pa G ( A ) , ch G ( A ) , ne G ( A ) , an G ( A ) , de G ( A ) , and mb G ( A ) to denote the union of the parents, children, neighbors, ancestors, descendants, and the Markov boundary of each vertex in a set A ⊆ V in G , respectively, and with the set A excluded after taking the set union. Note that, among these, ne G ( A ) is defined with respect to undirected edges, while the other five notions are defined in terms of directed edges. Formal definitions can be found in Appendix A. If A needs to be included, we simply capitalize the first letter of the corresponding symbol, e.g., Pa G ( A ) ≜ pa G ( A ) ∪ A .

UGs, DAGs, CPDAGs. If G consists solely of undirected edges, it is referred to as an undirected graph (UG), denoted as G = ( V, E ) . When G is composed entirely of directed edges and contains no directed cycles, it is referred to as a directed acyclic graph (DAG), denoted as ⃗ G = ( V, ⃗ E ) . A DAG encodes a set of conditional independence relations based on the notion of d-separation [11]. Two DAGs are Markov equivalent if they encode the same set of conditional independence relations. A Markov equivalence class of a DAG ⃗ G can be uniquely represented by a completed partially directed acyclic graph (CPDAG) [45], denoted by G ∗ = ( V, E ∗ ) . The undirected components of G ∗ are undirected and connected chordal graphs (UCCGs) [46], also known as chain components of G ∗ [7, 46]. We use M ( G ∗ ) to denote the set of all Markov equivalent DAGs represented by the CPDAG G ∗ .

2.2 Bayesian network and estimation-collapsibility

For a DAG ⃗ G = ( V, ⃗ E ) , suppose X V is a random vector indexed by the graph vertices. For A ( ⊆ V ) , let X A be the sub-vector indexed by the subset A . A joint distribution P over X V is compatible with ⃗ G if it holds that P ( x V ) = ∏ v ∈ V P ( x v | x pa ⃗ G ( v ) ) for any x V . This means that P factorizes according to the parents of each vertex v in ⃗ G . The family of distributions compatible with ⃗ G is denoted as P ( ⃗ G ) and the pair B = ( ⃗ G, P ( ⃗ G )) is termed a Bayesian network (BN) or DAG model .

For a BN, estimation-collapsibility can be formally defined via the commutativity of model fitting and marginalization [47]. Let R ( ⊆ V ) be the subset of nodes of interests, and let M ≜ V \ R represent the non-irrelevant variables we want to marginalize out. Denote ˆ P ( x R ) as the estimate of P ( x R ) obtained by marginalizing the maximum likelihood estimate (MLE) ˆ P ( x ) under the full DAG model ( ⃗ G, P ( ⃗ G )) . Meanwhile, let ( ⃗ G R , P ( ⃗ G R )) be the induced model, where the graph ⃗ G R is obtained by removing the vertex set M and all edges incident to any vertex in M . We let ˆ P ⃗ G R ( x R ) be the MLE of the joint probability under the induced model ( ⃗ G R , P ( ⃗ G R )) . Estimation-collapsibility means that the MLE obtained from the original DAG model, after marginalization, equals the MLE from the induced DAG model.

Definition 1. [see 27, Definition 1] Suppose B = ( ⃗ G, P ( ⃗ G )) is a Bayesian network, P ( ⃗ G ) is estimation-collapsible over a single vertex v (or onto V \ { v } ) if for any x V \{ v } , it holds that

AFF

APA

SAN

ALN

AIS

CDR

DET

PER

SUS

HOS

FTW

EGC

(a)

AFF

APA

SAN

ALN

AIS

CDR

DET

PER

SUS

HOS

FTW

EGC

(b)

AFF

APA

SAN

ALN

AIS

CDR

PER

(c)

ˆ P ( x V \{ v } ) = ˆ P ⃗ G V \{ v } ( x V \{ v } ) . Estimation-collapsibility over a set M (or onto R ≜ V \ M ) holds if ˆ P ( x R ) = ˆ P ⃗ G R ( x R ) .

When this estimation-collapsibility property holds, we only need to work with the induced sub-model ( ⃗ G R , P ( ⃗ G R )) to directly obtain the estimate ˆ P ( x R ) . The obvious benefit is a substantial saving of data collection and computational efforts while maintaining exact estimate and inference for a high-dimensional Bayesian network.

2.3 Existing criteria for checking estimation-collapsibility

2.3.1 Criterion for DAG . Kim & Kim [27] discussed estimate-collapsibility for directed acyclic graph (DAG) models of contingency tables [48]. Xie & Geng [28] further investigated for discrete and continuous variable DAG models, and introduced the important concepts below.

Definition 2. [see 28, Definition 3] A vertex v is c -removable from DAG ⃗ G if any two vertices in Mb ⃗ G ( v ) ≜ { v } ∪ mb ⃗ G ( v ) are adjacent, except when both vertices belong to pa ⃗ G ( v ) . A set M is sequentially c -removable if all vertices in M can be ordered such that they can be c -removed by that ordering.

Furthermore, relying on the mild assumption of model non-triviality [see 28, Definition 4]-a condition satisfied by most DAG models, including those for contingency tables and Gaussian distributions [28] and implicitly used by [27]-they established the equivalence between sequential c -removability and estimation-collapsibility for DAG models.

Proposition 1. [see 28, Theorem 2] Suppose that B = ( ⃗ G, P ( ⃗ G )) is a non-trivial DAG model, then M is sequentially c -removable from ⃗ G if and only if P ( ⃗ G ) is estimate-collapsible over M .

The above theorem enables us to check estimation-collapsibility via examining the sequential c -removability. The next example illustrates this procedure.

Example 1. Consider the DAG in Figure 2(a). Suppose we are interested in finding whether the model is estimate collapsible over M = { AFF , APA , SAN } . Checking sequential c -removability requires multiple rounds of iterations. In the first round, we find that neither AFF nor SAN is c -removable from ⃗ G , but APA is c -removable. After removing APA , the second round is repeated by examining if any node in { AFF , SAN } is c -removable from ⃗ G V \{ APA } . Upon checking, we find that no vertex in { AFF , SAN } is further c -removable. The procedure stops by concluding that the model is not estimate collapsible over M .

The above procedure quickly becomes infeasible as the graph size grows. Given a subset M with cardinality | M | , it requires O ( | M | ) iterations of meticulous vertex search. In each round, one must

scan the remaining vertices in M to identify the c -removable ones and then remove them. In the worst case-when the graph is dense and | M | is on the same order as | V | -the first iteration alone costs O ( | V | 3 ) , the next O (( | V | -1) 3 ) , and so on, leading to a total complexity on the order of O ( | V | 4 ) .

2.3.2 Criterion for UG . As our criterion also recover those for undirected graph (UG), we conclude this section by briefly reviewing some important results.

Definition 3. [see 50] In an undirected graph (UG) G = ( V, E ) , for a vertex v ∈ V , if the neighborhood ne G ( v ) induces a complete subgraph in G , then v is called a simplicial vertex in G . For a subset M ⊆ V , if the neighborhood ne G ( M ) induces a complete subgraph in G , then M is called a simplicial (vertex) set in G .

For multinomial and Gaussian undirected graphical models, the collapsibility over M is equivalent to the condition that each connected component of M is a simplicial set [23, 25]. The complexity of checking this simplicial property is O ( | V | + | E | ) .

3 Fast CPDAG-Based Verification of Estimation-Collapsibility

We now address the main query of the current work: how can we efficiently check estimationcollapsibility for high-dimensional Bayesian Networks using CPDAGs? In Section 3.1, we will establish an equivalent characterization of estimation-collapsibility from the perspective of CPDAGs. In Section 3.2, we develop an efficient and scalable algorithm to check the estimation-collapsibility.

3.1 Sequential c -simplicial set

For a Bayesian network (BN), checking estimation-collapsibility of the underlying DAG via the corresponding CPDAG G ∗ is possible, as this property is consistent across all DAGs within the equivalence class M ( G ∗ ) . In fact, for two Markov-equivalent DAGs ⃗ G 1 and ⃗ G 2 in M ( G ∗ ) , they encode the same set of conditional independence relations, ensuring that the families of distributions they encode are identical, i.e., P ( ⃗ G 1 ) = P ( ⃗ G 2 ) . Similarly, for Markov-equivalent subgraphs, the corresponding families of distributions are also identical. Therefore, when estimation-collapsibility holds for the underlying DAG, the property is preserved across all its Markov equivalent DAGs.

Building on this consistency, we re-state estimation-collapsibility (see Definition 1) from the perspective of CPDAG. The definition below formalizes this extension based on the family of distributions compatible with a CPDAG. For a CPDAG G ∗ = ( V, E ∗ ) , a joint distribution P over X V is said to be compatible with G ∗ if it is compatible with a DAG ⃗ G in M ( G ∗ ) . The set of all such distributions is denoted as P ( G ∗ ) .

Definition 4. For a CPDAG G ∗ = ( V, E ∗ ) and some M ⊆ V , P ( G ∗ ) is estimation-collapsible over the set M if for every DAG ⃗ G ∈ M ( G ∗ ) , P ( ⃗ G )(= P ( G ∗ )) is estimation-collapsible over the set M .

It is evident that, to examine the estimation-collapsibility for the underlying DAG model, we only need to check the estimation-collapsibility from CPDAG. Analogous to sequential c -removability in DAGs, we introduce the concept of a sequential c -simplicial (vertex) set for CPDAGs, which proves to be a key for addressing the problem.

Definition 5. Given a CPDAG G ∗ = ( V, E ∗ ) , for a vertex v ∈ V , if ch G ∗ ( v ) = ∅ and ne G ∗ ( v ) is complete, then v is called a c -simplicial vertex in G ∗ . Similarly, for M ⊆ V , if every vertex in M can be identified as a c -simplicial vertex during successive removals according to some vertex order, then M is called a sequential c -simplicial (vertex) set in G ∗ .

Remark 1. The set ch G ∗ ( v ) consists of nodes pointed to by directed edges from node v , i.e., ch G ∗ ( v ) = { w ∈ V \ { v } : v → w in G ∗ } , while ne G ∗ ( v ) refers to the set of nodes adjacent to v via undirected edges, i.e., ne G ∗ ( v ) = { w ∈ V \ { v } : v -w in G ∗ } .

Example 2. Consider the CPDAG in Figure 2(b). Since ch G ∗ ( APA ) = ∅ and its neighbor set ne G ∗ ( APA ) = { AFF, ALN, SAN } is complete, the node APA is a c -simplicial vertex in G ∗ . Similarly, HOS is also a c -simplicial vertex in G ∗ . The subset of nodes M ≜ { AIS, SUS, FTW, DET, HOS, EGC } forms a sequential c -simplicial vertex set in G ∗ , as M can be identified as a c -simplicial vertex during successive removals according to the vertex order DET, HOS, EGC, FTW, SUS, AIS .

The concept of a c-simplicial vertex in a CPDAG generalizes the notion of a leaf node (i.e., a node with an empty set of children) in DAGs and a simplicial vertex (i.e., a node whose set of neighbors is complete) [50] in undirected graphs. In particular, when G ∗ is a DAG, v is a c-simplicial vertex if and only if v is a leaf node . This is because when G ∗ is a DAG, it naturally holds that ne G ∗ ( v ) = ∅ , making the completeness of its neighbor trivially satisfied, and the only condition required is ch G ∗ ( v ) = ∅ . Meanwhile, when G ∗ is an undirected graph, v being a c -simplicial vertex in the CPDAG is equivalent to v being a simplicial vertex in the undirected graph G ∗ . In this case, ch G ∗ ( v ) = ∅ always holds, and the sole requirement is that ne G ∗ ( v ) forms a complete subgraph. The " c " in c -simplicial stands for compound , as the c -simplicial property (CPDAG) combines the features of a simplicial set (undirected graph) and a leaf set (DAG).

The notion of sequential c -simplicial set likewise generalizes sequential c -removable set for DAG. When G ∗ is a DAG and M is a sequential c -simplicial set, it directly follows from the previous discussion that M is also sequentially c -removable.

Furthermore, we can find the concept of sequential c -simplicial set generalizes that of simplicial set for undirected graph. When G ∗ contains only undirected edges (i.e. G ∗ is a chordal graph), recall a c -simplicial vertex reduces to a simplicial vertex. This means that, for a sequential c -simplicial M , its vertices can be ordered such that each vertex is a simplicial vertex at the time of its removal. Consequently, in a chordal graph G , we also term a sequential c -simplicial set M as a sequential simplicial set . Lemma 1 indicates that sequential simplicity (or sequential c -simplicity) is equivalent to simplicity for chordal graph.

Lemma1. For a chordal graph G = ( V, E ) , given a connected subset M ⊆ V , M forms a sequential simplicial set in G if and only if M is a simplicial set in G .

Remark 2. In a chordal graph G , if M is disconnected, its connected components can be analyzed independently. Then, M is a sequential simplicial set in G if and only if each connected component of M is a simplicial set in G .

Remark 3. Lemma 1 connects the classical concept of simplicial set [50] and our concept of sequential simplicial set for a chordal graph. The correctness of Lemma 1 follows directly from Proposition B.1 in Appendix B. For general undirected graphs, the relationship between the two concepts is also discussed in detail in Appendix B. Table C.1 in Appendix C consolidates the key concepts of simplicial, c -removable, and c -simplicial vertices and sets across UGs, DAGs, and CPDAGs to facilitate comparison and reference.

Example 3. Consider Figure 2(c), which we denote as G ∗ τ max . Clearly, G ∗ τ max is a chordal graph. Now, consider the subset M ≜ { AIS, CDR, SAN } . M is a simplicial set in G ∗ τ max because ne G ∗ τmax ( M ) = { AFF, APA, ALN } is complete. Additionally, M forms a sequential simplicial vertex set in G ∗ τ max , as M can be identified as a simplicial vertex during successive removals according to the vertex order CDR, AIS, SAN .

The preceding discussion underscores the profound connection between estimation-collapsibility and our notion of sequential c -simplicity in two classical situations. When the CPDAG is a DAG, if M is sequential c -simplicial (hence sequential c -removable), then P ( G ∗ ) is estimate-collapsible over the set M under the non-trivial model assumption [28, Definition 4]. Meanwhile, when the CPDAG is a undirected chordal graph, if M is sequential c -simplicial (hence each componnent is simplicial), then the estimate-collapsiblility also holds in multinomial and Gaussian undirected graphical models [23, 25]. These relations are summarized in Figure 1 and lead us to the question: beyond these two special cases, can we establish a connection between estimation-collapsibility and sequential c -simplicity for a CPDAG in general? Theorem 1 below provides an affirmative answer.

Theorem 1. Given a CPDAG G ∗ = ( V, E ∗ ) , let M ( G ∗ ) denote the set of all Markov equivalent DAGs represented by the CPDAG G ∗ . For M ⊆ V , the following four statements are equivalent: ∗

(1) M is a sequential c -simplicial set in G ;

(2) there exists a DAG ⃗ G ∈ M ( G ∗ ) such that ch ⃗ G ( M ) = ∅ ;

(3) for all DAGs ⃗ G ∈ M ( G ∗ ) , M is sequentially c -removable from ⃗ G ;

(4) under the non-trivial model assumption, P ( G ∗ ) is estimate-collapsible over the set M .

Theorem 1 establishes the connection between estimation-collapsibility and sequential c -simplicial sets. It clearly shows that our notion of a c -simplicial set naturally extends to the broader framework of CPDAG models, including DAGs and undirected chordal graphs as special cases. Specifically,

under the non-trivial model assumption [28, Definition 4], it can be employed to verify estimationcollapsibility for a high-dimensional Bayesian network. The proof of Theorem 1 is in Appendix D.

3.2 DSCS: fast CPDAG-based verification of estimation-collapsibility

Although Theorem 1 establishes CPDAG-based criterion for checking estimation-collapsibility, directly determining if a subset M is a sequential c -simplicial remains challenging. This is because multiple iterations are still needed to search for the c -simplicial vertices, which is a daunting task. Fortunately, for a CPDAG, we can find an equivalent characterization of sequential c -simplicity, which does not involve vertex order and eliminates the requirement of multiple algorithm iterations.

Remark 4. To highlight the essential concepts and intuitions, the following discussion focuses on the case where M is connected. For a disconnected M , the conclusion can be easily generalized as each connected component can be considered separately. In the Appendix, the theoretical derivations are developed for the general case where M has several connected components.

First, we establish a necessary condition for M to be a sequential c -simplicial set.

Lemma 2. Suppose G ∗ = ( V, E ∗ ) is a CPDAG. For any M ⊆ V , if M is a sequential c -simplicial set in G ∗ , then it holds that ch G ∗ ( M ) = ∅ .

̸

If ch G ∗ ( M ) = ∅ , then by Lemma 2, it follows directly that M is not a sequential c -simplicial set in G ∗ , thereby avoiding further effort for verification. However, the condition ch G ∗ ( M ) = ∅ alone does is not sufficient to ensure that M is a sequential c -simplicial set. Consider the CPDAG in Figure 2(b) for example, and let M = { AFF, SAN } . Although ch G ∗ ( M ) = ∅ , neither AFF nor SAN is a c -simplicial vertex in G ∗ . It follows that M is not a sequential c -simplicial set in G ∗ .

In addition to requiring ch G ∗ ( M ) = ∅ , additional conditions must be imposed for sequential c -simplicity to hold. We start by examining a special case where M is contained within a single chain component, and provide an equivalent characterization of its sequential c -simplicial property.

Lemma 3. Suppose G ∗ = ( V, E ∗ ) is a CPDAG with its chain components denoted by T = { τ 1 , τ 2 , . . . , τ K } . For a connected subset M ⊆ V , if there exists i ∈ [ K ] such that M ⊆ τ i , then the following two statements are equivalent:

(1) M is a sequential c -simplicial set in G ∗ ;

(2) ch G ∗ ( M ) = ∅ , and M is a simplicial set in G ∗ τ i , meaning that ne G ∗ τ i ( M ) forms a complete subgraph.

Remark 5. For a disconnected M , it is sufficient to impose the constraints separately on each of its connected components. In this case, statement (2) in Lemma 3 can be easily reformulated as follows:

(2 ′ ) ch G ∗ ( M ) = ∅ , and each connected component of M is a simplicial set in the subgraph G ∗ τ i .

We continue to consider the general case where a connected M is contained within multiple chain components of G ∗ . Let T = { τ 1 , τ 2 , . . . , τ K } be the set of chain components of G ∗ , and define α as a mapping from T to { 1 , 2 , . . . , K } such that if there is a directed edge from τ i to τ j , then α ( τ i ) < α ( τ j ) . This mapping α is called a topological ordering of the chain components in G ∗ . Without loss of generality, assume α ( τ 1 ) < α ( τ 2 ) < · · · < α ( τ K ) . For M to be a sequential c -simplicial set in G ∗ , in addition to requiring ch G ∗ ( M ) = ∅ , it suffices that each M i ≜ M ∩ τ i is a sequential simplicial set in G ∗ τ i for all i ∈ [ K ] . By Lemma 1, this is equivalent to requiring each connected M i in G ∗ τ i forms a simplicial set. This is because we can always remove the sets in the reverse order M K , M K -1 , · · · , M 1 . The specific conclusion is given by Theorem 2.

Theorem 2. Given a CPDAG G ∗ = ( V, E ∗ ) with its chain component set T = { τ 1 , τ 2 , . . . , τ K } , for connected M ⊆ V , let M i ≜ M ∩ τ i for each i ∈ [ K ] . The following two statements are equivalent: ∗

(1) M is a sequential c -simplicial set in G ;

(2) ch G ∗ ( M ) = ∅ , and M i in G ∗ τ i being a simplicial set for all i ∈ [ K ] .

Remark 6. For a disconnected M , statement (2) in Theorem 2 naturally extends to:

( 2 ′ ) ch G ∗ ( M ) = ∅ , and each connected component of M i in G ∗ τ i being a simplicial set for all i ∈ [ K ] .

Algorithm 1: DSCS Algorithm

̸

̸

The significance of Theorem 2 is that it provides an order-free way to check sequential c -simplicity. Recall from Section 3.1 that sequential c -simplicity for CPDAG implies estimation-collapsibility for the underlying DAG model. Therefore, Theorem 2 provides a convenient avenue to determine the estimation-collapsibility. The proof of Theorem 2 can be found in Appendix D.

Building on Theorem 2, we introduce the D etecting S equential C -simplicial S et ( DSCS ) algorithm in Algorithm 1. The algorithm proceeds in two steps. In the first step (Lines 1-2), we check whether ch G ∗ ( M ) is empty; if not, we can immediately conclude that M is not a sequential c -simplicial set in G ∗ . If ch G ∗ ( M ) is empty, we proceed to the second step (Lines 3-19). For each chain component τ i and each connected component M l j of M within τ i , we examine whether its neighboring nodes of M l j form a complete subgraph. Note that these checks are independent across chain components in parallel. The next example illustrates the execution of the algorithm.

Example 4. Let us re-examine Example 1 and recall we are interested in checking estimatecollapsibility with M = { AFF , APA , SAN } . Our DSCS algorithm works with the CPDAG in Figure 2(b). The CPDAG has chain components τ 1 = { AFF , APA , ALN , SAN , PER , CDR , AIS } , τ 2 = { SUS } , τ 3 = { FTW } , τ 4 = { EGC } , τ 5 = { HOS } , and τ 6 = { DET } . DSCS starts by checking if ch G ∗ ( M ) = ∅ , which is clearly true. Next, it checks whether each connected component of M ∩ τ i in G ∗ τ i for i = 1 , 2 , · · · , 6 is a simplicial set. From Figure 2(b), it is evident that M ⊂ τ 1 , so only the connected components of M in G ∗ τ 1 need to be checked for being simplicial. Since M is connected in G ∗ τ 1 , and ne G ∗ τ 1 ( M ) = { ALN , AIS , CDR } is not complete, we conclude the underlying DAG model is not estimate-collapsible over M .

The complexity of DSCS Algorithm is O ( | V | + | E ∗ | ) , where | V | and | E ∗ | denote the number of vertices and edges in G ∗ , respectively. While operations such as identifying the children subset ch G ∗ ( M ) , detecting connected components in M , and analyzing their neighborhoods have relatively low complexity, the chain component decomposition of G ∗ , requiring O ( | V | + | E ∗ | ) , constitutes the primary bottleneck. Following reviewer feedback, we can use the maximum cardinality search (MCS) algorithm [51-53] - which identifies all maximal cliques in a chordal graph and finds a perfect elimination ordering in linear time - to perform parallel verification of whether each neighborhood forms a clique across all chain components. This is done by checking whether the neighborhood is a subset of the maximal clique associated with the node (within that neighborhood) that appears earliest in the elimination ordering. Consequently, the complexity of Line 8 in the DSCS algorithm is O ( | τ max | + | E ∗ τ max | ) , where | τ max | is the number of vertices in the largest chain component τ max, and | E ∗ τ max | is the number of edges in this component. Thus, the overall complexity of the DSCS algorithm is O ( | V | + | E ∗ | ) .

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4 Simulation

Through numerical experiments, we examine the performance of the proposed DSCS Algorithm in Algorithm 1. Our algorithm is compared to the approach of directly detecting sequential c -removability of M , as proposed by [28]. The latter approach is abbreviated as DCR . The experiments were implemented with R and run on a computer with 2.20GHz CPU and 256 GB memory.

In the experiments, Erdös-Rényi graphs were randomly generated using the R-package pcalg [54], and the corresponding CPDAGs were obtained from the generated DAGs. We use n = | V | to denote the number of graph vertices and p to represent the edge creation probability. In each simulation replicate, we randomly generate a DAG with n ∈ { 500 , 1000 , 1500 , 2000 , 4000 , 6000 , 8000 , 10000 } and p ∈ { 0 . 1 , 0 . 05 , 0 . 01 , 0 . 005 } , and the vertex subset M was randomly selected from the graph with cardinality | M | . DCR (or DSCS) is then applied to the genrated DAGs (or corresponding CPDAGs) to check if the models are estimate-collapsible over M . For each combination of n , p , and | M | , the simulation was run 30 times. The average running time (in seconds) and the standard error were then calculated. Note that the proposed DSCS Algorithm can be run for each chain component of a CPDAG in parallel. In this experiment, only serial execution is implemented. Due to the extremely long computational time of the DCR Algorithm for large graphs, we only compared the average running time of DSCS with DCR for n ∈ { 500 , 1000 , 1500 } . When n ∈ { 2000 , 4000 , 6000 , 8000 , 10000 } , the running times for DSCS are reported in Appendix E. Moreover, to further highlight the advantages of the DSCS algorithm, we selected three real-world Bayesian networks from the R-package bnlearn -WIN95PTS (76 nodes, 112 edges), LINK (724 nodes, 1125 edges), and MUNIN (1041 nodes, 1397 edges). A detailed description of the three Bayesian networks can be found at https://www.bnlearn.com/bnrepository/ . For each network, we randomly sampled a vertex subset M of size | M | , then applied DCR (on the DAGs) and DSCS (on the corresponding CPDAGs) to test collapsibility over M . The running times are also reported in Appendix E.

For n ∈ { 500 , 1000 , 1500 } , the average running times of the two algorithms are shown in Figure 3. For different values of p , the corresponding curves are distinguished by color and point shape. Detailed results, including average running times and standard errors, are also provided in Table E.1 in Appendix E. In Figure 3, across various graph sizes n ∈ { 500 , 1000 , 1500 } and different values of p and | M | , DSCS consistently outperforms DCR in terms of runtime. For a fixed n , the average running time of the DCR Algorithm increases rapidly with p and | M | , while the DSCS Algorithm

exhibits much slower growth, demonstrating clear computational advantages. As n , p , and | M | increase, the advantage of the DSCS Algorithm becomes more pronounced. For example, when n = 500 , p = 0 . 1 , and | M | = 450 , the proposed DSCS is more than 530 times faster than DCR. When n = 1500 , p = 0 . 1 , and | M | = 1350 , DSCS achieves a speedup of more than 4000 times.

5 Conclusion

In this work, we have tackled a fundamental bottleneck in high-dimensional Bayesian network inference: how to efficiently verify a submodel's estimation-collapsibility and reliably collapse away irrelevant variables. Departing from the traditional DAG framework [27, 28], we show that one can directly work on a CPDAG learned from observational data. We introduce the concept of sequential c -simplicity and find it as a necessary and sufficient criterion to check estimation-collapsibility. This criterion generalizes and unifies previous characterizations in both DAG models [28] and undirected chordal graph models [23, 25]. Remarkably, based on CPDAG, we are able to develop an orderfree characterization (i.e. independent of the vertex permutation) and design an efficient Detecting Sequential C -simplicial Set (DSCS) algorithm. Beyond its theoretical value and computational efficiency, DSCS has profound practical consequences: it enables researchers to focus data-collection and computational effort on small subsets of interest to accelerate scientific discovery. We believe that incorporating collapsibility checks into researchers' preprocessing toolkit will significantly aid scalable probabilistic modeling, paving the way for robust and efficient inference in ever-expanding data regimes.

A limitation of this work is that we require the CPDAG to be accurately and fully learned from observational data. One could further relax the requirement of a fully accurate CPDAG by developing incremental or local CPDAG-learning strategies that suffice for collapsibility checks. Another direction is to extend DSCS to accommodate interventional data or dynamic Bayesian networks, broadening its applicability to sequential and adaptive decision-making.

Acknowledgments and Disclosure of Funding

We thank the anonymous reviewers and area chair for their insightful comments, which helped strengthen this paper. This work was partially supported by the Key Program of the National Natural Science Foundation of China (Grant No.12431009) and the National Key Research and Development Program of China (Grant No.2020YFA0714100, Grant No.2020YFA0714102). The first two authors contribute equally. Shiyuan He was partially supported by the National Natural Science Foundation of China (Grant Numbers 12571278).

References

Norman Fenton and Martin Neil. Risk Assessment and Decision Analysis with Bayesian Networks . CRC Press, 2018.

Baoping Cai, Yonghong Liu, Jinqiu Hu, Zengkai Liu, Shengnan Wu, and Renjie Ji. Bayesian Networks in Fault Diagnosis: Practice and Application . World Scientific, 2018.

Baoping Cai, Yonghong Liu, Zengkai Liu, Yuanjiang Chang, and Lei Jiang. Bayesian Networks for Reliability Engineering . Springer, 2020.

Olivier Pourret, Patrick Na, Bruce Marcot, et al. Bayesian Networks: A Practical Guide to Applications . John Wiley & Sons, 2008.

Steffen L Lauritzen. Some modern applications of graphical models. In Peter J Green, Nils Lid Hjort, and Sylvia Richardson, editors, Highly Structured Stochastic Systems , Oxford Statistical Science Series, pages 13-32. Oxford University Press, 2003.

Marloes Maathuis, Mathias Drton, Steffen Lauritzen, and Martin Wainwright. Handbook of Graphical Models . CRC Press, 2018.

Steffen L Lauritzen and Thomas S Richardson. Chain graph models and their causal interpretations. Journal of the Royal Statistical Society: Series B (Methodological) , 64(3):321-348, 2002.

Judea Pearl. Causal inference in statistics: An overview. Statistics Surveys , 3:96-146, 2009.

Peter Spirtes, Clark Glymour, and Richard Scheines. Causation, Prediction, and Search . MIT Press, 2001.

Zongming Ma, Xianchao Xie, and Zhi Geng. Structural learning of chain graphs via decomposition. Journal of Machine Learning Research , 9(Dec):2847-2880, 2008.

Judea Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference . Elsevier, 2014.

Xianchao Xie, Zhi Geng, and Qiang Zhao. Decomposition of structural learning about directed acyclic graphs. Artificial Intelligence , 170(4-5):422-439, 2006.

Sourabh Balgi, Adel Daoud, Jose M Pena, Geoffrey T Wodtke, and Jesse Zhou. Deep learning with DAGs. Sociological Methods & Research , page 00491241251319291, 2024.

Adrián Javaloy, Pablo Sánchez-Martín, and Isabel Valera. Causal normalizing flows: from theory to practice. Advances in Neural Information Processing Systems , 36:58833-58864, 2023.

Veronika Thost and Jie Chen. Directed acyclic graph neural networks. In International Conference on Learning Representations , 2021.

Peter Bühlmann and Sara van de Geer. Statistics for High-Dimensional Data: Methods, Theory and Applications . Springer Science & Business Media, 2011.

Robert G Cowell, Philip Dawid, Steffen L Lauritzen, and David J Spiegelhalter. Probabilistic Networks and Expert Systems: Exact Computational Methods for Bayesian Networks . Springer Science & Business Media, 2007.

Arthur P Dempster, Nan M Laird, and Donald B Rubin. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society: Series B (Methodological) , 39(1):1-22, 1977.

David Heckerman. A tutorial on learning with Bayesian networks. In Michael I. Jordan, editor, Learning in Graphical Models , pages 301-354. Springer, 1998.

Pradeep Ravikumar, Martin J Wainwright, and John D Lafferty. High-dimensional Ising model selection using ℓ 1-regularized logistic regression. The Annals of Statistics , 38(1):1287-1319, 2010.

E.H. Simpson. The interpretation of interaction in contingency tables. Journal of the Royal Statistical Society: Series B (Methodological) , 13(2):238-241, 1951.

G.U. Yule. Notes on the theory of association of attributes in statistics. Biometrika , 2(2): 121-134, 1903.

Søren Asmussen and David Edwards. Collapsibility and response variables in contingency tables. Biometrika , 70(3):567-578, 1983.

David Madigan and Krzysztof Mosurski. An extension of the results of asmussen and edwards on collapsibility in contingency tables. Biometrika , 77(2):315-319, 1990.

Morten Frydenberg. Marginalization and collapsibility in graphical interaction models. The Annals of Statistics , pages 790-805, 1990.

Vanessa Didelez and David Edwards. Collapsibility of graphical CG-regression models. Scandinavian journal of statistics , 31(4):535-551, 2004.

Sung-ho Kim and Seong-Ho Kim. A note on collapsibility in DAG models of contingency tables. Scandinavian Journal of Statistics , 33(3):575-590, 2006.

Xianchao Xie and Zhi Geng. Collapsibility for directed acyclic graphs. Scandinavian Journal of Statistics , 36(2):185-203, 2009.

David Maxwell Chickering. Optimal structure identification with greedy search. Journal of Machine Learning Research , 3(Nov):507-554, 2002.

Diego Colombo, Marloes H Maathuis, et al. Order-independent constraint-based causal structure learning. Journal of Machine Learning Research , 15(1):3741-3782, 2014.

Patrik Hoyer, Dominik Janzing, Joris M Mooij, Jonas Peters, and Bernhard Schölkopf. Nonlinear causal discovery with additive noise models. Advances in Neural Information Processing Systems , 21, 2008.

Jonas Peters and Peter Bühlmann. Identifiability of Gaussian structural equation models with equal error variances. Biometrika , 101(1):219-228, 2014.

Shohei Shimizu, Patrik O Hoyer, Aapo Hyvärinen, Antti Kerminen, and Michael Jordan. A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research , 7(10), 2006.

Peter Spirtes and Clark Glymour. An algorithm for fast recovery of sparse causal graphs. Social Science Computer Review , 9(1):62-72, 1991.

Peter Spirtes, Clark Glymour, Richard Scheines, Stuart Kauffman, Valerio Aimale, and Frank Wimberly. Constructing Bayesian network models of gene expression networks from microarray data. 2000.

Kun Zhang and Aapo Hyvärinen. On the identifiability of the post-nonlinear causal model. In Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence , pages 647-655, 2009.

Jonas Peters, Joris M Mooij, Dominik Janzing, and Bernhard Schölkopf. Causal discovery with continuous additive noise models. Journal of Machine Learning Research , 15(1):2009-2053, 2014.

Tom Claassen and Tom Heskes. A logical characterization of constraint-based causal discovery. In Proceedings of the Twenty-Seventh Conference on Uncertainty in Artificial Intelligence , pages 135-144, 2011.

Tom Claassen, Joris M Mooij, and Tom Heskes. Learning sparse causal models is not NP-hard. In Proceedings of the Twenty-Ninth Conference on Uncertainty in Artificial Intelligence , pages 172-181, 2013.

David Heckerman, Dan Geiger, and David M Chickering. Learning Bayesian networks: The combination of knowledge and statistical data. Machine Learning , 20(3):197-243, 1995.

Thomas Verma and Judea Pearl. Equivalence and synthesis of causal models. In Proceedings of the Sixth Annual Conference on Uncertainty in Artificial Intelligence , pages 255-270, 1990.

Thomas Verma and Judea Pearl. An algorithm for deciding if a set of observed independencies has a causal explanation. In Uncertainty in Artificial Intelligence , pages 323-330. Elsevier, 1992.

Xun Zheng, Bryon Aragam, Pradeep K Ravikumar, and Eric P Xing. DAGs with no tears: Continuous optimization for structure learning. Advances in Neural Information Processing Systems , 31, 2018.

Shengyu Zhu, Ignavier Ng, and Zhitang Chen. Causal discovery with reinforcement learning. arXiv preprint arXiv:1906.04477 , 2019.

Christopher Meek. Causal inference and causal explanation with background knowledge. In Proceedings of the Eleventh Conference on Uncertainty in Artificial Intelligence , pages 403-410, 1995.

Steen A. Andersson, David Madigan, and Michael D. Perlman. A characterization of Markov equivalence classes for acyclic digraphs. The Annals of Statistics , 25(2):505-541, 1997.

Joe Whittaker. Graphical Models in Applied Multivariate Statistics . Wiley Publishing, 2009.

Steffen L Lauritzen. Graphical Models , volume 17. Clarendon Press, 1996.

D. van Kampen. The SSQ model of schizophrenic prodromal unfolding revised: An analysis of its causal chains based on the language of directed graphs. European Psychiatry , 29(7): 437-448, 2014.

Gabriel Andrew Dirac. On rigid circuit graphs. In Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg , pages 71-76. Springer, 1961.

F˘ anic˘ a Gavril. Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph. SIAM Journal on Computing , 1(2): 180-187, 1972.

Donald J Rose, R Endre Tarjan, and George S Lueker. Algorithmic aspects of vertex elimination on graphs. SIAM Journal on computing , 5(2):266-283, 1976.

Robert E Tarjan and Mihalis Yannakakis. Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM Journal on computing , 13(3):566-579, 1984.

Markus Kalisch, Martin Mächler, Diego Colombo, Marloes H Maathuis, and Peter Bühlmann. Causal inference using graphical models with the R package pcalg. Journal of Statistical Software , 47:1-26, 2012.

Judea Pearl, Dan Geiger, and Thomas Verma. Conditional independence and its representations. Kybernetika , 25(7):33-44, 1989.

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A Graph terminology

A graph G = ( V, E ) is defined by a set V of nodes (or vertices), and a set E consisting of both directed and undirected edges. There is at most one edge between any pair of nodes u and v . We use ( u, v ) or u → v to denote a directed edge from u to v , and u -v to represent an undirected edge connecting u and v . The nodes u and v are adjacent if there is an edge connecting them, which is denoted by u G ∼ v . Otherwise, u, v are nonadjacent and denoted by u G ≁ v . A path connecting u and v in G , denoted by ℓ uv , is a sequence of distinct vertices ⟨ c 0 = u, c 1 , . . . , c m -1 , c m = v ⟩ such that c i G ∼ c i +1 for i = 0 , 1 , . . . , m -1 . The path ℓ uv is a directed path , denoted by ℓ ⃗ uv , from u to v if c i → c i +1 for all i = 0 , 1 , . . . , m -1 , and it is an undirected path if c i -c i +1 for all i = 0 , 1 , . . . , m -1 . Additionally, the path ℓ uv is a partially directed path from u to v if there are no edges of the form c i ← c i +1 for all i = 0 , 1 , . . . , m -1 . A cycle is a path from a vertex to itself. An undirected path from u to v , along with the edge v -u , forms an undirected cycle . A directed path from u to v , along with the edge v → u , forms a directed cycle . A partially directed cycle is formed by a partially directed path from u to v , together with the edge v → u or v -u . We note that both directed paths (cycles) and undirected paths (cycles) are partially directed paths (cycles).

When u → v , u is referred to as the parent node of v , and v is called the child node of u . If u -v , u is a neighbor node of v . For some u ∈ V , let pa G ( u ) denote the set of parent nodes of u , ch G ( u ) the set of child nodes of u , and ne G ( u ) the set of neighbor nodes of u . Specifically, these sets are defined as follows:

If ch G ( u ) = ∅ , then u is called a leaf node in G . For a subset A ⊆ V , the sets of its parents and children in G are, respectively, defined as

If ch G ( A ) = ∅ , then A is called a leaf set in G . We define the Markov boundary of u and A as mb G ( u ) = ch G ( u ) ∪ pa G ( u ) ∪ pa G ( ch G ( u )) \ { u } and mb G ( A ) = ⋃ u ∈ A mb G ( u ) \ A .

For simplicity, we also define

If there exists a directed path from u to v in G , u is referred to as an ancestor of v , and v is called a descendant of u . The set of ancestors of u in G is denoted as an G ( u ) , and the set of descendants of u in G is denoted as de G ( u ) . Specifically, these sets are defined as follows:

For a subset A ⊆ V , the sets of its ancestors and descendants in G are defined as follows:

We also define

If G consists solely of undirected edges, it is referred to as an undirected graph (UG). In contrast, if G is composed entirely of directed edges and contains no directed cycles, it is referred to as a directed acyclic graph (DAG). Let ⃗ G = ( V, ⃗ E ) denote a DAG, where V is the set of vertices and ⃗ E is the set of directed edges in ⃗ G . A path ℓ uv between u and v in ⃗ G is said to be blocked by a set S ⊆ V of vertices if and only if one of the following holds:

( i ) ℓ uv contains a chain x → z → y or a fork x ← z → y such that the middle vertex z is in S (i.e., z ∈ S );

( ii ) ℓ uv contains a collider x → z ← y such that the middle vertex z is not in S and no descendant of z is in S (i.e., De ⃗ G ( z ) ∩ S = ∅ ).

If all paths between u and v are blocked by S in ⃗ G , then u and v are d -separated by S in ⃗ G , denoted as u ⊥ ⊥ v | S [ ⃗ G ] . For disjoint subsets A,B,S ⊆ V , if every path between any vertex a ∈ A and any vertex b ∈ B is blocked by S in ⃗ G , then A and B are d -separated by S in ⃗ G , denoted as A ⊥ ⊥ B | S [ ⃗ G ] . The collection of all d -separation relations induced by ⃗ G is denoted as

For two directed acyclic graphs (DAGs) ⃗ G 1 = ( V, ⃗ E 1 ) and ⃗ G 2 = ( V, ⃗ E 2 ) , if I ( ⃗ G 1 ) = I ( ⃗ G 2 ) , they are said to be Markov equivalent , denoted as ⃗ G 1 ≈ ⃗ G 2 . For three distinct vertices u , v , and w , if u → w ← v and u is not adjacent to v in ⃗ G , the triple ( u, w, v ) is referred to as a v -structure collided on w . The skeleton of ⃗ G is an undirected graph resulted from turning every directed edge in ⃗ G into an undirected edge. Pearl et al. [55] showed that two DAGs are equivalent if and only if they share the same skeleton and the same v -structures. The Markov equivalence class of ⃗ G , or simply an equivalence class , denoted by M ( ⃗ G ) , contains all DAGs that are equivalent to ⃗ G . This equivalence class can be uniquely represented by a partially directed graph known as a completely partially directed acyclic graph (CPDAG) [45]. Let G ∗ = ( V, E ∗ ) denote a CPDAG, where V is the vertex set and E ∗ is the set of directed and undirected edges of G ∗ . We use M ( G ∗ ) to denote the set of all Markov equivalent DAGs represented by the CPDAG G ∗ . It can be shown that the skeleton of a CPDAG G ∗ is identical to the skeleton of every DAG in M ( G ∗ ) , and an edge is directed in a CPDAG if and only if it is directed in every DAG in M ( G ∗ ) [55]. As demonstrated by Andersson et al.[46], G ∗ is a chain graph composed of both directed and undirected edges, and it contains neither directed cycles nor partially directed cycles with directed edges. Additionally, the undirected components of a CPDAG G ∗ are undirected and connected chordal graphs. A chordal graph (also known as a triangulated graph) is an undirected graph in which every cycle of length greater than three contains a chord-that is, an edge connecting two non-consecutive vertices in the cycle. Equivalently, a graph is chordal if and only if it contains no induced cycles of length greater than three.

B Estimation-collapsibility for UG models

For undirected graph models, estimation-collapsiblity has been characterized via simplicial set [50]. Asmussen & Edwards [23] and Frydenberg [25] established the equivalence between collapsibility over M and the condition that each connected component of M is a simplicial set, in the context of multinomial and Gaussian undirected graphical models. Below, we provide a brief introduction to the core concept of simplicial (vertex) sets in UG [50]. By drawing an analogy with the concept of sequentially c -removable sets in DAGs, we introduce the concept of sequential simplicial (vertex) sets in UG and explore its relationship with the concepts of simplicial (vertex) sets.

Definition B.1. (Dirac [50]) In an undirected graph (UG) G = ( V, E ) , for a vertex v ∈ V , if the neighborhood ne G ( v ) forms a complete subgraph in G , then v is called a simplicial vertex in G . For a subset M ⊆ V , if the neighborhood ne G ( M ) forms a complete subgraph in G , then M is called a simplicial (vertex) set in G .

Similar to the concept of sequentially c -removable in a DAG, we introduce the concept of sequential simplicial (vertex) sets in an undirected graph. An illustration of the concept is provided in Example B.1.

Definition B.2. In an undirected graph G = ( V, E ) , for a subset M ⊆ V , if all vertices in M can be ordered such that each vertex is a simplicial vertex according to that ordering, meaning each vertex can be identified as a simplicial vertex during successive removals, then M is called a sequential simplicial (vertex) set in G .

Example B.1. Consider the UGs in Figures B.1 (a)-(c), and let's denote the three graphs as G 1 , G 2 , and G 3 , respectively. For the G 1 , we have ne G 1 ( a ) = { b, c } where b G 1 ∼ c . By the Definition B.1,

(a)

(b)

(c)

a is a simplicial vertex in G 1 . After removing a , the resulting UG G 2 has ne G 2 ( b ) = { c, d } with c G 2 ∼ d . As a result, b is a simplicial vertex in G 2 . Similarly, after removing node b , we obtain the UG G 3 , where c is clearly a simplicial vertex in G 3 . Therefore, the set { a, b, c } is a sequential simplicial set in original UG G 1 .

Note the classical concept of simplicial set and our proposed concept of sequential simplicial set are generally not equivalent. We illustrate this via Example B.2.

(a)

(b)

Example B.2. In Figure B.2(a), let M = { a, b } , with ne G ( M ) = { c, d } and c G ∼ d , which makes M a simplicial set in G . However, neither a nor b is a simplicial vertex on its own, meaning that M is not a sequential simplicial set. In Figure B.2(b), M = { a, b } , and both a and b are simplicial vertices in G , making M a sequential simplicial set. However, ne G ( M ) = { c, d, e, f } , where c G ≁ d , c G ≁ f , and d G ≁ e , implying that M is not a simplicial set in G .

Under additional conditions, we can establish the equivalence between simplicial sets and sequential simplicial sets for undirected graphs. In Figure B.2(a), it is given that M = { a, b } is a simplicial set in G , but M is not a sequential simplicial set. However, if an additional condition such as a G ∼ c or b G ∼ d is imposed, then M becomes a sequential simplicial set. In the more general case, the additional condition required is that G Ne G ( M ) is a chordal graph, as shown in Lemma B.1.

Lemma B.1. Let G = ( V, E ) is an undirected graph, for M ⊆ V , if M is a simplicial set in G and G Ne G ( M ) is a chordal graph, then M is a sequential simplicial set in G .

Proof. Here we consider using proof by contradiction to prove Lemma B.1. Let M = { m 1 , m 2 , · · · , m k } . Suppose that for any ordering, there exists a subsequence m i , m i +1 , · · · , m k , where 1 ≤ i ≤ k , such that none of these vertices are simplicial vertices in the induced subgraph G R ≜ G V \{ m 1 ,m 2 , ··· ,m i -1 } . This means that for all m j ∈ { m i , m i +1 , · · · , m k } , the neighborhood ne G R ( m j ) is not complete, i.e., there exist vertices x, y ∈ ne G R ( m j ) such that x G ≁ y . There are three possible cases:

(a)

(b)

(c)

Case (1) . When x, y ∈ V \ M ,

Since x, y ∈ ne G R ( m j ) and x, y ∈ V \ M , it follows that x, y ∈ ne G ( M ) . However, x G ≁ y , which contradicts the fact that M is a simplicial set in G .

Case (2) . When x or y is in { m i , m i +1 , · · · , m k } \ { m j } , Without loss of generality, assume x ∈ { m i , m i +1 , · · · , m k } \ { m j } and y ∈ V \ M . Since x is not a simplicial vertex in G R , there exists z ∈ ne G R ( x ) such that z G ≁ m j . ( i ) If z ∈ V \ M , let's consider whether y and z are adjacent:

( ia ) If y G ≁ z , since y, z ∈ ne G ( M ) , this contradicts the fact that M is a simplicial set in G ; see Figure B.3(a).

( ib ) If y G ∼ z , there exists a 4-cycle m j -x -z -y -m j , which contradicts the fact that G Ne G ( M ) is a chordal graph; see Figure B.3(b).

( ii ) If z ∈ { m i , m i +1 , · · · , m k }\{ m j , x } , since z is also not a simplicial vertex in G R , similar to the case with x , we need to analyze the adjacency of z 's neighbors. By applying this reasoning iteratively, we can construct a maximal connected component containing m j in the form m j -x -z -··· -m l -1 -m l . For this maximal connected component, there must exist v ∈ ne G R ( m l ) ∩ ( V \ M ) such that v G ≁ m l -1 (otherwise, m l would be a simplicial vertex in G R ). This situation is analogous to case ( i ) , leading to a contradiction; see Figure B.3(c).

Case (3) . When x, y ∈ ( { m i , m i +1 , · · · , m k } \ { m j } ) , Since x and y are also not simplicial vertices in G R , and given the arbitrary choice of m j , x and y will encounter similar situations as in cases (1) and (2).

Given the hereditary property of chordal graphs, any subgraph of a chordal graph is also a chordal graph. Therefore, for a chordal graph G = ( V, E ) , it is evident that G Ne G ( M ) is also chordal. Consequently, we have the following corollary.

Corollary B.1. Given a chordal graph G = ( V, E ) , for M ⊆ V , if M is a simplicial set in G , then M is a sequential simplicial set in G .

In Figure B.2(b), it is given that M = { a, b } is a sequential simplicial set. However, M is not a simplicial set in G . Nonetheless, if M is a sequential simplicial set and is connected, then M is a simplicial set in G .

Lemma B.2. Given an undirected graph G = ( V, E ) , if M ⊆ V is a connected subset and M is a sequential simplicial set in G , then M is a simplicial set in G .

Proof. Here we consider using proof by contradiction to prove Lemma B.2. Assume M is not a simplicial set in G , i.e., there exist x, y ∈ ne G ( M ) such that x G ≁ y . There are two possible cases:

Case (1) . There exists m ∈ M such that x, y ∈ ne G ( m ) .

Since x G ≁ y and x, y ∈ V \ M , m is not a simplicial vertex in G for any ordering. This contradicts the assumption that M is a sequential simplicial set in G .

Case (2) . There exist m x , m y ∈ M such that x ∈ ne G ( m x ) and y ∈ ne G ( m y ) .

Since m x and m y are connected within M , there exists a chordless path p = ⟨ m 0 = m x , m 1 , m 2 , · · · , m n , m n +1 = m y ⟩ where m i ∈ M for i ∈ [ n ] . Because p is chordless, for each m i (where i ∈ [ n ] ), m i -1 and m i +1 are not adjacent, meaning m 1 , m 2 , · · · , m n are not simplicial vertices. Since the vertices in M can be ordered such that each vertex is a simplicial vertex according to that ordering, either m x or m y must be simplicial. Without loss of generality, assume m x is a simplicial vertex. Thus, x G ∼ m 1 . Now consider the chordless path p ′ = ⟨ m 1 , m 2 , · · · , m n , m n +1 = m y ⟩ . In this case, x ∈ ne G ( m 1 ) and y ∈ ne G ( m y ) . By similar analysis, iterating this process will lead to the existence of some m ′ ∈ { m 1 , m 2 , · · · , m n } such that x, y ∈ ne G ( m ′ ) , which reduces to case (1); see Figure B.4.

In both scenarios, we encounter contradictions. Therefore, the assumption that M is not a simplicial set in G must be false. Thus, if M is a connected sequential simplicial set in G , then M must be a simplicial set in G .

By Lemma B.2, we obtain a sufficient condition for G Ne G ( M ) to be chordal.

Lemma B.3. Given an undirected graph G = ( V, E ) , if M ⊆ V is a connected subset and M is a sequential simplicial set in G , then G Ne G ( M ) is a chordal graph.

Proof. By Lemma B.2, we know that M is a simplicial set in G , which means ne G ( M ) is a complete subgraph. Suppose G Ne G ( M ) is not a chordal graph. For any cycle µ in G Ne G ( M ) with length greater than 3, since ne G ( M ) is complete, µ can contain at most two vertices from ne G ( M ) . Otherwise, µ would have a chord. Without loss of generality, assume that µ contains two vertices x and y from ne G ( M ) , and the remaining vertices are in M . Let m i be the first vertex removed from µ , then ne G µ ( m i ) is complete. Thus, µ must have at least one chord, i.e., there exists an edge connecting two neighbors of m i in µ . This contradicts the assumption that µ is a cycle of length greater than 3 without chords. Therefore, G Ne G ( M ) must be a chordal graph.

Based on the above series of lemmas, we can provide an equivalent characterization of a sequential simplicial set in an undirected graph that does not depend on the order, as stated in the following PropositionB.1.

Proposition B.1. Given an undirected graph G = ( V, E ) , for M ⊆ V , the following two statements are equivalent:

(1) M is a sequential simplicial set in G ;

(2) For each connected component C M of M , the subgraph G Ne G ( C M ) is chordal graph, and C M is a simplicial set in G .

Proof. (1) ⇒ (2) . Since M is a sequential simplicial set in G , it is evident that M 1 , M 2 , · · · , M K are also sequential simplicial sets in G . Therefore, for all i ∈ [ K ] , the vertices in M i can be ordered such that each vertex is a simplicial vertex according to that ordering. Additionally, since M i is connected, by Lemma B.2, M i is a simplicial set in G . By Lemma B.3, G Ne G ( M i ) is a chordal graph.

(2) ⇒ (1) . Given that for all i ∈ [ K ] , G Ne ( M i ) is a chordal graph and M i is a simplicial set in G , by Lemma B.1, for all i ∈ [ K ] , the vertices in M i can be ordered such that they are simplicial vertices in that order, i.e., M i is a sequential simplicial set in G . Furthermore, since M 1 , M 2 , · · · , M K are mutually disconnected, they can be ordered in any sequence, and they will still form a sequence of sequential simplicial sets. Therefore, M is a sequential simplicial set in G .

For chordal graphs, by hereditary property, any subgraph of a chordal graph is also a chordal graph. Proposition B.1 immediately implies Lemma 1 for chordal graph. It states that in a chordal graph, for any connected subset, a sequential simplicial set is equivalent to a simplicial set. Therefore, for a chordal graph model, being estimation-collapsible over M is equivalent to each connected subset of M being a sequential simplicial set.

C Summary of Key Concepts

To facilitate comparison and reference, this appendix consolidates the key concepts related to simplicial, c -removable, and c -simplicial vertices and sets across UG, DAG, and CPDAG graphical models. The detailed definitions are presented in Table C.1.

The concept of a c-simplicial vertex in a CPDAG generalizes the notion of a leaf node (i.e., a node with an empty set of children) in DAGs and a simplicial vertex (i.e., a node whose set of neighbors is complete) [50] in undirected graphs. In particular, when G ∗ is a DAG, v is a c-simplicial vertex if and only if v is a leaf node . This is because when G ∗ is a DAG, it naturally holds that ne G ∗ ( v ) = ∅ , making the completeness of its neighbor trivially satisfied, and the only condition required is ch G ∗ ( v ) = ∅ . Meanwhile, when G ∗ is an undirected graph, v being a c -simplicial vertex in the CPDAG is equivalent to v being a simplicial vertex in the undirected graph G ∗ . In this case, ch G ∗ ( v ) = ∅ always holds, and the sole requirement is that ne G ∗ ( v ) forms a complete subgraph.

The notion of sequential c -simplicial set likewise generalizes sequential c -removable set for DAG. When G ∗ is a DAG and M is a sequential c -simplicial set, it directly follows from the previous

discussion that M is also sequentially c -removable. Furthermore, we can find the concept of sequential c -simplicial set generalizes that of simplicial set for undirected graph. When G ∗ contains only undirected edges (i.e. G ∗ is a chordal graph), recall a c -simplicial vertex reduces to a simplicial vertex. This means that, for a sequential c -simplicial M , its vertices can be ordered such that each vertex is a simplicial vertex at the time of its removal. Consequently, in a chordal graph G , we also term a sequential c -simplicial set M as a sequential simplicial set . Lemma 1 indicates that sequential simplicity (or sequential c -simplicity) is equivalent to simplicity for chordal graph.

D Technical Proofs of The Main Results

Proof of Lemma 2

Proof. Assume that ch G ∗ ( M ) = ∅ , then ∃ m ∈ M such that ch G ∗ ( m ) \ M = ∅ . In this case, regardless of how the vertices in M are ordered and sequentially removed as c -simplicial vertices, m can never be a c -simplicial vertex. Therefore, by Definition 5, M is not a sequential c -simplicial set in G ∗ , which contradicts the known result.

Proof of Lemma 3

Proof. By Lemma 1, Statement ( 2 ′ ) is equivalent to:

(2 ∗ ) ch G ∗ ( M ) = ∅ , and M is a sequential simplicial set in G ∗ τ i .

We will now establish the equivalence between Statement (1) and Statement (2 ∗ ).

(1) ⇒ (2 ∗ ) . Given that ch G ∗ ( M ) = ∅ clearly holds, we now need to prove that M is a sequential simplicial set in G ∗ τ i . Since M is contained within a single chain component, and under the condition that ch G ∗ ( M ) = ∅ , we have ⋃ m ∈ M ch G ∗ ( m ) = ∅ . Therefore, determining whether M is a sequential simplicial set in G ∗ τ i is equivalent to determining whether M is a sequential c -simplicial set in G ∗ .

(2 ∗ ) ⇒ (1) . Since ch G ∗ ( M ) = ⋃ m ∈ M ch G ∗ ( m ) \ M = ∅ , and because M is contained within a single chain component, we have ⋃ m ∈ M ch G ∗ ( m ) = ∅ . In this case, M is a sequential simplicial set in G ∗ τ i

is clearly equivalent to M is a sequential c -simplicial set in G ∗ .

Proof of Theorem 2

Proof. By Lemma 1, Statement ( 2 ′ ) is equivalent to:

(2 ∗ ) ch G ∗ ( M ) = ∅ , where each M i forms a sequential simplicial set in G ∗ τ i for all i ∈ [ K ] .

We will now establish the equivalence between Statement (1) and Statement (2 ∗ ).

(1) ⇒ (2 ∗ ) . ch G ∗ ( M ) = ∅ clearly holds. Next, we prove that for all i ∈ [ K ] , M i is a sequential simplicial set in G ∗ τ i .

Without loss of generality, assume α ( τ 1 ) < α ( τ 2 ) < · · · < α ( τ K ) . Assume there exists a largest j , where 1 ≤ j ≤ K , such that the non-empty M j is not a sequential simplicial set in G ∗ τ j . Since M is a sequential c -simplicial set in G ∗ , M K is also a sequential c -simplicial set in G ∗ . Let G ∗ 1 ≜ G ∗ V \ M K . Then, it is evident that M K -1 is a sequential c -simplicial set in G ∗ 1 . Let G ∗ 2 ≜ G ∗ V \{ M K ,M K -1 } . Continuing this process, we deduce that M j is a sequential c -simplicial set in G ∗ K -j ≜ G ∗ V \{ M K ,M K -1 , ··· ,M j +1 } . By Lemma 3, M j must be a sequential simplicial set in G ∗ τ j , which contradicts our assumption.

(2 ∗ ) ⇒ (1) . Without loss of generality, assume α ( τ 1 ) < α ( τ 2 ) < · · · < α ( τ K ) . Given that ch G ∗ ( M K ) = ∅ and M K is a sequential simplicial set in G ∗ τ K , by Lemma 3, M K is a sequential c -simplicial set in G ∗ . Let G ∗ 1 ≜ G ∗ V \ M K . Then, ch G ∗ 1 ( M K -1 ) = ∅ and M K -1 is a sequential simplicial set in G ∗ 1 ,τ K . By Lemma 3 again, M K -1 is a sequential c -simplicial set in G ∗ 1 . Let G ∗ 2 ≜ G ∗ V \{ M K ,M K -1 } . Continuing this process, we find that M K , M K -1 , · · · , M 1 are sequential

̸

̸

c -simplicial sets. Therefore, the vertices in M can be ordered such that they are c -simplicial vertices in G ∗ , meaning M is a sequential c -simplicial set in G ∗ .

Proof of Theorem 1

Proof. (1) ⇒ (2) . Assume (2) does not hold. That is, for all ⃗ G ∈ M ( G ∗ ) , ch ⃗ G ( M ) = ∅ . Since G ∗ is a CPDAG, this implies that ch G ∗ ( M ) = ∅ or there exists an undirected connected component C of M such that ne G ∗ ( C ) is not complete. By Theorem 2, M is not a sequential c -simplicial set in G ∗ , which contradicts (1).

̸

(2) ⇒ (1) . Let T = { τ 1 , τ 2 , . . . , τ K } be the set of chain components of G ∗ . Partition M into each chain component, i.e., M = M 1 ∪ M 2 ∪ · · · ∪ M K , where M i ⊆ τ i for each i ∈ [ K ] . Assume (1) does not hold. By Theorem 2, there are two possible scenarios.

̸

( i ) ch G ∗ ( M ) = ∅ . By the properties of CPDAGs, for all ⃗ G ∈ M ( G ∗ ) , ch ⃗ G ( M ) = ∅ , which contradicts (2).

̸

( ii ) There exists j , where 1 ≤ j ≤ K , such that M j is not a sequential simplicial set in G ∗ τ j . Let M j = M j, 1 ∪ M j, 2 ∪ · · · ∪ M j,l , where M j,i ( i ∈ [ l ] ) are the connected components in G ∗ τ j . Since M j is not a sequential simplicial set in G ∗ τ j and G ∗ τ j is a chordal graph, by Lemma 1, there exists i ,where 1 ≤ i ≤ l , such that M j,i is not a simplicial set in G ∗ . That is, there exist x, y ∈ ne G ∗ τ j ( M j,i )

̸

such that x G ∗ ≁ y . In every DAG ⃗ G ∈ M ( G ∗ ) , x and y cannot both be parents of M j,i , otherwise a v-structure would be formed. Therefore, ch ⃗ G ( M j,i ) = ∅ , implying ch ⃗ G ( M ) = ∅ , which contradicts (2).

̸

(2) ⇔ (3) ⇔ (4) . By [28, Lemma 1] and Theorem 1, we have (2) ⇔ (3) ⇔ (4) is evident.

̸

E Additional Experimental Results

For the DSCS and DCR algorithms applied to random graphs with n ∈ { 500 , 1000 , 1500 } , the detailed average running times (in seconds) and the standard errors are presented in Table E.1. DSCS clearly shows a significant advantage over DCR, and this advantage becomes more pronounced as n and p increase. Furthermore, from Table E.1 and Figure 3, it can be observed that for a given n and | M | , the average running time of DCR increases rapidly with increasing p , whereas DSCS exhibits a slower and approximately linear growth. Additionally, the standard errors of DSCS's average running times are consistently smaller than those of the DCR Algorithm, indicating greater stability in DSCS's running times across various random graphs.

Due to the extremely long computational time of the DCR Algorithm for large graphs, for n ∈ { 2000 , 4000 , 6000 , 8000 , 10000 } , we only record the average running time of the DSCS Algorithm. The results are shown in Table E.2 and in Figure E.1. It can be seen that even for high-dimensional variables (e.g., n = 10000 ) and denser graphs (e.g., p = 0 . 1 ), the DSCS Algorithm can quickly determine the estimation-collapsibility of the underlying DAG model. For example, when n = 10000 , p = 0 . 1 , and | M | = 9000 , the average running time of the algorithm is just over 150 seconds. For a given n and p , the average running time increases approximately linearly with | M | .

̸

Based on these observations, it can be concluded that the DSCS Algorithm is capable of efficiently addressing the structure dimensionality reduction problem for high-dimensional Bayesian network with complex structure. It should be noted that some of the standard errors are relatively large due to the inherent randomness in the generation of the graphs and the random selection of the set M . For certain node subsets M , after finding ch G ∗ ( M ) = ∅ during the initial check (as per Lines 1-2 in the DSCS Algorithm), the DSCS algorithm can immediately stop, leading to a very fast determination of collapsibility. On the other hand, for some node subsets M with ch G ∗ ( M ) = ∅ , further checks are required (as per Lines 3-9 in the DSCS Algorithm), resulting in longer algorithm's running times.

To further highlight the advantages of the DSCS algorithm, we selected three real-world Bayesian networks from the R-package bnlearn -WIN95PTS (76 nodes, 112 edges), LINK (724 nodes, 1125 edges), and MUNIN (1041 nodes, 1397 edges). A detailed description of the three Bayesian networks can be found at https://www.bnlearn.com/bnrepository/ . For each network, we randomly sampled a vertex subset M of size | M | , then applied DCR (on the DAGs) or DSCS (on the corresponding CPDAGs) to test estimate-collapsibility over M . For every combination of network

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and | M | , we ran 30 simulations and recorded the average running time (in seconds), as reported in the Table E.3.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies the problem of when inference and marginalization in Bayesian networks can be performed by simply looking at the subgraph induced by the variables of interest rather than having to consider all variables. The authors design a generalization of existing algorithms for DAGs and undirected graphs by considering the completed partially directed acyclic graph (CPDAG) that encodes the Markov equivalence class of the Bayesian network. The main idea is to take advantage of the fact that CPDAGs are chains of undirected connected chordal graphs (UCCGs), since the authors observe that testing the property within UCCGs is straightforward. The running time is only quadratic in the number of vertices whereas previous work on DAGs required $O(|V|^4)$ time, and the speedup is evident is also from the empirical experiments.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n\nQuadratic speedup in testing estimation-collapsibility for DAGs.\n\nExperiments providing practical evidence on the speedup. \n\nConcrete examples that are associated with illustrations to assist the reader.\n\n**Weaknesses:**\n\nI think it would be desirable for the main manuscript to be self-contained; now, some terminology is defined only in the Appendix (see, e.g., Sec. 2.1, and related minor comments). Some further polishing could also be done (see minor comments for a non-comprehensive list of examples).\n\nThe experiments only consider Erdös-Rényi graphs and not any real-world instances.\n\n**Minor:**\n- Line 49: missing whitespace.\n- Line 92: $A$ is already a set, so it does not need braces around it.\n- Line 99: missing whitespace.\n- Fig. 2 caption: missing whitespace.\n- Line 121: grammatical error in \"a substantially saving of data collection and computational efforts\".\n- Lines 152, 153: maybe \"induces\" instead of \"forms\"?\n- Definition 5: I think you should emphasize that the endpoints of undirected edges of a CPDAG do not count as children. Similarly, it is unclear that children or parents aren't neighbors unless one reads the Appendix containing the definitions carefully.\n- Line 191: you have a set instead of an ordering.\n- Line 192: I think it is more common to call childless nodes sinks rather than leafs.\n- Line 265: Extra 'p'.\n- Line 529: The definition of a chord could be more precise (\"edge connecting two non-adjacent nodes\") since the nodes are, by definition, adjacent if the edge connects them. Alternatives include \"no induced cycle of length greater than 3\" or \"edge connecting two non-consequent nodes of the cycle\".\n\nQUESTIONS:\nYou write that \"Subsequent parallel verification of neighborhood completeness across chain components has complexity of order $O(|\\tau_{\\max}|^2)$, dominated by the largest component\". If I understood correctly, the issue is that you take quadratic time to evaluate whether each neighborhood is a clique? Have you considered if this could be done in linear time in the number of edges? I suspect that this might be doable by computing a perfect elimination ordering for each chordal component in linear time, since these enable characterizing the maximal cliques of the graph (see, e.g., [1]). Then, you would probably just need to check whether the neighborhood is a subset of the maximal clique associated with the node of the neighborhood that appears in the elimination ordering the earliest.\n\n[1] Fanica Gavril:\nAlgorithms for Minimum Coloring, Maximum Clique, Minimum Covering by Cliques, and Maximum Independent Set of a Chordal Graph. SIAM J. Comput. 1(2): 180-187 (1972)\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-11"}

comments: SUMMARY:
This work aims to propose efficient algorithms to verify estimation-collapsibility on a completed partially directed acyclic graph, or CPDAG, for Bayesian network inference. It introduces the concept of sequential c-simplicial sets and reveals the connection between the sequential c-simplicial sets and estimation-collapsibility. Further, it proposes a so-called detecting sequential c-simplicial sets (DSCS) algorithm as the verification algorithm with computational complexity analysis to show its efficiency. Empirical results on synthetic data are further presented.

STRENGTHS AND WEAKNESSES:
[strength]
- The research question considered in this work is important and challenging since the estimation-collapsibility can greatly improve the inference efficiency for many probabilistic queries.
- The introduced concept of sequential c-simplicial sets and its connection with the estimation-collapsibility is novel, which unifies the previous work.
- This work is overall well written, with a thorough discussion on previous work and theoretical results clearly elaborated. While this work is heavy in notations, I find the running example using Figure 2 very helpful.

[weakness]
- The experimental section will benefit from including more baseline methods for comparison, since now only one baseline is considered. It will also be improved a lot if more benchmarks are included, especially the real-world ones and those with diverse graph structures, since currently only results on Bayesian networks with Erdos-Renyi graphs are presented.

QUESTIONS:
- What is the complexity of the conversion from a DAG of a Bayesian network to a CPDAG?
- Can the authors provide more quantitative or qualitative comparisons with other baseline methods?

LIMITATIONS:
Yes.
TITLE: Official Review
CLARITY: 4
IMPACT: 4
ORIGINALITY: 4
DATE: 2025-07-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the problem of estimation collapsibility that is fitting a sub-model and ignoring non-essential variables. This is relevant in scenarios where the number of variables are much more than the relevant ones. The authors introduce sequential c-simplicial sets to characterize estimation collapsibility in CPDAG instead of DAG and introduce an algorithm for verifying estimation-collapsibility within CPDAGs.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\nAs mentioned, often in practice, the exact DAG is not available and using observational data, only up to Marko equivalence class can be revealed. Hence, the result of this work which characterises the estimation collapsibility for CPDAGs becomes important.\n\nThe proposed algorithm has complexity of quadratic from in the number of vertices which makes it scalable for large networks.\n\nWeakness:\nThe paper’s presentation has room for improvement. Some important definitions are missing such as non-triviality or chain components. \nIs the chain components are unique? If not how large the set of components, i.e., what is K in line 3 of DSCS algorithm?\nSince chain components are a major part of the proposed approach without (I could not find it) proper definitions and discussion on its cardinality, it is hard to read and understand the difference between the proposed method and the previous ones. \nWhy for example, other methods do not use the chain components Idea and reduce their complexity?\n\n\nAlthough the result of this work for CPDAGs are novel but given the previous definitions and characterisation of the c-removability, it is a limited contribution. In other words, it seems that given the previous results of estimation collapsibility in DAGs, its extension to CPDAGs is not trivial but a natural generalisation.\n\nQUESTIONS:\nPlease see the comments.\n\nLIMITATIONS:\nPlease see the comments.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-04"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper tries to provide a unifying framework of estimation-collapsibility for CPDAG (completed partially directed acyclic graph), DAG (directed acyclic graph) and UG (undirected graph). Estimation-collapsibility refers to the ability to remove a set of variables from the graphs while preserving the structure of the rest. The authors achieve this by focusing on CPDAG, which is an Markov equivalent class of DAG, with connections to UG.\n\nFollowing prior works, they give an equivalence relation between estimation-collapsibility and $c$-simplicial, a new notion proposed in this paper, under the non-trivial CPDAG assumption. On top of this equivalence result, this paper also gives a more efficient algorithm for detecting estimation-collapasibility via $c$-simplicial that improves the time complexity from $O(|V|^4)$ to $O(|V|^2)$.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. The problem is well-motivated: detecting estimation-collapasibility from CPDAG is a natural generalization of the DAG setting and it is interesting to see the connection from the UG perspective.\n2. This paper is well-written, clear and mostly self-contained. I find the graphs are pretty helpful in understanding the paper.\n3. Their algorithm is quite straight-forward given the equivalence theorem between estimation-collapasibility and $c$-simplicial.\n\nWeakness:\n1. I have some questions/concerns regarding applying $c$-simplicial criteria over the vanilla DAG setting -- see Question 3.\n2. Some terminology is missing and might affect reading -- see Question 1 and 5.\n\nOverall, I think this paper is solving a well-motivated problem that could be of interests to audience of this conference. Their techniques have a lot of connections to existing works, though I do have some doubts (perhaps due to my limited understanding).\n\nQUESTIONS:\nQuestion.\n1. On Line 196, the authors mention $\\textbf{ne}_{G^*}(v) = \\emptyset$, I don’t see why this is true, unless you define the neighbors as the ones connected via *undirected* edges. There’s no formal definition of neighbors other than on Line 89 (which had no formal definition on the main text). To avoid confusion, I think it's best to make a note mentioning the distinction between neighbors v.s. parents and children -- that they are disjoint.\n2. Does the $c$ in $c$-simplicial stands for conditional? I think it would be helpful to point out the meaning: easier to understand what's the difference between $c$-simplicial (CPDAG) and simplicial (undirected graph).\n3. Why does the definition of $c$-simplicial (for CPDAG) seem so different from the counterpart on DAG? Intuitively, do we not expect that DAG would be reduced back to CPDAG (viewing DAG as a subset of CPDAG, or simply transforming DAG into its CPDAG)? I notice the discussion on Line 200-202: it only describes reduction from CPDAG to DAG but not the only way around. Maybe it is obvious to the authors, but I don't quite see how to convert $c$-removable to $c$-simplicial (worth writing a bit more on this). Are they incompatible?\n   For example, consider the node $v$ in DAG: $v \\rightarrow a \\leftarrow b$. $v$'s $\\mathbf{Mb}_{G}(v)$ is $a$ and $b$. They ($a$ and $b$) are adjacent and does not being to $\\textbf{pa}_G(v)$. So it should be $c$-removable (and thus estimate-collapsible according to Proposition 1). But $v$ is not $c$-simplicial in CPDAG, since it is not a leaf node.\n4. Since the definitions for CPDAG, DAG, UG and its related key concepts on $c$-simplicial, $c$-removable, simplicial are so spread out, I would suggest collecting and adding them to a table for easy comparison and reference.\n5. What do you mean by \"non-trivial\", e.g., on Line 133? Perhaps should briefly explain it or put a footnote somewhere? Is there a definition for \"chain component\" which appears first on Line 263-264? I'm not entirely sure what it means in this context.\n6. If I understand correctly, the DSCS algorithm doesn't actually provide a $O(|V|^2)$ upper bound for DAG? See the incompatibility doubt I raised in Question 3 (also see related discussion around Line 76-80).\n\nTypos:\n1. On line 245, \"establishes CPDAG-based criterion for check estimation-collapsibility\", \"check\"->\"checking\".\n\nLIMITATIONS:\nyes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-08"}