On the Identifiability of Poisson Branching Structural Causal Model Using Probability Generating Function
Authors: Yu Xiang
Jie Qiao
Zefeng Liang
Zihuai Zeng
Ruichu Cai
Zhifeng Hao
Venue: NeurIPS 2024 spotlight (2024)
Keywords: Causal Discovery
Primary area: causal_inference

Abstract: Causal discovery from observational data, especially for count data, is essential across scientific and industrial contexts, such as biology, economics, and network operation maintenance. For this task, most approaches model count data using Bayesian networks or ordinal relations. However, they overlook the inherent branching structures that are frequently encountered, e.g., a browsing event might trigger an adding cart or purchasing event. This can be modeled by a binomial thinning operator (for branching) and an additive independent Poisson distribution (for noising), known as Poisson Branching Structure Causal Model (PB-SCM). There is a provably sound cumulant-based causal discovery method that allows the identification of the causal structure under a branching structure. However, we show that there still remains a gap in that there exist causal directions that are identifiable while the algorithm fails to identify them. In this work, we address this gap by exploring the identifiability of PB-SCM using the Probability Generating Function (PGF). By developing a compact and exact closed-form solution for the PGF of PB-SCM, we demonstrate that each component in this closed-form solution uniquely encodes a specific local structure, enabling the identification of the local structures by testing their corresponding component appearances in the PGF. Building on this, we propose a practical algorithm for learning causal skeletons and identifying causal directions of PB-SCM using PGF. The effectiveness of our method is demonstrated through experiments on both synthetic and real datasets.

On the Identifiability of Poisson Branching Structural Causal Model Using Probability Generating Function
Yu Xiang'; Jie Qiao'; Zhefeng Liang', Zihuai Zeng'. Ruichu Cai Zhifeng Hao
School of Computer Science, Guangdong University of Techhnoloy.. Guangzhou, China
PPngg Cheng Laboratory, Shenzhen, China
College of Science, Shantou University, Shantou, China
(thexiang2000, qiaojie.chn, Izfeng01 1021, zzhuaiii,, cairuichu) @ @gmail.com, haozhifeng @ stu.edu.cn
Abstract
Causal discovery from observational data, especially for count data, essential across scientific and industrial contexts, such as biology, economics, and network operation maintenance. For this task, most approaches model count data using Bayesian networks or ordinal relations. However, they overlook the inherent branching structures that are frequently encountered, e.g., browsing event might trigger adding cart or purchasing event. This can be modeled by binomial thinning operator (for branching) and an additive independent Poisson distribution (for noising), known Poisson Branching Structure Causal Model (PB-SCM). There is provably sound sumulant-baseee causal discovery method that allows the identification of the causal structure under branching structure. However, we show that there still remains gap in that there exist causal directions that are identifiable while the algorithm fails identify them. In this work, we address this gap by exploring the identifiability of PB-SCM using the Probability Generating Function (PGF). By developing a compact and exact closed-form solution for the PGF of PB- SCM, we demonstrate that each component in this closed-form solution uniquely encodes a specific local structure, enabling the identification of the local structures by testing their corresponding component appearances in the PGF. Building on this, we propose a practical algorithm for learning causal skeletons and identifying causal directions of PB-SCM using PGF. The effectiveness of our method is demonstrated through experiments on both synthetic and real datasets.
1 Introduction
Causal discovery from observational data, particularly for count data is a crucial task that arises in numerous applications, including biology (Wiuf and Stumpf (006) ) economic (WeiB and Kim 22044), network operation maintenance (Qiao et al. [2023], Cal et al. 2022) ). etc. Much effort has been made model and discover the causal structure from count data. One line of research models the count data as a type of Poisson Bayesian network (Park and Raskutti [2015] 2017)) or ordinal functional model (Ni and Mallick (2022). ). based on which various types of discovering methods are investigated. However, the Bayesian network and ordinal modeling ignore the inherent branching structure among the counting relationship, which is frequently encountered in real world (Weil (018)). Take Fig. (a) as an example, in online shopping, the purchasing event can be inherited from browsing, cart added, or promotion event, which exhibits branching structure such that website browsing (X1) may lead to either purchasing (X4) directly without adding to the cart, or adding
"Equal contribution.
I Cooressonniig author.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Purchase (X4)
User 1:
User 2:
Browsing (Xi)
Add cart
Promotion (X3)
(Xa)
(a) Causal Graph
(b) Branching Structure
Figure 1: (a) An example of the causal graph of count data in online shopping service (b) An illustration of the branching structure inherent in count data, where browsing events trigger subsequent events that are summarized counts.
to the cart (X2) followed by purchasing (Xa), depicted in Fig. (b). Accurately modeling and identifying the underlying causal mechanisms pcccanisms-partiiulally the branching structure inherent in these count variables-are of interest for providing valuable insights service providers.
Branching structure modeling is well studied across various domains, notably within the context of the integer-value value autoregressive model (WeiB) (2018), Al-Osh and Alzaid [1987] McKenzie 1985)). Most of these studies model the relationship among count variables utilizing a thinning operator 'o' (Steutel and van Harn 1979 ) For instance, the causal pair X1 + X4 in Fig. aaa can be modeled iid.
as follows: X4 a o X1 + where a o X1 = X1, cla) Sn ~ Bernouli(a)), e Pois(/s). That is, the thinning operator models the branching of an event, e.g., a website browsing event leads to a purchase with probability a, while the Poisson noise models the exogenous event. Such modeling can be formalized as the Poisson Branching Structure Causal Model (PB-SCM) (Qiao et al.) (024b)). ). Recently, Qiao et al. (224b) further explores the identifiability of PB-SCM and proposes likeeihhod-basse heuristic searching method for learning causal skeleton and a cumulant- based method for identifying causal direction. Although some identifiability results are developed by
Qiao et al. 2024b) using cumulant, we show that there still remains a gap in that there exist causal directions that are identifiable while the algorithm fails to identify them. For example, the current approach cannot fully identify the causal directions among X1, X2, and X3 Fig. (a). Moreover, the likelihood- base heuristic searching method requires extensive coefficient estimating and structure searching suffering from computational complexity and potential convergence issues.
In this paper, we aim to develop the identifiability of the Poisson Branching Structural Causal Model (PB-SCM) using the Probability Generating Funetion (PGF) as the characterization of the distribution. Specifically, we first develop the closed-form solution for the PGF of PB-SCN which establishes the connection with the causal structure. With this connection, we find that each component this closed-form solution is directly associated with a unique local structure, enabling us to learn the skeleton and analyze causal asymmetries without the need for the entire structure. Based on this, we provide an exact poouddopooyoomall time causal structure learning algorithm (i.e., polynomial in the magnitude of the observed variables). We demonstrate the effectiveness of the proposed causal discovery method using synthetic data and real-world data.
Our contributions are threefold. We develop the closed-form solution for PGF of PB-SCM and establish connection between the closed-form solution of PGF and the graphical patterns, 2) We exploit the closed-form solution of PGF to identify the causal skeleton as well the causal direction of PB-SSCC PB-SCM, which allows us to identify the udjacency and causal direction directly. 3) We propose a practical structure learning algorithm and demonstrate its efficiency and effectiveness through synthetic and real-world data experiments.
Related work
For brevity, we review causal discovery methods that are applicable and fully identifiable on ob- servational discrete data. Numerous methods and theories have been developed for learning causal structure from observational data (Spirtes et al. 22000, Zhang et al. (2018) Glymour al. (0019] Cai
et al. [00 8b). Qiao et al. [2024a]). In particular, for discrete data, one can employ constraint-based
methods (Pearl 2009), J. Spirtes al. (1995)), score-based methods (Chickering 22002. T T'aamardinos
et al. (2006) ) to identify the causal structure by exploring the conditional independence relation

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among variables, but the conditional independence relation can only identify up to the Markov equivalent class (Pearl 2009]). Recently, numerous approaches have been developed for categorical
or ordinal data that can identify beyond Markov equivalent class (Cai et al. 2018a| Ni 2022]. Ni and
Mallick 2022]. Peters al. 2010). Leonelli and Varando (2023]. Figueiredo and Oliveira (023]). However, these methods are rarely suitable for count data. Although some recent works explore the Poisson Bayesian network for the Poisson data (Park and Raskutti [2015 2017)), and further extend
it to the zero-inflated Poisson data (Choi et al. [2020), there still remains gap in the identifiability of the Possion branching structure model.
3 Poisson Branching Structural Causal Model
In this section, we first introduce the Poisson branching structural causal model (PB-SCM), and then we introduce the preliminary of PGF.
3.1 Problem Formulation
Let X {X Xa) denote d-dimensional random count vector, of which the causal relationship
consists of causal directed acyclic graph (DAG) G(V, E) with the vertex set V {1, 2, .... d} and edge set E. We use Pa(i) = {اlj - E E}. Ch(i) = Gike + j E E} and Des(i) = Úili ~ j E E} denote the sets of parents, children and descendants of vertex in G(V, E). We assume that any variable in X satisfies the following Poisson Branching Structural Causal Model (PB-SCM):
Definition (Poisson Branching Structural Causal Model), For each random variable X₃ E X, ler 4 Pois( (p:) be the noise component of X3, X; is generated by:
X;
Σ jENa(S)
ag o Xj + ci.
where Oz,i E (0, 1] is the coefficient from vertex j i, and ( o X; is a Binomial thinning operator Bernoulli(a).. independently of X,.
such that a X₁ = Σx= <(0)) where g/() t.d.
The formulation is consistent with the existing literature, such as Qiao et al. 2024b AI-Osh and Alzaid [1987]. We further assume that the faithfulness assumption, the causal Markov assumption, and the causal sufficient assumption hold. These assumptions are commonly used in constraint-based causal discovery methods, e.g., PC algorithm (Spirtes et al. [2000)). With these assumptions, we formalize our goal as follows:
Goal. Given i.i.d. samples D (1
m from the joint distribution P(X) generated by 2d
PB-SCM, our goal is to identify the unknown causal structure G' from D.
3.2
Preliminary
To develop the identifiability of PB-SCM, we resort to using the probability generating function as the proxy to analyze the distribution and its asymmetry property. Here, we recall the definition the probability generating function:
Definition 2 (Probability Generating Function). Given discrete random vector X = [XI,..., Xa)T
taking values in the non-negative integers g²0 the probability generating function of X. is defined as:
(x(z)=
X1
(2)
where p is the probability mass function X. The power series converges absolutely at least for all complex vectors [2],..., zd. E Cd with max[[ | 1.
Since the PGF is uniquely defined by the probability mass function, such uniqueness of the power series expansion will in turn define the probabilities, meaning that the analysis of the distribution can be conducted on PGF proxy (Johnson et al. (2005))
3

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G
1 212
112
1 dga
2,1 2x,) (0)
X
22
22
62a
(1 (11
(1 (1
22 14
X
2(1 Te (1)
4=3) (0)
2 (1(1 m Pi,
2 M2gs Puts [1
6
X1
X1
c
X1
Starting X₃
422
3,S
d1s
Xg
① %
2 (t a12) 2, a (1)
@ d1 222) (12
4, (1)
T22(1)
(a) Starting X1
(c) Starting at X3
Gx(2) = explu, ①
ex
3 + ④ 14) explux (S ( - U2) [xp 1 pg
Tx, (1)
(12
Tr
(d) The closed form of PGF of the triangular structure where Xx, (0) = Tx, Tx, Tx2 =
Figure 3: Illustration of the graphical implication in the closed-form solution for the PGF of PB-SCM.
Here, Tx (1) indicates that X, is reached from X,, while (1 axx, (0) indicates the opposite.
4 Structure Learning of PB-SCM Using Probability Generating Function
In this section, demonstrate how to utilize PGF to identify causal structures. We first develop the closed-form solution for the PGF of PB-SCM, which has a compact and exact representation capturing the global structure information. Building on this, delvv into the components of the expansion of the PGF's closed form, demonstrating that each component in PGF is connected to a specific local structure. To effectively capture these components, we develop local PGF, Finally, based on such a connection, we present theoretical results regarding the graphical implications of the component captured by local PGF, which can be used discover causal structures.
4.1 Motivating Example
Before the formal discussion, we first present a motivating example illustrating how PGF helps analyze the identifiability of PB-SCM.
01,2
11,3
Taking the triangular structure in Fig.
example, the corresponding closed form of PGF is given in Fig. 3 We can observe that the closed form of the PGF can be separated into three parts, starting X1, X2, and X3, respectively. Each part of PGF encapsulates the path information in the overall causal structure. For example, considering the term 2122 2 in ④ the 2 represents that there are two paths from X1 to X3 if the path 01,2, 022,3, 01,3 are 'open'. By this, one can deduce that for the reserved direction, e.g., X3- + X2, such a term 23 will not exist, which justifies that the causal direction between X2, X3 is identifiable.
2
02,33
Figure 2: Example triangular structure.
Another attractive property of the PGF is the ability to analyze the local behavior of the structure. For example, by properly setting 21 approach 0, can deduce whether the exist of adjacency between X2, X3 by testing whether the term 2243 exists as shown in term 6. Furthermore, as shown in Fig. [ by analyzing the local PGF, can show that the causal direction X2 - X₂ +- X3 is identifiability while the previous cumulant- based method (Qiao et al. 2024b) can not.
In light of the connection between PGF and the causal structure, we can develop a PGGF-ased method to effectively identify the causal structure. the following section, we first demonstrate how the PGF encapsulates the causal structure by developing the closed-form solution of the PGF Then, benefiting from the local analysis of PGF, we address the identifiability gap of PB-SCM and present a more general identifiability result.

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4.2 Closed Form Solution of Probability Generating Function
In order analyze the identifiability of PB-SCM, we first establish fundamental theorem that provides the closed-form solution for the PGF of PB-SCM:
Theorem 1 (Closed-form solution for PGF of PB- PB-SCM), Given a random vector X = [X1,..., X,.""
following PB PB-SCM, let z(j) { Des(j) U {5)), the PGF of P(X) is given by Gx(z)
IIEE Ge 2 C (()(a))))
where
Gzy(Z())) = GB(ass) (2) x IINECHS) Gau(z())) (B(as,))
Ch(j) +0
(3)
Otherwise
in which (.) is the PGF of Poisson noise G and G B(ona) (.) is the PGF of Bernoulli distribution with parameter Oz.j-
The main idea of the proof is to decompose the expectation of PGF according to the causal structure, e.g., conditional expectation in PGF can be derived as E [ 1
[4] ()(o since
01,3 o c Binomial(n C (1,3)) Applying the above rule recursively with the law of expectation, we can obtain Theorem
1 The overall proof is given in Appendix A.
Theorem I reveals the inherent relationship between the causal structure and the close-formed solution of PGF. To see this connection more concretely, we further develop the following theorem:
Theorem 2. Given random vector X [X1, .. Xa) following PB-SCM, the PGF of P(X) can be expressed by:
Gx(z)= II
exp expl14 x (Tx, 1)),
(4)
(E(4]
where Tx, (1) 24 (0, I[ Tx (s,), Tx, (0) and oi o4. if 1 and
0. = 1 Oij if = 00
Theorem 2 provides more compact closed-form solution for the PGF of PB-SCM. Intuitively, consists of different path information starting at different vertex. Taking the triangular structure in Fig. as an example, the closed-form solution PGF can be expressed as follows:
Gx = exp 11 x (1 (1--0.0 (1-0).2) - -1 01.3211 + +0.2(1 - 01 02.3) + 012202.111 01
- -1,,(1a1,,))2133 21,201,311 02,3) + 01,202,331,,221 pn
eepfNe M2 x [(1 - 023)22 02.382225 mae x explps x -1)
(5)
which is the combination of three main terms Tx, (1), Tx, (1), and Tx, (1) given in Theorem 2 Here Tx, (1) contains all possible path information from X, to all its descendants while Tx, (0) denotes the vertex is 'close" which will not be reached at the current path. Theorem shows that for each Tx, (1), it enumerates all possible combinations of 'open' and "close' of the children of X; recursively. For example, in terms @ we have X₂ 'close' and X3 'open' such that only 23 appears in this term and obtain 0,1,3(
(11,2)1122 23. Formally, provide the graphical implication of the closed-form solution of PGF as follows:
implication of closed-form solution of PGF). Given a random vector X = [X1,.., Xa) following PB-SCM, for any subgraph G(L, E) with the subset of vertices L S V such that the i-th vertex is the root vertex in G(L, E), a component Cz IiEEA(1) 2j
Proposition (Graphical
2P5 with constant
C O, exponent pj Zt exits in Tx, (1), and only if there exists subgraph G(L,, E') with subset of the edges E' C E such that for each j L L,', (i), there are at least Pj directed paths from X to X, in the subgraph GLL, E').
Proposition D reveals the graphical implication of each term in the closed-fome solution of PGF that each term is related to number of directed paths in certain subset graph. In particular, for the
5

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highest order, in terms Q 21 indicates that there are two paths from X1 to X3 and one path from X, to X2 this triangular structure. which implies that X1, X2 is the cause of X3 offering a way to identify the causal direction by detecting whether 2 exists. In fact, the mamulant-basee method is exactly the method that identifies the causal direction by detecting the highest order of 2; using cumulant (Qiao et al. (2024b)).
However, one drawback of detecting the highest order is that the highest order of 2 does not always contain full identification for PB-SCM. For example, in Fig. (a), since ž 21222] 2223=2 has the order of while 21 22, 222s only has an order of 1, we can only conclude the causal direction from X2, to X1 but not causal relation among X1 X2. X3 using merely the highest order. Fortunately, the causal structure X1 + X2 t X3 remains identifiable since the term 22 does not exist in the PGF. This indicates that there is no directed path starting from any vertex that passes through all three vertices X1; X2, X3. In this case, we can conclude that it must be the collider structure and allows us to identify the causal structure by detecting whether such a term exists. Another drawback of detecting the highest order term that it requires multiple derivatives of PGF or complex construction with higher order cumulant, hindering the analysis of the identifiability. In this work, benefiting the local property of the PGF, we explore a consistent way to identify the causal structure while only lower-order information is required.
4.3 Local Probability Generating Function
One of the attractive properties of PGF is the ability to isolate and examine some specific components by strategically setting the other variables 2 approach zero within the probability generating function, which allows us to investigate the local structure and devise a practical structure learning algorithm. Formally, we define the following local PGF:
Definition 3 (Local Probability Generating Function). Given random vector X = [ Xa)T following PB-SCM and its PGF (z) The local PGF of vertices C [d] is the pointwise limit of (z as the Z,z. approach 0, 1,1. = {2 E [d] L}, denoted as: G (z) = lim -0 Gx (z).
The local PGF can be constructed using the original PGF by setting the limit of z. By this, similarly to Theorem 2 we have the following closed-form solution for the local PGF:
Theorem 3. Given random vector X = [X1..., x,! following PB-SCM and its PGF Gx(z). For a subser L C [d] of the set of vertices, and the set of the children of vertex within L, denoted by (i) = Ch(i) O L, the local PGF of L can be expressed by:
II iEL
exp {m x (T+, (1)
1)}
II <E[d]LL
exp( -xh
(6)
where (1) Zj
Σ sc (0. 1) tl jEChu())
II a T (8)
II jECH))NOL))
(1 (iss) and T (0) = 1.
Taking Fig.
as an example, the local PGF of vertices L = Xa} is c(X2.Xs)
exp(-<1} (xpp]/2[11 و2,3)22 + 02, - U22 exp(μg²3 - Ha Such local probability gener- ating function allow us to analysis and to identify the local structure, e.g., the term #2 in the local PGF G (2) represent the adjacent relation between X2 and X3.
In the following sections, we will further explore the identifiability of several specific structures that serve fundamental local structures within graph.
4.4 Identifiability
In this section, we address the identifiability of the PB-SCM using the PGF. Our focus is on identifying three fundamental local structures to reconstruct the causal graph from causal skeleton to causal direction: (i) the adjacent relation, (ii) the local triangular structure, and (iii) the local collider structure.
Adjacent relation. We first address the identifiability of the adjacent relation, which is involved in identifying the causal skeleton. For each pair of vertices X,, X, X, the component 2ız appears in the PGF if and only if X, and X, are adjacent, indicating that there exists a path either from X; to X, from X, to X.. Therefore, can detect the adjacency relation by testing whether the second

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partial derivative 8 log Ba,t2y equal to zero in local structure. In order to construct a hypothesis test for such condition, formulate the condition rank condition. Formally, the adjacent relation can be identified using the following thcorem:
logg Gig)
Theorem 4 (Identifiability of adjacent vertices), Let Xi, X3 E X be two arbitrary verlices with the
ack (2)
(z)
corresponding local PGF Glid) (z). Define the matrix A (1.) =
Be
with
D2g
24, 2j approach 1, the condition Rank(AUl.)) if and only if X4 is non-adjacent to X3-
Intuitively, Theorem identify the adjacent relation by detecting the existence of 2;zj in the logarithm of the PGF, i.e., the Tx, (i) inside the exponential function. By this, the identification of the causal skeleton is given. Next, show the identifiability of local triangular structure.
Triangular structure. For any three vertices X,, Xj, Xx E X forming a triangular structure. Such a structure must have one and only one vertex with an in-degree of 2. This asymmetry is captured by the second-order derivative of local PGE, e.g., 2ı25z where X has an in-degree of 2. By this, the causal direction X; - X2 and X3 X2 in the local triangular structure is identifiable as follows: Theorem (Identifiability of local triangular structure). Let X,, Xx X form a trian- gular structure with the corresponding local PGF ((i. (z). Define the matrix B(i.))) a (z)
G (lkssih(2)) (z)
dee of Pok
with Ess Rgs approach 1, the condition Rank Rank(B((...)) = 2 if and
OG (GGL (x)
Ozk
0ct
only if Xg is the vertex with an in-degree of in this triangular structure, iee, X X
Note that the causal direction in the triangular structure is not fully identifiable because it does not exhibit asymmetry and can always construct an equivalent PGF in the reversed direction which also discussed in Qiao et al. [2024b]. Next, we consider the identifiability of the local collider structure, which is also referred to as the unshielded collider structure.
Local collider structure. Given three adjacent vertices x--x--X; If they form a collider structure X, Xj+ & Xx,, the corresponding pattern in the closed- form solution of PGF is 2,25 2j2x. but no Zzzzzk, reflecting the absence of non-block path among these three vertices. Thus we have 8 log (() 0. Similarly, 02.0at
construct rank condition for identifying such local collider structure using the following theorem:
Theorem (Identifiability of local collider structure). Let Xi, X,, Xx X be three adja-
cent verlices with the corresponding local PGF G(3.M))) Define the matrix Cli...)
G (k....k) (z)
oafeii (x)
Bes with 24, 2j, 2 approach 1, the condition Rank ((iJ.))) = 1 if and Rank((((J.A))
aG( (2)
(((
Use
D2,0ak
only if the vertices X,, X,, Ng. form the structure X; - X, <- Xg.
Combining the identifiability in Theorem 5 and Theorem of the local structure, we conclude the following graphical implication of the identification of PB-SCM.
Theorem 7 (Graphical implication of identifiability). Given a pair of adjacent vertices X;, X, E X following PB-SCM, the causal direction of X; + Xj is identifiable if there exists verter X E X (X,, X,) such that X₃ + X3.
4.5 Learning Causal Structure of PB-SCM Using Probability Generating Function
In this section, we propose a practical algorithm for learning the causal structure of PB-SCM using PGF. Note that the probability generating function can be estimated by employing the empirical probability generating function (Nakamura and Pérez-Abreul (1993). The algorithm is given in Alg.
I Our algorithm involves two steps: learning the skeleton of DAG G using the result developed in
Theorem and inferring the causal direction using the results developed in Theorem 3 and 6
Learning Causal Skeleton. To learn the causal skeleton, following the Theorem 4 construct
the matrix A (td) for each pair of vertices X,, X, E X, and assess its rank to determine adjacency
7

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between the vertices (Line 2-4). Notably, learning the causal skeleton is efficient because each pair of vertices requires only one examination.
Learning Causal Direction. Given the skele- ton, we orient the causal direction following the Theorem S and Theorem [ Our initial focus on orienting within triangular structures. For each triangular structure, we enumerate the ma- trix ( to test whether X5 has indegree of two based on Theorem 5 and orient X; X and X, - X if detected (Line 5-7). After the focus on orienting the remaining undirected edges fol- lowing Theorem
Algorithm 1: Causal Discovery for PB-SCM
Using Probability Generating Function
Input: Data set
Output: Causal Graph G
1 G + emply graph;
// Learning Causal Skeleton do
for each pair of vertices i (1a))) a then if Rank condition Rank(A A (+5)
orientation in the triangular structure,
Add undirected edge "X; X," in G
O Specifically, for each undi- rected edge X; x=x j. we first consider testing the causal direction in a pattern like X; -x, < Xx. Then, we consider testing the causal direction in pattern X; X, Xx. Such test is con- ducted by constructing the matrix C(iik)) and to test whether the rank is 1, and then orient X; + X, f Xx detected (Line 8-10).
based on Theorem
Learning Causal Direction for each triangular X., if Rank condition Rank ((..k(B)).))) o then
X. X,,
E do
Orient "X, Xx X," based
on Theorem 5
s for each structure X. - X3 t- X2 E or
X, X3 Xx E G do
Rank Test. To test the rank of A 6., B and ((i.k) we employ rank test by testing 10 whether the second (minimum) eigenvalue X2 is zero, i.e. Ho A2 0 v.s. H1 X2 * O. Since 11 Return G the trace of the matrix converges to a normal distribution based the central limit theorem, and thus the sum of eigenvalues is also Gaussian. Thus, if Hy true, we approximate the eigenvalue as a normal distribution with zero mean. We further
if (ank((((63.))) = then
Orient "X; + +, + Xx based
on Theorem
employ bootstrap method (Efron and Tibshirani [1994) to estimate the variance of such distribution
by calculating the bootstrapping statistic 22 from N resampling dataset D E (Di [D C D, and estimate the variance of Ag. Building this, the p-value of X2 from the original dataset can be obtained.
Complexity Analysis. In the step of learning skeleton, we determine whether each pair of vertices adjacent by testing the rank of the matrix following Theorem 4 and hence the complexity of skeleton learning is ([jdddd 1)). In the step of learning causal direction, we consider the complete
graph, where there are (3) d(d-1) triangular structures, and for each triangular structures, test the rank of the matrix for three vertices following Theorem and hence the complexity is o( ((=ld-1)(d2)). Similarly, orienting collider structures from a complete graph, we choose three 2
adjacent vertices and test the rank once following Theorem hence the complexity is O ddd-1)(d2)),
Therefore, the total complexity is ( - 1) + 2d(d
3
Experiment
5.1 Synthetic Experiments
In this section, we test our proposed method on synthetic data. We first conduct control experiments on synthetic data evaluate the sensitivity of our method to sample size and different indegree rates. Following this, we present case studies involving 3, 4, and 5 vertices to further illustrate the identifiability our approach. The baseline methods include the cumulant-bacee
method (Cumulant)
Qiao et al. (022bb ) OCD (Ni and Mallick (2022). ), PC (Spirtes et al. (2000). ) GES (Chickering 2002
Sensitivity Experiment In the sensitivity experiment, we synthesize data with fixed parameters while traversing the target parameter. The default settings are as follows, sample size=30000, number of vertices=10, indegree rate=3.0. range of causal coefficient Os.j E [0.1, 0.3), range of the mean of Poisson noise μ4 E [0.05, 0.15). Each simulation is repeated 15 times.
8

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Table 1: Sensitivity to Avg. In-degree Rate.
SHDL
Avg. In-degree
2.0
2.5
3.0
3.3
20
330
3.5
Ous
0.74
0.81 0.07
0.86 0.00
0.89 = 0.04
9.07
1.99
8.33
2.28
6.40
1.08
5.27
1.39
Comulant
0.73+
0.77
0.00
0.83 0.03
13:40
14:10
13.00
13.20
2.23
PC
0.00
0.62 0.11
0.51
0.122
9.90
= 3.45
11.80
2.48
15.90
3.91
16.10
4 3.21
GES
0.48
0.48
0.11
0.41
0.37 0:010
14.90
4.48
19.50
4.61
25.90
4.18
30.5
4.06
OCD
0.23 0.22
0.27 4 0.23
0.28 0.166
0.37 2 0.14
16.10
70
19:40
5.50
23.60
1.62
24.30
4.67
Table 2: Sensitivity to Sample Size.
F11
SHD1
Sample Size
5000
15000
30000
50000
5000
15000
30000
50000
Ours
0.76 + 0.09
0.82 = 0.04
0.86 = 0.03
0.87 + 0.04
11.50
+ 3.34
9:27
2.37
6.40
1.68
5.87
1:25
Cumulant
PC
0.45 1: 0.11
(154 0.11
0.54 0.13
0.66 1: 0.09
19.50
4 4.90
15.70 4: 3.77
15.90
1: 4.12
13,00
2.79
GES
0.30 + 0.10
0.44 + 0.20
0.41
0.11
0.43 +
23.70
4.14
22.70
7.85
25.90
4.41
8.84
OCD
0.30 1: 0.12
0.35 1: 0.18
0.28
0.16
0.38 1 0.20
21.90
3.35
20.80
4.57
23.60
4.62
20.90
5.74
the control experiments on the average in-degree given Table as the average in-degree controls the sparse of causal structure, the higher the in-degree rate, the less sparse in causal structure leading decrease of performance of the baseline methods, PC, GES and OCD. In contrast, our method and Cumulant show improved performance as they benefit from the denser structure which provides more identifiable structures. Additionally, our method outperforms Cumulant by overcoming identifiability limitations through leveraging PGF, demonstrating better performance.
In the control experiments on sample size presented in Table 2 our method's performance improves with increasing sample sizes, consistently outperforming all baseline methods. Furthermore, it surpasses Cumulant under the same conditions due its efficient identification of directions in local structures without relying on high-order statistics, thus enhancing accuracy.
Case Study To demonstrate the identifiability of the proposed PGF-based method, present studies using causal graphs with 3, 4, and vertices. The results of our method and the baseline methods are summarized in Table
Table 3: Case studies of causal graphs with in total 3, 4, and 5 vertices, respectively. Red undirected edges indicate that adjacency has been learned but the direction cannot be determined, while red directed edges indicate incorrectly learned directions.
Causal Graph
Ours
Cumulant
GES
OCD
Generally, our PGF-based method successfully identifies almost all the edges, except for X; + X2 in the graphs with 3 and 5 vertices. This because there is neither a vertex with indegree of to provide causal asymmetry. nor an additional vertex to form the local collider structure, which aligns with our theoretical results. Notably, the cumulant-basse
method fails to identify the structure X1 + X2 t X3 in the 4-vertex graph and the edge X, + X, in the 5-vertex graph, as it cannot leverage low-order information. This limitation. however, is successfully addressed by the PGF-d method, which can utilize such information effectively. Regarding other baseline methods, PC and GES encounter difficulties in identifying the Markov equivalent class, while OCD illustrates identifiability issues in PB-SCM.

---

5.2 Real World Experiments
In this section, we evaluate the performance of our proposed method on two real-world datasets to assess its effectiveness in realistic scenarios.
Football Events Dataset We test the proposed method on a real- world football events datase which includes 941,009 events from 9,074 games across Europe. The experiment focused on analyzing the causal relationships between specific events such as Foul, Yellow card, Second yellow card, Red card, and Substitution, as depicted Fig. The goal is to identify causal relationships from the counts of these events.
F
R
S
Figure 4: Football Event
Graph (F: Foul, Y1: Yellow
In detail, we orientate in the local triangular structures Y, - Y/2 - F, Y/ R- F, and the local collider structure Yi - S R. As a result, Our method successfully identifies adjacent vertex Foul First Yellow Card and other six causal directions, which consistent with our theoretical result.
card, Y2: Second yellow card, R: Red card, S: Substitution)
Shopping Mall Paid Search Campaign Count Data We also evaluate our method using shopping mall paid search campaign datase This dataset contains information from a five-month paid search campaign for U.S. shopping mall, spanning from July to November 2021. In this analysis, we focus on the count variables Impressions, Clicks, and Conversions, which represent fundamental count data in e-commerce scenarios. These variables exhibit the following causal relationships Impressions + Clicks, Clicks + Conversions, and Impressions + Conversions, forming triangular structure.
Our experimental results demonstrate that the proposed method successfully identifies the adjacent relation between Impressions and Clicks, and the other two causal directions. These results align with our theoretical findings, suggesting that the method has the potential to be applied to real-world scenarios involving count data.
6 Conclusion
In this work, investigate the identifiability of the Poisson branching structural causal model using the probability generating function. We derive a nontrivial closed-form solution for the PGF of PB-SCM and further establish the connection between the closed-form solution of PGF to the causal structure, showing that the closed-form solution of PGF encompasses the path information with various subgraphs. With this connection, we employ the local property of PGF and propose simple yet efficient way to identify the local causal structure of PB-SCM by constructing matrix with rank test. By this, we provide a practical algorithm and a hypothesis test approach for testing the causal structure and verifying the effectiveness of the algorithm via synthesis and real-world data. The proposed theoretical results take a meaningful step understanding the causal mechanism and completing the identifiability result of PB-SCM.
The main limitation of this work is that the explicit estimation of the probability generating function does not scale well to the number of nodes. Developing sample-efficient estimating method could be a promising direction. In addition, causal faithfulness and sufficient assumption may restrict the usage of this work in a border scenario, which needs to be processed with extra detection steps to eliminate the effect.
7 Acknowledgments
This research was supported in part by National Key R&D Program of China (2021ZD011 11501), National Science Fund for Excellent Young Scholars (62122022), Natural Science Foundation of China (61 (61876043, 61976052, 62406080), the major key project of PCL (PCL2201A (2.2011112)) A12). We sincerely appreciate the comments from anonymous reviewers, which greatly helped to improve the paper.
mall-paiddddlllaaaee -ai---paaachb-aaaa search-aampaign-dataaa
10

---

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---

Appendix
Table of Contents
A
Proof of Closed-Form Solution for PGF of PB-SCM
13
A Proof of Theorem
13
Proof of Theorem
17
Proof Theorem
Proof Proposition
20
B
Proof of Identifiability
21
B
Proof of Theorem
22
Proof of Theorem 5
22
B.3 Proof of Theorem 6
23
4 Proof Theorem
24
B.5 Proof Lemma
24
C
Additional Experiment Details
32
Additional Discussion
32
D.i Discussion on the Benefit of Local PGF
32
Discussion on the connection cumula ant based method
E Broader Impacts
33
A Proof of Closed-Form Solution for PGF of PB-SCM
A.1 Proof of Theorem
Theorem 1 (Closed form of PGF within PB-SCM), Given random vector X = [X1,.., Xnn
following PB-SCM, let Z()) {aill E Des(j) (i), the PGF of P(X) is given by Gx(z)
IeTt Ge 24 x (IsecnosGis(Zss))
where
GB(as,s) (zg x IINECRG) G))(((())) GB(as,))
Ch(j) #
Giz(z())
(A.1)
Otherwise
in which (.) is the PCF of Poisson noise e; and G Goaz)) (-) is the PGF of Bernoulli distribution with parameter oi.
A.1.1 An Illustrative Example for Deriving the Closed Form
Before formal proof, we first provide an intuition of proof through detailed example. Deriving the closed form of multivariate PGF involves decomposing the expectation using the law of total expectation. This ensures that each decomposed expectation involves only one random variable, each of which has a closed-form PGF, as most univariate PGFs admit a closed form. We first introduce some closed forms of univariate PGFs used in the proof. Specifically, the univariate PGFs of the Poisson and Bernoulli distributions are given by: Gz(z) = E[2*] cxp(μ(z - 1)) when + az where B(a), the Bernouli distribution
Poisson(() and GB(a) (z) = E[z"] with parameter a.
Consider the triangular structure in Fig. 5 the corresponding PGF is expressed as follows:
Gx E
1: 01.20111111000. 41 22 +o2 t02
22
(A.2)
13

---

X1 C1
v1.2
0133
X2 a1,20X1
X3 01 t
Figure 5: Example triangular structure.
Since PB-SCM is kind of additive noise model, we can represent cach variable by the noise of ancestors with the corresponding coefficient along with the causal path. Next, since noises are independent of each other, by rearranging the components that share the same noise, can further decompose the expectation:
Gx (z) E [1 22
20,2 00,3001 02,3001 2001
23
23
E
2002
E [23
[2
(A.3)
This form reveals the underlying causal mechanism, where the power of 2j in the expectation involving = is the noise component from X, to X, with the corresponding path coefficients. That is, each expectation captures the noise's potential influences, along with their respective causal paths.
To analyze this expression more closely, we focus on the first expectation involving 21' in Eq. A.3 By applying the law of total expectation, we decompose this expectation, ensuring that each decomposed inner expectation depends on only one variable:
E
E1 2002 01,390 _,02,900,2001
22
Zg
2g
01,300
[c]
[22
01,200 02,3001,
2g
[a]]
(A.4)
= =E
[
F3
31,399
la]
E
[22
[22 E
[2g
02,3001, L00
1012 o a] [q]].
An immediate observation from Eq. A.44is is that, to establish conditional independence and decompose the expectation, condition the variables following the causal order (e1 corresponds to X₁ and 01, o 6j corresponds to X2). thereby encapsulating the graph structure within the PGF.
Next, each decomposed expectation involves only one variable, e.g., E [2] 0 [G] is the PGF of 01,3 o fjle1 Binomial X, a)., has closed form PGF (2)))) ) Similar other expectations, we finally obtain the closed form of (x(z):
Gz) E [1] ( (zs) x E [12 GB(ozs) G (23))1120111))
E [|2] G GB(a (zs) GB x GB (33))]
(A.5)
G., (2) x G GB( (2a) x GB(01))
GB(oza) (23)))
Note that, [[2 GB( ()) (2s)] [a] is PGF of 012 a// where the input is 22
G GB(a,,)) (23). Similar to this, we have:
L2 02.3062 22 Fg
[
02. [ea]] = E
[ G ((ans) =
(A.6)
and E [2 Ge, (23). Then the close form of PGF of Fig. 5 is as follows:
Gx(z) =G., (z) x (B(o)ss) x G(a x (Bo)))
(A.7)
x Ge (22 x G(( (23)) (23).
Therefore, to derive the closed form of PGF, we need to apply the law of total expectation following the causal order, which constructs a recursive formula. To formalize this in a recursive formula, we further define some graphical concepts.
Notation. We use pi-j_ = pimj
denotes the set of all directed paths from vertex to j.
where pi-) (i, k1, 12, ... kp: p k- | denote the k-th directed path from vertex i j. For each directed path Pi we use A (a A2i 0xk denote the corresponding coefficients sequence of path Pi-j. We let pi--i (Pimi) also be valid directed path for simplicity. Besides, we use A)-) X; aK o .... o akk A2 Oki,bs X, denote to perform consecutive thinning operation X, based on the path sequence.
14

---

A.1.2 Formal Proof
Following the definitions, the PGF of joint distribution P(X1, ((X1,.. X,,) under PB-SCM is given by Gx(z) = F Fl=x" =. Since PB-SCM kind of additive noise model, we can represent each
variable by the noise of ancestors with the corresponding coefficient sequence along with the directed path:
phoni
X;
Ap honi o En + Eis
(A.8)
AEAn(S)
pml
where An(i) denotes the set of ancestors of vertex i.
Next, by rearranging the components that share the same noise, can further decompose the expectation as noises are independent of each other:
Enerei Ejetmi ocn tel
Eneanin)
Slphiny AB pmd
semeecan ten
Gx(z)= = E 1
x x Zn
21(pfm51 Af oes
p
(A.9)
= E
E
3j
[E(n]
jeDex(
xE(ni [n]
jeDes(() p=1
where (is(1) is the descendent set of vertex i. We start by introducing the following Lemmas.
Lemma 1. For each
ts Il
11, 2j
we have:
pi-t
E
3j
A p ore or
= Ga
2i
of
II
2
A
oEn
(ars)
jEDes(s)
p=1
jECh(()
keDee(()
p=1
(A.
Proof. We first condition & using the law of total expectation and obtain:
pinti
E
II p=1
1/
oc,
E
2
II p=l
2j
p
oc:
C2
ieDes(()
jeDes())
|
E
EE
II kgDes())
24
A
oar 060
Ci
LiEC(())
p=1
(A.11)
Now, regarding the definition of thinning operation ai,i X Bernoulli(a) are i.i.d. Bernoulli random variables, Eq. A. can rewritten as follows:
Σx
(o..))
where (ars)
1
p
II
II
2j
sivis
p
Des
II JECK(()
E
(ang)
II
II p=1
2k
De
A bi
Ei
JEDs(()
p=1
k=Dos(j)
|
E
II E
61
II
A
of
JECK(()
n=1
D=DDs(f)
p=1
n=1
|
(a
(
E
E
2
6;
JECK(()
A Des(,)
p=
(A.12
15

---

Since we condition the e,, which can be regarded as constant in the expectation, we have:
E
2
II
E
II
II p=1
2
A
(ois))
C2
(i)
p=1
jECN(i)
D=Des())
E
2/
II
E
25n
II p=1
om
JECN(S)
D=DDs()
|p/
E
II
2
A
okn
(ain)
JECK(()
D=s
p=1
AA.13)
E (m
A A
on
(ai.))
which is the PGF of et, with the input being 24 [LECC))) Therefore, we finally have:
II==De(())
-1p=1
2R
| II
1
of
E
II
= Gas
II jECh(()
2
Des(s)
A DeD()))
p=1
(A.14)
This completes the proof of Lemma]
L
Lemma 2. Let Gt,(Z()) = E oj IR
we have:
GB(an)) (z) x INECCO Gga(2(x)) G)(ans)
Ch(j) +
Giz(z())
(A.15)
Otherwise
Proof. (i) Cf(j) 0, which means vertex j has descendant, have:
ppi-ly
Giz
E
2j
Em
(mL
II D=Des())
II p=1
2k
# of
(asa))
E
(A.16)
where c(ans) is the Bernoulli distribution with parameter ai,j. Therefore, E 25n
the PGF of
E (ora)
Bernoulli (a..)) with input 2j. we have G. (z())
G (())
(ii If Ch(j) A 0, we condition the Bernoulli variable (ass)) and obtain:
GLy(2()) =
Af fonk
osn
c
Zi
Sti
keDes()))
p=1
pirnd
(mi.)
0 (oes)
A
oa
(ans)
c(ars)
2j
Zk
21
Sn
RECh(j)
reDes(k)
p=1
(A.17)
16

---

Similar to Lemma as follows:
have aj,t (ars)
E
(oin)) ((),a))
Sn
Therefore, Eq.
A.17 can be rewritten
En doup
ppletl II p=1
Ap-log, (nsiee)
Gin(Z()) = E
{
5n
II
E
2g
Sn
clara)
keCo(()
(eDes(k)
Sn 2.2
E
]
()
II E
Ea
1.a
j.
II (eDes(k)
21
A
of (.)
kECh()
p=1
ppkntt
A s (
og
E
E
2k
kECh())
IEDes(R)
p=
(A.18)
which is the PGF of Bernoulli variable (aaia)
En
Therefore, we have
[pkmt
Ap A (a3,a)
Grz(Z()) G((a))
2j II E
II (EDos(K)
II p=1
2
obn
(A.19)
kECH())
where E
*k
HEEDES(R)
I1,
p=1
A
agn
is Gz,((Z())) according to the definition, that is
Gz)(((()) = B(
2j ILLECmO) G3
(A.20)
which completes the proof of Lemma]
Regarding to the Eq. A.9
by leveraging the Lemma 1 we have:
pinsi
jimior
Gx(z)=
II E
2
II jEDes(t)
II p=1
2j
(E(m]
(A.21)
p
II Ge.
II
E
(arg)
II
2j
1/
te (ans)
sE(m]
(ECh(s)
kEDes())
p=1
Next, according to the Lemma
we have:
pp II p=1
Gx(z) II Ge,
II
II
ko
(01,))
II Ge,
& II Gis(Z())
2
2x
KE(n)
jECb(E)
kEDes(j)
RE(m]
jECb(() (A.22)
Grion (2) (IIEEC Gza(Z(w))
Ch(j) #
where Gi, (z.())))
This completes the proof.
Otherwise
A.2 Proof of Theorem 2
Theorem 2. Given a random vector X [X1,.., Xa) following PB-SCM, the PGF of P(X) can be expressed by:
Gx(z) II
explu x (Tx, (1) - -1)},
(A.23)
[E(
where Tx, (1) 2; 2. (0,1) Hs a Tx (8;). Tx, (0) and o ai. if and
a aijj if = 0.
17

---

Proof. According to Theorem 1 we have
Gx(z) = II injn) Ge (x x II SECRO Gi)(R()))
(A.24)
Since (-) is the PGF of Poisson noise es, we have:
Gx(z) II [Ein]
exp {
24 II Ga)(e())-1)
(A.25)
Therefore, to complete the proof, it suffices to show that:
(1) x II ILSECNC Gi)(Z())
(A.26)
holds for each vertex in the recursive process.
We proceed by structural induction. Specifically, first show that Eq. A.26 holds for the leaf vertices. Then, assuming that the expression holds for each child vertex of parent vertex (i.e., j E Ch(i)), demonstrate that the expression also holds for the parent vertex
Base case Given an arbitrary leaf vertex since Ch(l) 0 and by the definition of (1, we have Tx, (1) 2 x 1. Therefore, Eq. A.26 holds for the leaf vertex.
Inductive Step Next, Assuming that for each child vertex j of the following holds: Tx, (1) 2j x IIRECRS) G),(Zz()) Our goal is to prove the Tx, (1) 2, I[JoC(() Gty (z()).
When Ch(i) * 0, according to the definition of Gi.(Z()) we have:
II SECo Gig(z()) II jECM(I) GB(o,,)
2j II G3a((()))
(A.27)
II SECH())
(1- Faggg I Gualew))
By expanding Eq. A.27 we have:
G,A(((0)) ] (A.28)
Il SECO G(y(Z()=
Σ
II
(1 --, II
II
ass
2j kECY()
SCON(1) (JECK()))
jES
where S represents all subsets of Ch(i), corresponding all possible combinations of child vertices of vertex
8s Next, regarding the definition of Tx, (1), since 0g
[oi.j Cij
8j =
we have:
$11
8j 0"
(1, =
Σ
II
5 T (sj)
BE (E (xe) jECh(i)
Σ
0 Tx Tx, (0)
II JES
a Tx,
(A.29)
SCCN()
= 2, Σ
II
(1 (1-)
II
Oii
II kECA())
Gz,((Za))
SCCh()
(iECH(O)S
JES
given (1) 2j x IILECHN) (z))) Then, comparing Eq. A.28 and Eq. A.29
we find that:
(1) 24 II JECA(() Gis(Ze)).
(A.30)
Substituting back into A.25 we have:
Gx(z)
II (E(n)
exp H4 2 II LEcmC) (()(=())-1)
(A.31)
II
explu: (1) 1)),
(E(n)
which completes the proof.
18

---

A.3 Proof of Theorem 3
Theorem 3. Given random vector X [X X,,", following PB-SCM and its PGF Gx For a subset L C [d] of the set of vertices, and the set of the children of vertex within L, denoted by Ch Ch(i) O L, the local PGF of L can be expressed by:
(z
II EL
exp 14 x (1) 1)}
II exp(-Mi3:
(A.32)
<E[d]LL
where T (1) 2z
Σ
II
O (s,)
II
(1 (1-ai) ai,) and xx2 1.
se (0. (16,)) (ECh..()
Proof. According to the definition of Local PGF, we have:
(x(
lim 2.1. -0
ILLEN Ge 24 [ Gas(zn))
lim II 2,-
Gro
Zi II SECCCO)
II (is(E)))
II jE(dXL
GO zi x II IIyECnS Gis (as(zu))
VEL
Ge lim 2,1
2 II II Gi)(Z())
II jE(d]L
lim Ge 2 II I Gis(z())) 20-0
(A.33)
Since 2j approaches zero for each [d] L, we have:
II JE 20
lim Ge
x II, JECK(()
(ESECME) Gi)(z()))
(A.34)
{ = II lim exp 1ij (1))2))) x ILLECNO) GRa(Z(n)) 1 -1)} jEIdAL #0
II jEIdAL
(A.35)
Therefore, we have:
log (z) = lim II Ge Ge
II
II JEldAL According to Theorem it can be
exp(-m)
(A.36)
20
EL
Next, we consider the lim Ge 24 x IIjecco (()(()))) #--0
expressed as follows:
lim Gr. zi x
II
Gig((()) G
=0-00
jECH(1)
= lim (Tx; -1)
=0-+0
Tx, (xx(1)-1)
(A.37)
= exp
Ui
lim
20
ofj (3,) (1))
= exp
Mi x
lim
Σ
II
(aM)) jECh(i)
Next, we are going to show that, when lim
Tx, (8j) = T (8z), we have
20
lim (1x = Σ II
a Tk
II
(1 (1-ot))= (1). (A.38)
=0-00
BE LE (EChL())
JECh(i),CCu(1
Let (hz = Ch(i) n L, we have:
lim Go 2i x
G(o(((s))
:)} (A.39)
lim exp =
Mi X zi
Σ
II
(x)))(()))
II
(x,x)(,)) -1
sE(0,1) (EChg())
(ECH(i)(CL(()
19

---

In scenarios where E Ch(i) \ (t) and s, 1, considering that 2j Z,I. and 2g - 0, (that is, 2j approach to zero for those vertices j $ L) we have:
lim (xy lim 2j -0
Σ II
(sk))
(A.40)
Be (0,1)1)) kECh(j)
Otherwise, when j Chy (i), we have:
lim (85) = lim 2j #L-0
Σ
II
3, Tx, (8k)
X,L 0
(E, (((g)) kECb(j)
Σ
II
ajj lim Txx (sk) z4-0
(A.41)
BE (0,1)10m)) kECh())
2g
Σ
II
ajk-xx Xx (sk),
(E001)) kECh())
and when E Ch(ii) \ (t) and 8, = 0, we have lim Tx, (0) 1. Therefore, we have: z0-0
lim
Ga
II G(L(R()))
#L 00
II (1-ow,) (1-oo)-1)) (A.42) JECh(1)(Ch,(8)
exp
14
Σ
II
lim Tx,(8,) = +0
=E(0,1) ((())) (9) (50h)()
Σ
II
(5)
II (1 (1-aw)- (ECh(()OO(())
exp
#E(0.1) (01 (8)
According to the definition of T we obtain:
lim Ge 2 II jECH()
(i.(( exp {m x (Tx 1)}
(A.43)
LL
By substituting Eq. A.43 into Eq.
A.36 we finally have:
log G (z)
exp {a (1) II exp(-14} (E(d)L
(A.44)
iEL.
which completes the proof.
A.4 Proof of Proposition] 1
Proposition 1 (Graphical implication of closed-form PGF). Given random vector X [Xi,.., Xal following PB-SCM, for any subgraph G(L, E) with the subset of vertices L C V such that the i-th vertex is the root vertex in G(L, E), component Cz [L 2/ with constant C + O, exponent Pj E Zt exits in Tx, (1), if and only if there exists a subgraph G(L, E') with subset of the edges E' E such that for each X, in the subgraph G(L,E")
{i}, there are least Pj directed paths from X to
Proof. We consider the subgraph G(L, E') as a local structure associated with the node set According to Theorem 3 we have;
T =1)=2
Σ
II
II (1-ay) (A.45)
ChC (EChL()
We select a subset of edges E' C E, indicating that edges in E' are 'open' and edges in E | E' 'closed' within the local structure containing the vertex included in E' are considered non-existent in the subgraph. Consequently, in Eq. A.45 we have: (1) 8, = =1 Tx (s; 1) if the edge j is in E', and (2) a T (s, 0) if the edge i + is not a
L. This means that edges not
20

---

in E', That is, each subset E' corresponds to an element in {0, (0.1)10)(0)) representing a possible combination of whether each edge 'open' or is 'close'.
We describe the term Cz 41 2) through recursive process. Initially considering the children of X,, there exists term x (1) as follows:
II
(1)
II
ax (0)
II (ECh
(1-aus)
(t) 1-JEE
(EC
=
Σ
II
ojk
2 T (oa)
(xx())
II (1-a4))
II
a1.)
(E)
se(0,))ich 511 kECh),))
JECH,
JECR()
Constant coefficient
CC
II
3j
E wEfo, Gil kcChE()
II
Txe (8k)
(EChL()
(A.46)
Continuing from the Eq. A.46 for each term in the product expansion, similarly expand based on the children of Xy. This expansion includes the following term:
CX II
II (EChLU)) j-SkEE
II AEChL J-KEE
0 o Tx, (0)
II kECHG)ChL()
(1- (jj))
(EChL(()
(
Comsiant coefficient
= C x 3
II
2j
II kEChy() j-tkEE
2k
E
II
II JECK(H)(Chy
(1 - ax.i)
(EC
sE(0,1)iCh,, (k)
i-J
Cxa
II (EChL())
2j
II kEChL()) J-KER
26
Σ 10 h (kC)L(k)
II
At T (BL)
II (ECK(H)(CNL())
(1 11-a0 oxe
i-JER'
(A.47)
The term "z: [LEch INECHLD zx" describes how X; can reach its descendants X through i-jEB'
j-tkEE the children Xy. This shows that all reachable vertices are expressed in the form of the product of z variables, indicating that in the subgraph G(L, E'), the number of paths X, takes to reach its descendants & Des(i) is given in the power of zg- Thus, the corresponding term for the subgraph G(L, E'), Cz, I[ 2j is included in Tx, (1), where represents the number of paths from X; to X, in the subgraph G(L, E'). Conversely, if such subgraph G(L, E') does not exist, then Tx, (1) will not contain such a term C2, [[JEL 2j since the number of directed paths from to
Ps
j is less then Pj. This completes the proof.
B Proof of Identifiability
We first introduce the following necessary Lemmas for our proof.
Lemma 3. Given two vertices i,j E V and the corresponding local PGF ci:3) (z). log cfi- if and only if vertex non-adjacent vertex j.
02,05,
Lemma 4. Given three vertices i,j.k k E V and the corresponding local PGF Glhaikt
8 log G (())))))()))) 0 and only if vertices i, form the structure + +- k.
Oz,
Lemma 5. Given three vertices i,j.k E V and the corresponding local PGF (hi.i (z). 8 log Cli-sik) + if and only if vertices i,j, form the structure i + + j and vertex is adjacent vertex j.
21

---

The proofs of Lemma Lemma and Lemma 5 are deferred to section B.3
B.1 Proof of Theorem 4
Theorem (Identifiabilityy of adjacent vertices), Let x, X, E X be two arbitrary vertices with the
(i.i)) (z)
ack (x)
corresponding local PGF GASS)) (z). Define the matrix (i.))
8 2 ( (x)
Ba
with
Dzg 2j approach 1, the condition Rank(A = if and only if X, is non-adjacent to Xj.
dz,02;
Proof. Considering the partial derivative of log G (KJ) (z), we have
i log Gli
aG* (z)
1
(B.1)
Ozi
(t
(z)
Ozi
8 log G (z)
1
G
0
fist (z)
1
ack
(z)
aox
(ij) (z)
(B.2)
02,02,
GX
fi,s)
(z)
0z,02j
G
(z))
Oz,
Ozj
log cf (x)
If part that is
If vertex i in non-adjacent to vertex j. according to Lemma
3 have
0.
Ø2,82;
(z)
aGfwi (z)
x
= O.
(B.3)
G {ij)
02,02,
c+ (z))
Oz,
Øzj
By rearranging the equation, we obtain:
PQ (z)
aG (z) aG 0.
cl
(B.4)
O2g02z
Oz
Ozj
cllii G (
Notably, Eq. B.4 is the determinant of matrix A (6.)
and the
dag
det(A (A.)) 0 means that the rank of matrix A (6.) is 1, i.c., Rank (A (1r) = 1. This com-
pletes the proof of the if part.
Only If part Rank(A (A (151)
have det(A (151) O. which means:
(bd) (z) 2 ( (z)
actli actiu 0
(B.5)
θz,0z
Ozi 0z,
(att (z) we have:
Divide both sides by
0 PQG (iit (z)
actin (z) acth 0
1
1
(B.6)
Glidl (z) dz,0z,
clio)(())
Oz,
2 log cff (2)
Comparing Eq. B.2 we have
Therefore, according to Lemma
X2 is
35,02,
non-adjacent to X,. This completes the proof of Theorem
B.2 Proof of Theorem 5
Theorem 5 (Identifiability of triangle structure). Let X,, Xx X form triangu-
lar structure with the corresponding local PGF cdi.k)(2)) Define the matrix B (b.a)
22

---

G (z)
oak aatisi) (x)
oa (x)
R po (x)
with 2ss 2j. zn approach the condition Rank 2 if and
Osk
azt
only if X5 is the vertex with an in-degree of in this triangular structure, i.e, Xi -xx Xj.
Proof. Considering the second order partial derivative of log G ((i.k)) (z) with respect zko we have
GliJ. i log
G
0G
(z)
(B.7)
azk
GX
2
Zzk
@ log GLU.M (z)
G
0
(z)
1
(GG
(B.8)
02
Gx
Ozk
Ozk
If part If X,, x, X form the triangular where X is the vertex with indegree of 2, according to 8 lox Glisk.) (x)
Lemma 5
we have
*
0,
that is
8%
8d G (.J. (z)
[ (z) 2
* 0.
(B.9)
G
(...k)
Ozk
By rearranging the equation, we obtain:
che.h &. (z)
/acff (z)
(z)
X O,
(B.10)
Ozk
Ozk
Goft.ikk (z) acki (z)) Oek
aG aati (z)
ac O=t det(B (1.AA) * means the rank of matrix B ((AAA) is full rank, i.c., Rank (B (1..hi)) completes the proof of the if part.
Notably, Eq. B.10 is the determinant of matrix B (f..k)
Seffiiio (2)
Beg
and the
= 2. This
Only if part If Rank (B())))) = 2, we have
O Pa
(1s.M)
(z)
/ac (z) Ozk
2
det (B(().))) G. (1.ih))
(z)
# 0.
(B.11)
Ozb
Divide both sides by (aftsiti)) (z)
we have
82 82 glGii (z)
(ackhdii (2)
2
* 0,
(B.12)
G ((iik)
(z)
O23
(G
(z)) (z))²
Ozk
and hence we have 8 log (0 (z)
4 the vertex X;, Ni, Xa form the triangular where X₂ is the vertex with in-degree of i.e., X; X--XI-X, This completes the proof of the only if part.
f 0. According to Lemma
Oug
B.3 Proof of Theorem 6
Theorem 6 (Identifiability of collider structure), Let X1, X be three adjacent
vertices with the corresponding local PGF G (() Define the matrix ((i..A) =
G clissA)(2) (z)
aGta.3 (x)
Oz occ
with 2,, 3js 2 approach 1, the condition Rank (liijAA))))))) = 1 if and
Ozz
0s,0en
only if the vertices X1, X2, Xx form the structure X; + X2 t- X
23

---

Proof. The proof of Theorem is similar to that of Theorem 4 Glisikk) log Gx
Consider the partial derivative of
(z), we have
d log G
Cla.Ah)((2 (z)
1
aG
(z)
aG
(z)
02,0zk
GX
(z)
Gdi (x))
8z:
azk
(B.13)
If part If vertex i, k form the structure i - k. according to Lemma 8 log (li.)))) 0, that
we have
8 PGGis (z)
1
0G (i.iiitt (z) (z) aG
(z)
(B.14)
X cl (z)
Ō2,0zz
((is,))) (z)
Ōz.
θzu
By rearranging the equation, we obtain:
Gx ((..))) 8 &
(1.ik))) (z)
ac (z) (h.i (z)
(z)
(B.15)
02,0zk
dzg
Ozk
G (()
aafiai (x)
Notably, Eq. B.15 the determinant of matrix C(.AR)
and the
aG (2)
Pott (x)
O5k
det(C = 0 means the rank of matrix CliJA)) is i.e.e. Rank (CHid.)) = 1. This completes the proof of the if part.
Only if part If Rank( 1, we have det det(C()))))) 0, which means:
8 ( (z)
actiuk) (z) ac (k..k) (z)
Gl cli.h (z)
(B.16)
O2,0zk
Oz
Ozk
Divide both sides by (ot(di)t)2) we have:
j Cll.idhk (z)
aathiit (z) acts.dt (z)
(B.17)
Glaakt (z)
02,02z
G
(ha.h (z)
2
0z,
O2k
we have 0 log G (.f...M) (x)
Regarding to Eq. B.13
0, according to Lemma vertices i, & form the
dz.
structure i + - k:
B.4 Proof of Theorem 7
Theorem 7 (Graphical Implication of Identifiability). Given a pair of adjacent vertices Xx, E X following PB-SCM, the causal direction of X; + Xj is identifiable if there exists verter X X (X., X,) such that X x X,.
Proof. Given a pair of adjacent vertices such that X; - X,, suppose there exists X E X (X,, such that X Xj- We consider two cases: (i) X; is adjacent X₁ and (ii) X; is non-adjacent to X5: (i) If X; is adjacent to Xx. then X,, Xj, X5 form a triangular structure where X; + X, + X According to Theorem 5 we can identify that X, is the vertex with an indegree of 2, confirming that X; + Xy; (ii) If X, is non-adjacent to Xt. then X,, X,, Xg form a collider structure X, + X, +
Xx. This structure can be identified according to Theorem 6 which completes the proof.
B.5 Proof of Lemma and
We first introduce some notations and necessary lemmas concerning the limits of the local PGF.

---

Notation. For simplify, we define G2, (z()) == G.. 24 [LECCC))) (()(z))) as the component
involving & of PGF. According to Theorem A.1 have Gx(2) [liE(1] G (2(1))-
Lemma Considering the limit of function Gü, (2(3)) as 2; - 0, we have lim, Gū (2(1)) = e eTM
Proof. Since G22 (Z()) Ge (zs [[EEN())) Ge)(Z())) where G.)()) is the PGF of Poisson noise 6i, we have:
(2(2()) = lim Gr, (3) x II,ECCO) ( = (0) = =
(B.18)
=0
which completes the proof.
Lemma 7. Considering the limit fanction Gzz(Z()) as 2, - 0, we have lim G5,5 (2())
1 1-asj- as.j
Proof. Since Gz)(Z())) = GB(ass) (z) x ILEECGG) C3A(((())) where GB(ass) is the PGF of
Bernoulli distribution with parameter 0i,,, we have:
lim GB(a))) 20
2j
Gal(e()) G GB(a,,) (0)= = aij.
(B.19)
which completes the proof.
B.5.1 Proof Lemma
Proof. According to the definition of local PGF, we have
G(i (z) = lim Gx(z) = lim
II G (2(1)
2((0)) +0
2((00) +0
(B.20)
LE/d]
According to Lemma
we have lim G. (z(r)) = e for + i. consequently, we have:
G
(is) lim
G (Z(3) (2())
II LE(d
lim
G (Z(a)) (2())
II
lim 2000-
G² (2(1))
(B.21)
lim
G (Z()) (Z())
II mTM lE(d((t))
2655 00
If part.
Suppose vertex is not adjacent vertex have:
lim G (z))
lim Ge 24 Il TECOMO G.a(Z(s))
(B.22)
(6,5) 0
2,655 -0
Since j & Ch(i), therefore zy E Z(fi.} for all k E Ch(i), we have:
lim 4 (2())) = Ges
z x II IILECAOO masss -0
lim Gial(r(s))
(B.23)
where
lim ((,(((()) lim GB(a..)) 20(1))+0
2x IIIECRR GRr(aw)) (((a..)) (B.24)
-0
Substituting Eq B.24 into Eq. B.23 we have
lim
G (2(1)) = Ge
zi x II II GB(or))())
( +0
(B.25)
piizs x II II (1 ai, 14
= exp
25

---

The derivation of (G3)
lim
Giy (2)) is similar to
lim
G. (Z()), it follows that,
+0
( +0
IREEMSS (1 (1- az,x) (1-03,)) Ht/ 13}
lim
G (()) = exp Pyzj x
(B.26)
(4,51
Substituting Eq.
B.25 and Eq. B.26 into Eq B.21
have
GX
lim
G (1)) G (2())
II t
-0
LE (d]])
Mizi II II (1 (.i,)) + 1Jjj x I (1-a5,)) (
Σ
exp
MM
Ic(d)
(B.27)
Taking the logarithm of Eq. B.27 obtain:
log G ((id)
(z) 24 x
II kECN(O) (1 as,k) + My23 x II (1 -1 az,
Σ
(B.28)
iE|d
Consequently, Taking the partial derivative of Eq. B.28 we finally have:
i log Gfis) (z)
p4
kECH()) (1 di,k),
(B.29)
Ozi
8 log ((L (z)
0,
(B.30)
0zz0zj
which completes the proof of the if part.
Only if part. We prove by contradiction, i.e., suppose that is child vertex of vertex i, aim to
8 log c(:n) (x) + 0.
prove that
dz,day
We first consider G (Z()) in Eq. B.21 since j E Ch(i), we have
lim ((.)) -0
G. (Z,()()) =
lim Ge,
23 x (z x
II kECh())
(Z())
(((5 -0
Ge
24 x
Gss(a))) x
II
lim
Gi,
(B.31)
kECH()
Ga,
2
lim
G.s(Z())
II
GBlor))
kECM())
We next focus
lim
Gz)(Z())):
(x()))
lim Ge)(Z()) lim GB(aa))
2j II GgL((()))
2x(6)) -0
GB(aa))
2j II
lim GjLe(a99)
mi(e)) +0
(B.32)
GB((a))
2j II GBlag.)()) (0)
1 - oni + Di II II GB(a).)
Substituting Eq. B.32 into Eq. B.31 we have:
1 an, + as.s II ILEEAAG GBio)n)(0)) x II GB
lim G2 (2(3)) = Ga
21 x
(x.))) 00
RECKOM
II (1 0..
II (1-0 II (1- (0-00,))00 - 14
exp
kECh(s)
kECk(J
kECh()
(B.33)
26

---

Next, we consider G (Z))) in Eq. B.34 as i & Ch(j), therefore 2n (ish) for all & E Ch(j). Similar to the if part, we have
lim
G (z()) exp
II kECN()
- (1-a3,))1333 - Mi
(B.34)
( -0
Substituting Eq. B.33 and Eq. B.34 into Eq.
B.2 and taking the logarithm of it, we have
log GR. (z) =
II kECh(s)
(1
1222 +
II (1 (1- (5,)))))
kECh(i)
(B.35)
II a4,))
II kECH())
(1 - 1,,002222
Σ pg.
kECh(()
1E[4]
Since the presence of 2z2j, we have
a log G (iJ) (z) θz,0z,
II
(1 a(,))
II (1- (3-a);))4;; #0,
(B.36)
kEC
kECH()
which completes the proof of the only if part.
B.5.2 Proof of Lemma 4
Proof. The proof of Lemma is similar to that Lemma 3
We aim show that the component Z42z is absent in log G Glbakk (z) if and only if the vertices i,j, k form the structure + + k.
According to the definition of local PGE, we have
cliddi)(2)) (2) lim
Gx(x)= =
lim II G (2())
(B.37)
2. -0
24650 +0 1E(4]
According to Lemma we have lim C2 (Z(1)) - for + i, k. consequently, have:
G
(h..k)
(z)
lim
G ((()) (Z(1)) (Gt (2(M)
II (ESG))
G (2()
(B.38)
(x(1a))) +0
lim
G (1)) G (116))GF (Z(A))
II (E(d)((JAM)
e e
(B.39)
2,00 -0
Taking the logarithm of Eq.
B.39 we have:
log Gle... (z) lim
log G2, (K()) + log C2, (2()) log G2, (2(01)
Σ (E.4)
Mi (B.40)
8² log Glask) (2)
If part Suppose i, j. & form the structure i + + k. we aim to prove that by showing that none of the functions lim G (2()), lim C,,, (26)). lim G (2()) in Eq. (6..k))))
(((ijik)
B.40 involve the component 2;2k, which implies that there no a directed path between and k.
Since the j is the child of and k, and j has no child, similar to proof of Lemma
we have:
lim Gi (2()) = exp 1322 x II III (1- (1-a5,) 14 A. 0
(B.41)
lim (7,()()) exp
II (1 oa. maat (1ECK())
II (1-a) (1- II 4εCK()
(1 - and )isss pi
-0
(B.42)
lim Go (Z(H)) = exp
II (1-0H.2))p4
II (1 o II (1-0) (1-00,,))4000.1 1b6
12ECh(k)
LECK(N)
(2ECh()
(B.43)
27

---

Eq. B.43 contain the term involving 2, the partial derivatives of the logarithms these equations with respect to 2; and & are all zero. Therefore we have
Since none of Eq. B..1 Eq. B.42
0 log G(i (z) = O,
(B.44)
0z,0zk
which completes the proof of the part.
Only if part We prove by contradiction. Suppose that i. k: do not form the structure f- k. This implies j, & form either (i) the chain structure + - k: or (ii) the fork structure + - k.
We aim to prove that 8 log G()..))(() (x) f O.
02,02k
We first discuss the case (i), i, ho form the chain structure + + k. we prove that
8 log (x) * by showing that the lim log C. (2(1) in Eq. B.40 contains the com- Abaay) 0
05:05t
ponent 2
According the Theorem and Lemma have:
lim (2 = lim
Ge
Zi x
II [EcCn Gis((())
Wiaik -00
(((.j)k) 00
= =Ga
Zi x
lim
Gir(Z()) IIECMMOO lim Gu(z())
200A +0
= Ga
2
x
lim
Gig(Z()) x ILLEMMOOS
(1 (1-a.))
(6.58)
(B.45)
Consequently, we expand the Gi, (z,) and apply the Lemma 7 we have
lim
(z)()())
*(tiskk) -0
=GB(ang)
2j x GBlaza)
2 II ae
lim GLa(((m)) x II MECCGGN(EE) (ee.e
lim Gy.n (Z(m))
=B(aagg)
2g GBiaga)
26
II (1- (1-aua)
II (1- as.m) (ECHG))()
nECh(A)
-B(ans)
(1 ay-ay,)e + +aya2fa4
II [IEECR (1 ann)
II mECHO)((K)
(1 a)m))
:=82
= GR(m. ((1 a.,n) + +
= =1 ai + (1 a3,)),,,, + a.,, 01
(B.46)
For simplicity, the constant coefficients in the equation are replaced with 82, Substituting Eq. B.46 into Eq. B.45 have:
lim
C,, (2()) Ce, 2; x asj + (1 233 91 ILCOMmOeS (1 as.1)
(1-as.))
((65)))) 0
C..
[[1 (15)= + (a + 01 /
x TIECMONG (1 (1-as;)) as,1) (B.47)
Clearly, the equation contains the term involving žızyzk. We have:
82 log G (iiik) (z)
(1 a1,,a II [L az,) x 2g # 0.
(B.48)
0z,0zk
Then we consider the case (ii). If j. lo form the fork structure i k. we prove that 8 log Glf.5.4) (xx
# by showing that the lim log G (ZG) contain the component 2,2y2h- (
28

---

According the Theorem | and Lemma] 7 we have:
lim (2 (1.1))
lim *16122133
G.g
II secnc) G()((()))
= -G
2s x
lim Gus(26)) x lim
( -0 (Z(A() x TIEEAAO m.ess)) +0
lim (2.((()))
2) +0
G.
lim
Guz(Z()) x lim
Gi x IIECAOOAIN (1 as.))
(1-ai.)
(3 +0
(B.49)
Consequently, we expand the Gi.(Zz)) (z,e(zz) and apply the Lemma
lim (6.ik)) +0 Giz(Z())) = G(
2j x Bxtiisa)
lim
II IImECIO) Gjm(Z(m))
(B(a)))
2j x II ImMECAG) (1 --ii,m)
(B.50)
= -iiy + ais II ImmECra) (1 ai,m) Zjj
and
lim (uakk
Gi,x(Z(x)) B((oa)
2k
lim
II IILECMOD Gk.(((())
*ttikk +0
(B((a)))
2 IIECCAN (1 a.k.)))
(B.51)
[IEECAA)) (1- akin) Zk.
- 0ii
Oiik
:
For simplicity, the constant coefficients in the equation are replaced with 32, Substituting Eq. B.50 and Eq. B.5T into Eq. [B.9 we have
G. (z() G., *((LA)))
lim
2: x (1 oig + B123) x (1 0s, - B22%))
II [LconorcS
(1 -...)
(B.52)
Clearly, the equation contains the term 2k2 Then we have
82 log G (...k)) (z) = 3132 II TEcoOor (1 (1- as,)) x #0,
(B.53)
02,0zk
which completes the proof of the only if part.
B.5.3
Proof of Lemma 5
Proof. According the Theorem
we have
log ((..A)) (z) log
exp {mr x (T (1) (1)))))))1)) II expl-mi
(E(IJ.))
VE(d]Vi
(T()))((11) (1) - 14 (T(ich 11 + (T(ikk)) (1) (1)-
Σ
Xx
(E(d)((1.AA))
= (1) 137 1 (6...) (1) + me To (1..A)) (1)
Σ p..
[e/d
(B.54)
If part Suppose that i,j, form the structure + + and + ý where vertex has two directed path lead to vertex j. We aim to demonstrate that ((..kk)) (1) contain the term involving
2/ 2 such that x 2, 22,2y # 0. Consequently, it follows that
0² log (f().*)
# 0.
0=
0z
29

---

According to the Theorem 3
T (ig.kt Xi
(1) is expressed as follows:
T ((..kk) (1) =1)=2;
II
a T (1.k)
(8))
II
(1 (1-ai))
(E (a (ECh(L
JECh())
(B.55)
where Chi.. (1) = {5 in this structure. For simplicity, let 31
II jECh(())Ch(ssan
(1- (,,))))
have:
T
(...k)
Σ
oie T
((. (1. (sk).
(B.56)
X1
sE(0.132
We consider the term B.56 where both 8j = and 8; which given by:
T (ijkk (1)a
(a (1) a T Thijkh (1)T(
(B.57)
X
Consequently, since vertex j has child it and vertex is has no child in this sturcture, we have:
Xx
(1) 2 (ECH(K)
II (1 axa)
=1 (((
Σ
86
T (...tt
(8k) (8k)
II
(1-13,)
j.k X1
=0(0,1)
(ECHG))(ch.l)
(.....) = T Xx
(1) (1 (1- a(,x) (0)
II (1-a3,) (EC
(B.58)
1 a aj,k + 2z
II (1 ak.))
II (1-a3,)).
lECR(K)
(ECMG)CCCLG)
By combining Eq.
B.56 B.57 and B.58 we can observe that 2,2,2 is present. Therefore
8 log Glissk)
* O. this complete the if part.
Only if part We prove by contradiction. Suppose that vertices i, do form the triangular
structure that k + j and is adjacent to j. we aim to prove that 8 log G (x) 0. This setup
d=t
implies two cases: (i) j, & do not form triangular structure, (ii) vertices j, A: form the triangular structure where the vertex has the indegree of is not
We first consider the case (i). When i, & do not form a triangular structure, they could instead form collider, chain, or fork structure. According Lemma 4 and the intermediate results derived in its proof, in each of these structures, the local PGFs do not contain term involving Zzžzzğ such that
8 log al 1:)) (x) 0, which creates contradiction.
0=
Then we consider the case (ii). Suppose that vertices i, form the triangular + i +- j and +
Recall the Eq. B.54 the log (i.j.k) (z) is given by:
log G (05.8) (z) T k. (1) + T,, (1..h) (1)
M X
(15.)
(1)
Σ Ph-
(B.59)
IE|4
Our goal is to show that none of the terms T (Hdikk nent involving =;
xx (1), Tx (i..it)
(1) and (id.A))
(1) include compo-
We first consider T (15.A) (1). Since vertex i has no child in this structure, we have:
T (1)
II
(1 - -aa)))
(B.60)
(ECR(()
which only involves 24-
30

---

Next, we consider the T ((.jk))
(1), since vertex & has only one child in this structure, we have:
X1
T.....) Xx
(1) = 2
Σ
(8t)
II
(1 (1- ax.)
oe (0,1)
LECH(NNE
(1) +x -a ax. (1))))
II (ECh(R)V())
(1-ax.)
(B.61)
=x(1 (1k.)) aus
II (1-au,))ze)
II !ECA(HM
(1- a.a)),
IECH()
which leads to terms involving z and Z
Finally, consider T TT... (1). the if part, have:
(1). Since vertex has two children i and & in this structure, similar
Tx (1
(1) (1) 2j
Σ
(iJik) (s))a
II
(1-ai,)
(B.62)
Xx
(E(0.))2
In analyzing the equation, will focus the terms 2j
Σ
Xx (s,)T
Td.s 1o.. (sx), treating
sE(U,112
the other components as constants for simplicity. We have:
Σ
XX. (s:) (13.))
(1.jMM)
(sk)
2,Tx (Bj.)) (O)I (0) Tx T
(I)I (...
(0)
2j
8E(0,132
+ +2 (1.1 (.)
(0)T Xx
(1) + 2,
(1..h)
(1)T
(13,)) (1)
X1
(B.63)
Since (biikkk X,
(0) = Tx (1.i)) (0) = we have:
2j
(1i.k) X. (s:) (i..i) (Sx) 2j + 2, (1.i (1) +
T Xx
(1..M)
(1)
T,
(1j.))
(1)T
(..M) (1)
X
(0.1))
(B.64)
According to previous discussion, we have x; (t.j.k)) Consequently, derive the following result:
(1) involves 2,, Ix (h..h
(1) involves 24 and 2
2, 2,Tx ((.A)) (1) involves 2,2;:
Tiiii (i.i.) (1) involves ε,z4 and 242
(1J.A) (I)T(( (1) involves 222 and 2f Zk.
We can observe that 24 appears only to the first power within Eq. B.61 and B.62 In other words,
log G (z) does not contain any term involving zf Therefore, the second partial derivative of
log G (cJ.k) (z) with respect to yn equal to zero, which creates a contradiction. This completes the only if part.
31

---

C Additional Experiment Details
Each experiment reported in the main paper was conducted 12th Gen Intel(R) Core(TM) i3-12100 CPU with 16GB RAM, without the use of a GPU. The runtime for each experiment provided to facilitate reproduction under similar conditions. The significance level (alpha) for the rank hypothesis test used in these experiments is set at 0.01.
Additional Metrics The main paper presents the Fl scores and Structural Hamming Distance (SHD) from synthetic data experiments. This section extends these results by providing Precision, Recall, and runtime metrics for each experiment, as detailed in the following Table Table. and
Table. 6
Table 4: Sensitivity to Avg. Indegree Rate.
Recall
Avg. Indegree
2.0
2,5
5.0
3,5
20
2.5
3.0
3.5
Ours
0.07
0.08
0.04
0.05
0.63
Cumulant
0.96 0004
0.98
0.02
0.90::0.0
0.98 0.001
0.58
0.03
0.63 = 0.08
0.69 0.00
0.72 0.04
PC
0.64 0.20
0.66 0.163
4
0.133
0.56 0.16
0.58 = 0.10
0.52 0.112
0.58 0.10
GES
0.55 0.15
0.56 0.16
4.12
0.111
0.43 0.14
0.43 0.11
0.37 2 0.10
0.34 0.09
OCD
0.24
0.22
0.27
0.23
0.28 0.16
0.36 + 0.14
0.23 + 0.21
0.27 - 0.23
0.27 + 0.17
0.37 + 0.14
Table Sensitivity to Sample Size.
Recallt
Precisiont
Sample Size
15000
30000
15000
Ours
0.78 + 0.11
0.87 0.05
0.51 + +0004
0.92 : 0.05
0.72 1 0.08
0.77 + 0.04
0.81 -: 0.004
0.82 1: 0.04
Cumuulant
0.03
0.97
0.69
0.04
PC
0.13 0.10
0.31110.11
0.57 0.11
0.71
0.46 0.13
0.53
0.11
0.12
0.62 0100
GES
0.20
0.47 0.13
0.48
0.38+
+ 0.20
0.37 0.10
0.22
OCD
0.28 + 0.10
0.35 +..5+0..18 0.18
0.28 + 0.16
0.39 + 0.21
0.34 0.144
0.35 + 0.19
0.27 0.17
0.36 +-220
Table 6: Runtime of each method under the default setting.
Ours
Cumulant
PC
GES 6.90
OCD
Runtime (second)
7.94 + 0.75
77.07 4.94
5.00 + 1.35
1.86
9216 + 1368
D Additional Discussion
D.1 Discussion on the Benefit of Local PGF
In this section, we illustrate the benefit of introducing local PGF through a toy example. First, is important note that, based the closed form of the PGF, it is theoretically possible to identify causal structures using the original PGF, as the closed form encapsulates the entire graph structure. This could be achieved by exhaustively analyzing all terms present the PGF However, such an approach is intractable because the search space for these terms grows exponentially with the number of vertices. Additionally, the method would involve high-order differentiations, which are difficult to estimate accurately.
To illustrate this point, provide toy example with causal structure of 5 vertices, shown in Fig. 6 In this example, we focus on identifying the edge Xa + X5. For the correct structure where X4 Xs, the terms C x 21 = =3 and C2 x 22 o 2ī exist in the PGF. Conversely, for the reverse direction X4 <- X5, the terms Cf x 21 2122202 og and C2 x 21 21222 appear in the PGF.
Notably, shown in Fig. 6 (c) and (f), the term 2 2222232 exists in the PGF for both structures. This is because there are always at least two directed paths from X1 to each of X., and Xs, regardless of the direction between X, and Xş. This implies that taking the second derivative with respect 24 does not reveal any asymmetry between the structures. Therefore, we want to identify the direction using the global PGF, we have to apply test involving the third derivative that distinguishes the correct structure by showing the absence of terms with 2 in the PGF for the correct direction.
In conclusion, when the graph structure becomes more complex and the number of paths between vertices increases, higher-order derivatives are required, making implementation challenging. Local PGF offers an alternative approach, enabling us to avoid these difficulties.
32

---

Causal graph
Branching structure and its corresponding term PGF
Correct causal graph
(b) 2,22,,224
C2 2,22232425
(d) Causal graph with reverse direction X3 X
C Zzggz
Figure 6: Illustraiion of causal graphs and their possible branching structures with the corresponding terms in PGF. (a) Correct causal graph where X, + X5. (b)-(c) Branching structures of (a) with corresponding PGF terms: (b) includes all paths; (c) excludes path from X4 to Xs. (d) Causal graph
with reversed direction X5 + X4. (e)-(f) Branching structures of (d) with corresponding PGF terms: (e) includes all paths; (1) excludes path from X5 X4.
D.2 Discussion on the connection with the cumulant-bass -based method
In this section, we discuss the connection between the cumulant- baasd method and our method, as well as the advantages of our method.
The cumulant-based method (Qiao et al. 2024bb) introduces the concept of k-path cumulants summation, denoted as (x; ~ Xj)- Here, Ax(X; ~ Xj) * 0 indicates the existence a common ancestor m for vertices and j. with & directed paths leading from m to j. Such path information, specifically the number of directed paths from one vertex to another, is encoded within the PGF. If there are directed paths from vertex to j. then terms involving 2m will appear in the PGF. Therefore, detecting the number of directed paths to vertex j is equivalent to identifying the order of 2j the PGF.
However, only the highest non-zero order of A& ~s X,) useful for identifying the causal direction, as it does not yield asymmetry at lower orders. For example, consider vertices X,, Xj. and X,, where X; + X,. and there
directed paths from X,, to X;, and h + directed paths from Xmm to X, where 2 1. In this scenario, has A (X, ~4 X2) + 0 and A& ~4 X,) * O, and A++1(X; (X, ~4 X,) * 0 while AX, ~ X.) = 0. The latter shows an asymmetry that the former does not. This means that the cumulant-based method must detect the highest non-zero order of (X, X2), i.e., the highest order of 2; in the corresponding terms in the PGF, reveal the asymmetry. Moreover, certain unshielded collider structures, such as X1 + X2 and X3 - X₂ in Fig. ((a) of the paper, are non-identifiable using this method. This is because both the correct and reverse directions result in Ag=1 * and Á 0, which leads non- inn-idnniiiiablitty.
In contrast, our method can fully leverage lower-order information to identify causal directions due to the local property of the PGF. By removing redundant directed paths involving vertex through setting the corresponding to approach zero in the PGF, we can focus on identifying within a small local structure, thereby avoiding the need for high-order information.
E Broader Impacts
identify several important societal impacts of our proposed method, including both positive and potential negative impacts:
33

---

1) This paper introduces advancements in the modeling and identification of causal relationships from count data, thereby revealing the causal mechanism of Poisson count data.
2) Inadequate data and training can result in inaccurate causal graphs, potentially leading a misunderstanding of the underlying causal relationships. Such misunderstandings may prompt inappropriate risky decision-making by stakeholders relying these insights.
To mitigate the potential negative societal impacts mentioned above, encourage research and practice to follow these instructions:
1) Integration human oversight is recommended, where domain experts should verify and complement model outputs with their expertise to guide decision-making effectively.
2) Continuous monitoring and updating of model parameters should be implemented to align with real-world data and expert feedback, ensuring the accuracy and applicability of causal predictions.
34

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NeurIPS Paper Checklist
1. Claims
Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Question:
Answer: [Yes]
Justification:
We emphasize our contributions in the abstract and introduction.
2. Limitations
Question: Does the paper discuss the limitations the work performed by the authors?
Answer: [Yes]
Justification: We have discussed the limitations of our works in the Section 6
3. Theory Assumptions and Proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
Justification: We have provided the full set of assumptions and a complete proof.
4. Experimental Result Reproducibility
Question: Does the paper fully disclose all the information needed to reproduce the main ex- perimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
Answer: [Yes]
Justification: We provide detailed experimental parameters used in synthetic experiments and the web address where the real-world data can be accessed.
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Question: Does the paper provide open access to the data and code, with sufficient instruc- tions to faithfully reproduce the main experimental results, as described in supplemental material?
Answer: [Yes]
The code will be available as open access through a GitHub repository. We will also provide detailed instructions for reproducing the experimental results, ensuring that all materials necessary for replication are accessible.
Justification:
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Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer: [Yes]
Justification: We provide the alpha value of the hypothesis test for learning the causal structure the Appendix C
7. Experiment Statistical Significance
Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Answer: [Yes]
Justification: The paper appropriately reports the standard deviation (std) for each experi- ment.
8. Experiments Compute Resources
Question: For each experiment, does the paper provide sufficient information the com- puter resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
Answer: [Yes]
35

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Justification:
Yes, our paper provides comprehensive details on the computer resources required reproduce the experiments. All pertinent information is included in the Appendix
9. Code Of Ethics
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Answer: [Yes]
Justification: The dataset used in the paper is properly cited.
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36


Reviews:
[{"review_id": "NJOILawRAx", "reviewer": "Reviewer_w346", "summary": "This paper investigates the causal structure learning on the Poisson Branching Structural Causal Model (PB-SCM)and its identifiability using probability-generating function (PGF). The identifiability is established by developing the closed-form solution of the PGF which can be utilized to identify the causal structure of PB-SCM. The proposed methods are validated through synthetic experiments and real-world experiments.", "questions": "See weakness", "limitations": "See weakness", "rating": 7, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- This work proposes a PGF-based method for identifying the causal structure which is interesting.\n- The authors propose the closed-form solution of PGF and establish the connection to the graph. By this, this work shows that there still remains a gap in the identifiability of the PB-SCM and addresses this gap using the local property of the PGF. This is a novel and significant contribution.\n- The proposed method is overall sound with a theoretical guarantee.\n- This paper is well-present and well-motivated.", "weaknesses": "- The authors propose to use the local PGF for identifying the causal structure. It seems that the global PGF can also be used for identifying the causal structure and it would be more beneficial if more discussions could be involved.\n- What is the benefit of using the local property of PGF? Is it possible to set the z_i not approach to zero?\n- The authors propose to use the Gaussian distribution for the rank test. It would be better to illustrate why the trace of the matrix converges to a normal distribution.\n- The experiments only contain some random experiments. It would be more beneficial if there are some case studies. \n- The experiments should explain the arrow in the table after the metric.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> **Q1**: The authors propose to use the local PGF for identifying the causal structure. It seems that the global PGF can also be used for identifying the causal structure and it would be more beneficial if more discussions could be involved.\n\n**A1**:Thank you very much for this suggestion to improve the completeness of our paper. It is indeed theoretically feasible to identify causal structures through the global PGF. However, using the PGF directly to determine causal directions involves computing higher-order derivatives, which can complicate the implementation and thus affect its feasibility.\n\nTo illustrate this point, we provide a toy example with a causal structure of 5 vertices, as shown in Fig. R1 of our attached PDF. In this example, we focus on identifying the edge $X_4 \\to X_5$. For the correct structure where $X_4 \\to X_5$, the terms $C_1 \\times z_1z_2z_3z_4^2z_5^4$ and $C_2 \\times z_1z_2z_3z_4^2z_5^2$ exist in the PGF. Conversely, for the reverse direction $X_4 \\leftarrow X_5$, the terms $C'_1 \\times z_1z_2z_3z_4^4z_5^2$ and $C'_2 \\times z_1z_2z_3z_4^2z_5^2$ appear in the PGF.\n\nNotably, as shown in Fig. R1 (c) and (f), the term $z_1z_2z_3z_4^2z_5^2$ exists in the PGF for both structures. This is because there are always at least two directed paths from $X_1$ to each of $X_4$ and $X_5$, regardless of the direction between $X_4$ and $X_5$. This implies that taking the second derivative with respect to $z_4$ does not reveal any asymmetry between the structures. \nTherefore, if we want to identify the direction using the global PGF, we have to apply a test involving the third derivative that distinguishes the correct structure by showing the absence of terms with $z_4^3$ in the PGF for the correct direction. Following your suggestion, we have added this toy example and discussion to the paper to enhance the understanding of the connection between the PGF and causal structures.\n\n> **Q2**: What is the benefit of using the local property of PGF? Is it possible to set the $\\displaystyle z_{i}$ not approach to zero?\n\n**A2**: Thank you for this valuable question to emphasize our contributions. As demonstrated in the answer to Q1, it is possible to use the PGF directly without setting $z_{i}$ to approach zero when identifying directions. However, when the graph structure becomes more complex and the number of paths between vertices increases, higher-order derivatives are required, making implementation challenging. \n\nTherefore, we propose using the local PGF, which simplifies the process by isolating specific terms in the PGF. This is done by setting other $z_i$ to approach zero, effectively reducing the complexity of the derivative calculations. In the example above, by letting $z_2$ and $z_3$ approach zero, we can remove terms in the PGF that involve $z_2$ and $z_3$, such as $C_1 \\times z_1z_2z_3z_4^2z_5^2$ and $C_2 \\times z_1z_2z_3z_4^2z_5^4$. This reduction allows us to focus on examining the local triangular structure formed by $X_1$, $X_4$, and $X_5$, which only requires estimating the second derivative.\n\n> **Q3**:  The authors propose to use the Gaussian distribution for the rank test. It would be better to illustrate why the trace of the matrix converges to a normal distribution.\n\n**A3**: Thank you for this question. As mentioned in Section 4.5, we use the empirical PGF (ePGF) $\\bar G_{\\mathbf{X},n}(\\mathbf{z}) = \\frac{1}{n}\\sum_{i=1}^{n} z_{1}^{X_{1}^{(i)}} \\cdots z_{d}^{X_{d}^{(i)}}$ to estimate the PGF $ G_{\\mathbf{X}}(\\mathbf{z})$, which is an unbiased estimator of the PGF [1][2]. According to the central limit theorem, the quantity $ n^{1/2}\\\\{\\bar G_{\\mathbf{X},n}(\\mathbf{z}) - G_{\\mathbf{X}}(\\mathbf{z}) \\\\}$ converges in distribution to a normal distribution $ N(0, \\sigma^2)$ with zero mean and variance $\\sigma^2$, which can be estimated using the bootstrap method. Therefore, in matrix $\\mathbf{A}^{\\\\{i,j\\\\}}$, the estimations of the local PGF $ G_{\\mathbf{X}}^{\\\\{i,j\\\\}}(\\mathbf{z})$ and its partial derivative $\\frac{\\partial^2 G_{\\mathbf{X}}^{\\\\{i,j\\\\}}(\\mathbf{z})}{\\partial z_{i} \\partial z_{j}}$ also converge in distribution to normal distributions. \nSince the sum of two normal distributions is also a normal distribution, the traces of matrices $\\mathbf{A}^{\\\\{i,j\\\\}}$ converge to a normal distribution. The same reasoning applies to matrices $\\mathbf{B}^{\\\\{i,j,k\\\\}}$ and $\\mathbf{C}^{\\\\{i,j,k\\\\}}$.\n\n[1] Esquível, Manuel. \"Some applications of probability generating function based methods to statistical estimation.\" Discussiones Mathematicae Probability and Statistics 29.2 (2009): 131-153.\n\n[2] Nakamura, Miguel, and Víctor Pérez-Abreu. \"Empirical probability generating function: An overview.\" Insurance: Mathematics and Economics 12.3 (1993): 287-295.\n\n> **Q4**: The experiments only contain some random experiments. It would be more beneficial if there are some case studies.\n\n**A4**: Thank you for your valuable suggestion, which allows our experiments more comprehensive. We have added three case studies involving causal graphs with 3, 4, and 5 vertices respectively. The results are presented in Table R1 in our attached PDF. \n\n> **Q5**: The experiments should explain the arrow in the table after the metric.\n\n**A5**: Thank you for your suggestion. We have added a description of the arrows in the table: ↑ indicates that a higher value is better, while ↓ indicates that a lower value is better.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ROSSz7w18H", "reviewer": "Reviewer_Gqam", "summary": "This paper addresses the identifiability of Poisson Branching Structure Causal Model (PB-SCM) using the probability generating function, during which a compact, exact closed-form PGF solution is developed and the identifiability of PB-SCM is complete based on such relation. A practical algorithm for learning causal skeletons and causal directions using PGF is proposed, with effectiveness demonstrated on synthetic and real datasets.", "questions": "- What is the connection between the cumulant-based method and the proposed PGB-based method?\n- Can you further improve Theorem 7?", "limitations": "The authors had adequately addressed the limitations in their conclusion and the broader societal impacts in their appendix.", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "- This paper addresses the identification of the PB-SCM model using a probability-generating function. Compared with the previous cumulant-based, the identifiability of PB-SCM is further complete. \n- The theory releases an interesting connection between the probability-generating function and the graph of PB-SCM which allows us to further explore the identifiability of the PB-SCM.\n- Based on the theoretical results an efficient algorithm is proposed, and the effectiveness of the proposed method is also verified in the experiments.\n- The writing is clear and well-organized and the intuition of the proposed method is effectively conveyed.", "weaknesses": "- This paper states that the cumulant-based method is exactly the method that identifies the causal direction by detecting the highest order of $z_i$. Can you explain why? It would be interesting to see the connection between these two methods.\n- The local collider structure used in section 4 seems to be referred to the unshielded collider structure which should be stated clearly to avoid ambiguity.\n- Theorem 7 states that the identifiability of the causal direction is given the known causal skeleton. Can it be extended to address the general identifiability of the whole causal structure?\n- Since the paper is quite dense, more illustrations can be provided in the toy example to illustrate the connection between PGF and the graph and how it leads to the identifiability. \n- The citing format should be improved as it is mixed with the main text. \n\nTypos:\n- Line 204 similarly", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> **W1:** This paper states that the cumulant-based ... connection between these two methods. \\& **Q1:** What is the connection between the cumulant-based method and the proposed PGF-based method?\n\n**A1:** Thank you for your question allowing us to further clarify the connection between the cumulant-based method and our method, as well as the advantages of our method.\n\nThe cumulant-based method [1]  introduces the concept of $k$-path cumulants summation, denoted as $\\tilde\\Lambda_{k}(X_{i} \\leadsto X_{j})$. Here, $\\tilde\\Lambda_{k}(X_{i}\\leadsto X_{j})\\neq 0$ indicates the existence of a common ancestor $m$ for vertices $i$ and $j$, with $k$ directed paths from $m$ to $j$. Such path information is encoded within the PGF. If there are $k$ paths from vertex $m$ to $j$, terms involving $z_m z_j^k$ appear in the PGF. Thus, detecting paths to vertex $j$ is equivalent to identifying the order of $z_j$ in the PGF.\n\nHowever, only the highest non-zero order of $\\tilde\\Lambda_{k}(X_{i} \\leadsto X_{j})$, involving $k+1$ order cumulants, helps identify the causal direction, as it does not show asymmetry at lower orders. For example, consider vertices $X_i$, $X_j$, and $X_m$, where $X_i \\to X_j$, and there are $k$ paths from $X_m$ to $X_i$, and $k+p$ paths from $X_m$ to $X_j$ where $p \\geq 1$. Here, $\\tilde\\Lambda_{k}(X_{i} \\leadsto X_{j}) \\neq 0$ and $\\tilde\\Lambda_{k}(X_{j}\\leadsto X_{i}) \\neq 0$, and $\\tilde\\Lambda_{k+1}(X_{i} \\leadsto X_{j}) \\neq 0$ while $\\tilde\\Lambda_{k+1}(X_{j} \\leadsto X_{i}) = 0$. The latter shows an asymmetry that the former does not. This means that the cumulant-based method must detect the highest non-zero order of $\\tilde\\Lambda_{k}(X_{j} \\leadsto X_{i})$, i.e., the highest order of $z_i$ in PGF terms, to reveal the asymmetry. Moreover, certain local (unshielded) collider structures, such as $X_1 \\to X_2$ and $X_3 \\to X_2$ in Fig. 1(a), are non-identifiable using this method because both directions result in $\\tilde\\Lambda_{k=1} \\neq 0$ and $\\tilde\\Lambda_{k=2} = 0$, leading to non-identifiability.\n\nIn contrast, our method fully leverages lower-order information due to the local property of the PGF. By removing redundant paths by setting the corresponding $z$ to zero in the PGF, we focus on a small local structure, avoiding the need for high-order information.\n\n[1] Qiao, Jie, et al. \"Causal Discovery from Poisson Branching Structural Causal Model Using High-Order Cumulant with Path Analysis.\" Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 38. No. 18. 2024.\n\n> **W2:** The local collider structure ... stated clearly to avoid ambiguity.\n\n**A2:** Thank you for your insightful question, which helps us clarify the terminology used in our paper. Exactly, the local collider structure in this paper can be referred to as the unshielded collider structure. Following your suggestion, we have added the definition of a local structure: Given a DAG $G(\\mathbf{V},\\mathbf{E})$, a local structure $G'(\\mathbf{L},\\mathbf{E_L})$ is a subgraph of $G(\\mathbf{V},\\mathbf{E})$ with vertex set $\\mathbf{L} \\subset \\mathbf{V}$ and edge set $\\mathbf{E_L} = \\\\{i \\to j \\mid i,j \\in \\mathbf{L}, i \\to j \\in \\mathbf{E}\\\\}$.\n\nGiven vertices $i, j, k$, if $i \\to j \\leftarrow k$ and $i$ and $k$ are not adjacent, they form an unshielded collider structure, which we refer to as a local collider structure. If there is a direct edge between $i$ and $k$, this edge will be included in the local structure formed by $i, j, k$, known as a local triangular structure. We will clearly define these terms to avoid ambiguity.\n\n> **W3:** Theorem 7 states ... identifiability of the whole causal structure? \\& **Q2:** Can you further improve Theorem 7?\n\n**A3:** Thank you for your valuable suggestion. \nIndeed, it is theoretically possible to identify the whole causal structure without relying on the causal skeleton because the PGF uniquely encodes distribution, corresponding to a unique causal structure. However, the complexity of such an approach can be exceedingly high. \n\nLet us clarify what this identifiability issue is and why learning the causal skeleton is essential for addressing the identifiability of the whole causal structure.\nAs a foundation of identifiability, the closed-form solution of the PGF is directly associated with the causal structure, \nwhich allows us to recover the causal structure by exploring whether different structures correspond to the form of PGF.\nHowever, if we try to directly analyze the whole causal structure, the number of possible forms of PGF we have to examine can be large since the number of structures significantly increases with the number of vertices. The large search space makes structural identifiability difficult. \n\nTherefore, to address this issue, we introduce the local PGF, which enables us to examine its form in a small local structure, i.e., local triangular structure and local (unshielded) collider structure. This approach reduces the complexity of the search space by focusing on identifiable local patterns. Therefore, we require the causal skeleton and prioritize the identifiability of adjacent vertices in the Identifiability section.\n\n> **W4:** Since the paper is dense, more illustrations ... identifiability.\n\n**A4:** Thank you for your suggestion, we will add more illustrations to the toy example to enhance understanding. Specifically, we have enhanced the description of $T_{X_i}$ in the caption of Fig. 3: ``Here, $\\alpha_{i,j}T_{X_j}(1)$ indicates that $X_j$ is reached from $X_i$ in this branching structure, while $(1-\\alpha_{i,j})T_{X_j}(0)$ indicates the opposite.'' This aims to provide an intuitive understanding of the connection between the PGF and the graph structure.\n\n> **W5:** The citing format ... main text.\n\n**A5:** Thank you for your suggestion, which will help enhance the clarity and readability of our paper. We will change the citation format to use square brackets to clearly distinguish them from the main text.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ir82YQb6OI", "reviewer": "Reviewer_nGrv", "summary": "The paper focuses on causal discovery from observational count data, proposing a method to address the identifiability gap in Poisson Branching Structural Causal Models (PB-SCM) using Probability Generating Function (PGF). The authors develop a closed-form solution for the PGF of PB-SCM, enabling the identification of local causal structures by testing component appearances in the PGF. The effectiveness of the method is demonstrated through experiments on both synthetic and real datasets.", "questions": "n/a", "limitations": "Yes", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "The paper has good presentation and motivation. \n\nThe authors derive a compact and exact closed-form solution for the PGF of PB-SCM, establishing a connection between the PGF and the causal structure. \n\nThe paper proposes a practical algorithm to learn causal skeletons and identify causal directions in PB-SCM using the derived PGF. \n\nThe proposed method is validated through experiments on synthetic and real-world datasets, demonstrating its effectiveness and superiority over existing methods in identifying causal structures in count data.", "weaknesses": "The proposed method cannot handle high-dimensional covariates.\n\nThe orientation of edges is not detailed.\n\nFigure 4, the caption should be correct. \n``F: Foul, Y1: Yellow card, Y2: Second yellow card, R: Red card, S: Substitution, H: Hand ball'' -> ``$F$: Foul, $Y_1$: Yellow card, $Y_2$: Second yellow card, $R$: Red card, $S$: Substitution, $H$: Hand ball''\n\nThe soundness of the proposed method requires the causal sufficiency assumption, which limits the applications of the method in real-world scenarios.\n\nMinor: \n\nBuilding on this, We -> Building on this, we", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> **W1:** The proposed method cannot handle high-dimensional covariates.\n\n**A1:** Thank you for pointing out this issue; this will be an important problem for us to address in future work. As mentioned in the conclusion section, the performance of our method can be limited by the scale of the dimension. \nWhile the empirical PGF is effective in estimating the PGF, it encounters challenges when dealing with high-dimensional data. The main reason is that the product of multiple $\\displaystyle z_{i}^{X_{i}}$ can result in extremely small numbers, which affects numerical stability and accuracy. Therefore, a promising future direction is to develop a sample-efficient method for estimating the PGF to better handle high-dimensional covariates.\n\n> **W2:** The orientation of edges is not detailed.\n\n**A2:** Thank you for pointing out the need for more detail on edge orientation. According to the proposed closed form of PGF, the terms in the PGF correspond directly to the graphical structure, thus we can perform orientation by checking whether specific terms exist in PGF. \n\nFor instance, Eq. (5) in the paper presents the closed form of the PGF for the triangular structure in Fig. 2. The presence of the term $z_1z_2z^2_3$ in Eq. (5) indicates that there are two directed paths leading to $X_3$ from either $X_1$ or $X_2$. Here, the order of $z_3$ is 2, meaning that $X_3$ is the vertex with two in-degrees in the triangular structure. Consequently, we can orient the edges as $X_1 \\to X_3$ and $X_2 \\to X_3$. Building on this, we orient each edge by considering a local triangular structure with this edge and examining whether this edge points to a vertex with two in-degrees, as described in Algorithm 1 (Lines 5-7) in the paper.\n\nOtherwise, if an edge $X_i - X_j$ cannot be part of a triangular structure, we attempt to orient it in a pattern such as $X_i - X_j \\leftarrow X_k$ or $X_i - X_j - X_k$ with another vertex $X_k$. Specifically, by examining whether the term $z_iz_jz_k$ does not appear in the local PGF involving $X_i, X_j, X_k$, we can conclude that there is no directed path between $X_i$ and $X_j$ through $X_k$. In such cases, we orient the edges as $X_i \\to X_j$, and $X_j \\leftarrow X_k$ as described in Algorithm 1 (Lines 8-10).\n\nFollowing your suggestion, we will provide a more detailed description of the orientation process in the method section.\n\n> **W3:** Figure 4, the caption should be correct. \n\n**A3:** Thank you for pointing out the typos and providing the correction. Following your suggestion, we have corrected the caption of Figure 4: Football Dateset Result ($F$: Foul, $Y_1$: Yellow card, $Y_2$: Second yellow card, $R$: Red card, $S$: Substitution, $H$: Handball)\n\n> **W4:** The soundness of the proposed method requires the causal sufficiency assumption, which limits the applications of the method in real-world scenarios.\n\n**A4:** Thank you for your valuable suggestion. Relaxing the causal sufficiency assumption will indeed be a focus of our future work. Nevertheless, this assumption is a common assumption in many causal inference methods. Several prominent works in the field also utilize this assumption. For instance, Spirtes et al. in their seminal work [1] on causal discovery algorithms assume causal sufficiency to ensure accurate inference of causal structures. Similarly, Pearl discusses the importance of causal sufficiency in structural causal models [2]. These references highlight that while the causal sufficiency assumption can limit applicability in some real-world scenarios, it is a foundational assumption that underpins many existing causal inference methods. In future work, we will further explore the identification of latent confounders to address the limitations posed by this assumption.\n\n[1] Spirtes, Peter, Clark Glymour, and Richard Scheines. Causation, prediction, and search. MIT press, 2001.\n\n[2] Pearl, Judea. Causality. Cambridge university press, 2009.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qP3rgHBdWM", "reviewer": "Reviewer_CtKZ", "summary": "In order to achieve causal structure learning on PB-SCM, this paper explores the identifiability of PB-SCM using the Probability Generating Function (PGF).  Furthermore, this paper enables the identification of the local structures by testing their corresponding component appearances in the PGF. Building on this, this paper proposes a practical algorithm for learning causal skeletons and identifying causal directions of PB-SCM using PGF.", "questions": "1. What are the accuracies of learning the Probability Generating Function and the subsequent causal structure learning? What are the bottlenecks of the proposed method?\n2. The experiments on real-world data provide insufficient information. How do other methods perform on the real datasets?\n3. How does the proposed method perform on networks that are not purely Poisson Bayesian networks or on mixed networks of general Bayesian and Poisson Bayesian networks? Additionally, does this method require prior knowledge that the data originates from a non-Poisson Bayesian network?", "limitations": "This paper briefly touches on the limitations of the proposed method. This paper has no negative social impact.", "rating": 6, "confidence": 5, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "The use of Probability Generating Function (PGF) to address the identifiability of the Poisson Branching Structural Causal Model (PB-SCM) is a novel approach. The proposed practical algorithm for learning causal skeletons and identifying causal directions of PB-SCM using PGF is well-detailed. The authors provide a clear and systematic approach, making it easier for practitioners to implement the algorithm. This paper makes several theoretical contributions and has a strong theoretical basis.", "weaknesses": "1. While these experiments demonstrate the effectiveness of the method, additional experiments on a wider variety of real-world datasets from different domains would strengthen the argument for the method's generalizability.\n2. The symbols T and the corresponding formulas that appear in Figure 3 are explained later in the text, which may cause significant confusion for readers.\n3. The term ch(i) in Formula 3 lacks a definition. It is possibly intended to be Des(i) as defined earlier in the text.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "> **W1:** While these experiments ... argument for the method's generalizability.\n\n**A1:** We appreciate the suggestion that makes this paper more complete and closer to realistic applications. We further examined our method using a shopping mall paid search campaign dataset [1]. This dataset contains data from a five-month paid search campaign for a U.S. shopping mall, spanning from July to November 2021. In this dataset, we focus on the variables *Impressions*, *Clicks*, and *Conversions*, which are fundamental count data in e-commerce scenarios. These variables follow causal relationships: *Impressions* $\\rightarrow$ *Clicks*, *Clicks* $\\rightarrow$ *Conversions*, and *Impressions* $\\rightarrow$ *Conversions*, forming a triangular structure.\n\nOur method successfully identifies the adjacent vertices between *Impressions* and *Clicks*, as well as the other two causal directions. This outcome is consistent with our theoretical result, which means that our method can be applied to more real-world scenarios.  Following your suggestion, we have added these experimental results to our paper.\n\n[1] https://www.kaggle.com/datasets/marceaxl82/shopping-mall-paid-search-campaign-dataset\n\n> **W2:** The symbols T and the corresponding ... which may cause significant confusion for readers.\n\n**A2:** Thank you for your suggestion, which will help enhance the clarity and readability of our paper. We have added an explanation of $T_{X_j}$ in the caption of Figure 3: ``Illustration of the graphical implications in the closed-form solution for the PGF of PB-SCM. Here, $\\alpha_{i,j} T_{X_j}(1)$ indicates that $X_j$ is reached from $X_i$ in this branching structure, while $(1-\\alpha_{i,j})T_{X_j}(0)$ indicates the opposite.\"\n\n> **W3:** The term ch(i) in Formula 3 lacks a definition. It is possibly intended to be Des(i) as defined earlier in the text.\n\n**A3:** Thank you for your careful review and for pointing this out. In the context of a given DAG $G(\\mathbf{V}, \\mathbf{E})$, $Ch(i)$ is defined as $Ch(i) = \\\\{j \\mid i \\to j \\in \\mathbf{E}\\\\}$, representing the set of children of vertex $i$. We have added this definition alongside the definitions of $Pa(i)$ and $Des(i)$ to ensure clarity.\n\n> **Q1:** What are the accuracies of learning the Probability Generating Function and the subsequent causal structure learning? What are the bottlenecks of the proposed method?\n\n**A4:** Thank you very much for this suggestion to improve the completeness of our paper. \nThe accuracies of our method are supported by the fact that the empirical probability generating function (ePGF) $\\bar G_{\\mathbf{X},n}(\\mathbf{z}) = \\frac{1}{n}\\sum_{i=1}^{n} z_{1}^{X_{1}^{(i)}}\\cdots z_{d}^{X_{d}^{(i)}}$ is an unbiased estimator of the PGF $G_{\\mathbf{X}}(\\mathbf{z})$ and almost surely converges to $G_{\\mathbf{X}}(\\mathbf{z})$ as the sample size $n\\to \\infty$, according to the strong law of large numbers [2][3].\n\nHowever, as mentioned in the conclusion section, the performance of our method can be limited by the scale of the dimension. While the ePGF exhibits effectiveness in estimating PGF, it probably encounters challenges when dealing with high-dimensional data, where the main reason is that the product of multiple $ z_{i}^{X_{i}}$ can result in an extremely small number. Thus, a promising future direction is to develop a sample-efficient method for estimating PGF.\n\n[2] Esquível, Manuel. \"Some applications of probability generating function based methods to statistical estimation.\" Discussiones Mathematicae Probability and Statistics 29.2 (2009): 131-153.\n\n[3] Nakamura, Miguel, and Víctor Pérez-Abreu. \"Empirical probability generating function: An overview.\" Insurance: Mathematics and Economics 12.3 (1993): 287-295.\n\n> **Q2:** The experiments on real-world data provide insufficient information. How do other methods perform on the real datasets?\n\n**A5:** Thanks for your valuable suggestion to improve the completeness of our experiments. We have further examined the performance of other methods on the real datasets. The cumulant-based method successfully identified the correct direction of edges, except for $ F\\rightarrow R$. This can be attributed to the fact that only a few fouls directly cause a red card, thus leading to estimation issues with the cumulant method. As for PC, GES, and OCD, they struggle to recover the structure, with their F1 scores being 0.53, 0.18, and 0.44 respectively. Following your suggestion, we will add these experimental results to our paper.\n\n> **Q3:** How does the proposed method perform on networks that are not purely Poisson Bayesian networks or on mixed networks of general Bayesian and Poisson Bayesian networks? Additionally, does this method require prior knowledge that the data originates from a non-Poisson Bayesian network?\n\n**A6:** Thank you for this insightful question, which allows us to clarify the scope and applicability of our method. The main contribution of our paper is to propose a more comprehensive identifiability result for PB-SCM using PGF. Despite that our PGF-based method may generate an incorrect graph structure when applied to data with inappropriate distributions - since different distributions have different PGFs - We want to emphasize that the assumptions of PB-SCM are widespread in many real-world scenarios.\n\nBy leveraging the thinning operation, PB-SCM explicitly models the inherent branching relationships between discrete variables, particularly for count data. Intuitively, the influence from a parent vertex to a child vertex is directly represented by the change in event counts, as the thinning operation outputs an integer representing the contribution of the count from the parent event. This directly reflects the dynamics of count data, effectively modeling the natural generating process in many real-world systems. Therefore, although our method relies on model assumptions, these assumptions are both common and typical.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
Dear Reviewers w346, Gqam, nGrv, and CtKZ,

Thanks for the thoughtful and constructive reviews, which improve the completeness and readability of our paper. It is encouraging that reviewers think that the proposed PGF-based method for identifying PB-SCM is novel (w346, Gqam, CtKZ) and interesting (Gqam), our theoretical result is sound (w346, CtKZ), our paper is well-present and well-organized(w346, Gqam, nGrv), and our experimental results show the effectiveness (Gqam, nGrv) of our PGF-based method. We here provide a general response to summarize the modifications of the paper.

- To Reviewer w346, we have added an additional discussion on using the (global) PGF to identify the causal structure.
- To Reviewer w346, we have explained the benefits of using the local properties of the PGF.
- To Reviewer w346, we have clarified why the trace of the matrix converges to a normal distribution.
- To Reviewer w346, we have included three case studies in our attached PDF involving causal graphs with 3, 4, and 5 vertices, respectively.
- To Reviewer w346, we have added descriptions of the arrows in Table 1 and Table 2 of the paper.
- To Reviewer Gqam, we have illustrated the connection between the cumulant-based method and our PGF-based method.
- To Reviewer Gqam, we have clarified the terminology used for the local collider structure.
- To Reviewer Gqam, we have explained the necessity of the causal skeleton in addressing the identifiability of the entire causal structure.
- To Reviewer Gqam, we have improved the description of the toy example and the citation format.
- To Reviewer nGrv, we have discussed the limitations of our method and the reasonableness of the causal sufficiency assumption.
- To Reviewer nGrv, we have added a detailed description of how to orient the edges.
- To Reviewer nGrv, we have corrected the caption of Fig. 4.
- To Reviewer CtKZ, we have added a discussion on the accuracies and bottlenecks of our PGF-based method.
- To Reviewer CtKZ, we have included an additional real-world experiment, and provided the performance of other methods in the real-world experiment of the paper.
- To Reviewer CtKZ, we have added an explanation of $T_{X_i}$ in the caption of Fig. 3 and included the definition of $Ch(i)$.
- To Reviewer CtKZ, we have discussed potential issues our method may encounter when assumptions are violated.
- To all reviewers, we have polished our paper and corrected the pointed-out typos.

Thanks again for your time dedicated to carefully reviewing this paper. We hope that our response properly addresses your concerns.

With best regards,
Authors of submission 17086

Meta-review: This paper  has received very positive feedback from reviewers, who praised its novel approach to identifying causal structures in Poisson Branching Structural Causal Models (PB-SCM) using Probability Generating Functions (PGF). The reviewers highlighted the paper's strong theoretical contributions, including the development of a closed-form PGF solution and an efficient algorithm for learning causal structures.  The authors provided a comprehensive rebuttal, addressing concerns by clarifying theoretical connections, improving explanations, adding case studies, and discussing the method's limitations and future directions. Overall, the paper is considered technically sound and impactful, with reviewers recommending its acceptance with minor revisions to enhance clarity and completeness.
Decision: Accept (spotlight)