Extendable and Iterative Structure Learning Strategy for Bayesian Networks
Authors: Hamid Kalantari
Russell Greiner
Pouria Ramazi
Venue: ICLR 2025 Poster (2025)
Keywords: Structure learning
Bayesian networks
Causal discovery
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

Abstract: Learning the structure of Bayesian networks is a fundamental yet computationally intensive task, especially as the number of variables grows. Traditional algorithms require retraining from scratch when new variables are introduced, making them impractical for dynamic or large-scale applications. In this paper, we propose an extendable structure learning strategy that efficiently incorporates a new variable $Y$ into an existing Bayesian network graph $\mathcal{G}$ over variables $\mathcal{X}$, resulting in an updated P-map graph $\bar{\mathcal{G}}$ on $\bar{\mathcal{X}} = \mathcal{X} \cup \{Y\}$. By leveraging the information encoded in $\mathcal{G}$, our method significantly reduces computational overhead compared to learning $\bar{\mathcal{G}}$ from scratch. Empirical evaluations demonstrate runtime reductions of up to 1300x without compromising accuracy. Building on this approach, we introduce a novel iterative paradigm for structure learning over $\mathcal{X}$. Starting with a small subset $\mathcal{U} \subset \mathcal{X}$, we iteratively add the remaining variables using our extendable algorithms to construct a P-map graph over the full set. This method offers runtime advantages comparable to common algorithms while maintaining similar accuracy. Our contributions provide a scalable solution for Bayesian network structure learning, enabling efficient model updates in real-time and high-dimensional settings.

Published as a conference paper at ICLR 2025
EXTENDABLE AND ITERATIVE STRUCTURE LEARNING STRATEGY FOR BAYESIAN NETWORKS
Hamid Kalantari & Russell Greiner
Pouria Ramazi
Department of Computing Science
Department of Mathematics Statistics
University of Alberta Edmonton, Alberta T6G 2R3, Canada
Brock University
St. Catharines, ON L.2S 3A1, Canada pramazi hammazz@brocu..oom
(hkalant2, rgreiner) Qualberta.ca
ABSTRACT
Learning the structure of Bayesian networks from a dataset of instances is a funda- mental yet computationally intensive task, especially the number of variables grows. Traditional algorithms require retraining from scratch when new variables are introduced, making them impractical for dynamic or large-scale applications. In this paper, we propose an extendable structure learning strategy that efficiently incorporates new variable Y into an existing (P-map) Bayesian network graph G over variables X' resulting in an updated P-map graph & on X = X U {Y}. By leveraging the information encoded in G. our method significantly reduces computational overhead compared to learning G from scratch. Empirical evalu- ations demonstrate runtime reductions of up 1300 without compromising accuracy. Building on this approach, structure learning over X. Starting with a small subset U C X, we iteratively add the remaining variables using extendable algorithm to construct a P-map graph over the full set. This method achieves runtime advantages compared to common algorithms while maintaining similar accuracy. Our contributions provide scal- able solution for Bayesian network structure learning, enabling efficient model updates in real-time and high-dimensional settings.
introduce a novel iterative paradigm for
INTRODUCTION
Dependency between random variables can be represented by a directed acyclic graph (DAG), where for some directions, a link from variable A to B signifies that A causes B. When the DAG is coupled with the conditional probability distribution (CPD) of each variable given its parents, forms Bayesian network, which enables both probabilistic and causal queries. The joint probability distribution of the variables then factorizes according to the DAG, meaning it becomes the product of the associated CPDs.
Estimating the DAG from observational data, known as structure learning. is typically approached using either constraint-bas based or acore-based algorithms (Kitson et al.) 2021). Constraint- based methods, such as PC (developed by Peter Spirtes and Clark Glymour) (Spirtes et al. (000) and Fast Causal Inference (FCI) (Spirtes et al., 2000), rely on detecting dependencies between variables using conditional independence (CI) tests (Guo et al. 2020). In contrast, score-based methods search for
DAG that maximizes a score function, such as the Bayesian Information Criterion (BIC) (Koller &
Friedman 2009).
notable gap exists current approaches: no algorithm efficiently updates an existing DAG when new variables are introduced. This issue is particularly relevant in fields such social science (Card) 1999), psychology (Primack et al. (207)). and financial studies (Bollen et al. 2011). where important variables may be omitted in the initial stages of research but later recognized critical. For instance, in stock market prediction models, analysts might begin with historical stock prices, trading volumes, and economic indicators, only to later discover the significant impact of social media sentiment (Bollen et al. 2011). Incorporating such new variables would traditionally require
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re-leamnigg the entire DAG, process that becomes computationally prohibitive the number of variables grows. While existing online (Kocacoban & Cussens 2019) and incremental (Alcobe (005) structure learning algorithms address scenarios where datasets are updated over time by adding new instances (over the original set of variables), they are not designed to efficiently incorporate a new variable into a learned DAG, they would, instead, simply discarding the original structure.
We propose an extendable structure learning algorithm that avoids the need to re-learn the entire graph when a new variable is added. Specifically, we investigate the effect of adding a node Y to an already learned correct structure G over a set of variables X. We present two algorithms to obtain the extended structure for X U {Y). Our key findings are that adding a new variable results only in the deletion of edges between the original variables, not the addition of new ones, and analyzing how the added node can affect the existing edges between nodes in X.
Consequently, the search for the highest-scoring DAG is confined to a reduced space, rather than the full space of all possible DAGs over the extended set of variables. This reduced search space informs the development of an extendable score-based algorithm, as well as constraint-based algorithm that leverages the existing CI tests from the learned structure. By significantly reducing the number of CI tests compared to re-learning the structure from scratch, we achieve a more computationally efficient solution. The complexity of the extendable constraint-bate based algorithm is O(N 2T where N is the cardinality of X, d represents the maximum degree of nodes in G. is the degree of node Y in true DAG, and m < K S N. This represents a substantial improvement over the PC algorithm's complexity of O((N benchmark datasets in Bayesian network demonstrate a runtime reduction of up to 1300-fold, while also improving the accuracy of the learned structure in terms of structural Hamming distance.
1) where M max d, m)- Simulation results over some practical
Furthermore, we also develop an iterative strategy to learn the structure of Bayesian networks using the proposed extendable algorithms. This algorithm initially learns the structure over two randomly- selected random variables, and at each iteration, one of the remaining variables is added to the set using the extendable algorithms. The accuracy and speed of this iterative algorithm are comparable to, and sometimes better than, those of the PC algorithm.
BACKGROUND
Bayesian network is a probabilistic graphical model that represents a joint probability distribution over set of random variables X' = {X,, ..p......! In general, the joint distribution P(X) can be factorized using the chain rule as - P(X, Xi..... X,-1). Given a network structure, this factorization can be simplified by exploiting conditional independencies among the variables.
Each such factorization corresponds to Directed Acyelic Graph (DAG) G, where the nodes represent the random variables in X, and edges indicate direct dependencies. Specifically, for each conditional 3 are the parents of X,, G includes a edge
term P(X; Pax.). ). where Pax. S. from each parent of X, to X; in G. The concept of d-separation in a DAG formalizes the notion of conditional independence among variables. A trail (or path) between two nodes X and Y in G is a sequence of nodes (X Xo, XI...., Xn, = Y) such that each pair (x+1) is connected by an edge (regardless of direction). The length of a trail is defined as the number of edges it contains, and the distance between nodes is the minimum length among all trails connecting them. A node Z on a trail is called a collider if the edges on the trail meet at Z as X + Z + X
Consider the DAG G with node set X. A trail T between two nodes X and Y in X is active relative to a set of nodes Z if. (i) every non con-collider on T is not member of Z, and (ii) every collider on T is an ancestor of some member of Z. The node subsets X and y are d-separated given the subset 2, if there is no active trail between any node X E X and any node Y E y given 2.
Definition 1 (d-separation) (Koller & Friedman 2009)
If X and y are d-separated given Z, denoted d-sepa (X, 2), we say that the paths between X and y are blocked by Z. Define I(G) as the set of all d-separations in DAG G. Let ((P) denote the set of all conditional independencies implied by the distribution P. The Markov condition imposes that I(G) C I(P) and the distribution P is said be faithfull to the DAG G if I(P) C I()). If I(g) = I(P), as implied by the two assumptions, then G is called a P-map sperfect-map for P.
has been proven that almost all distributions P admit some P-map g (Koller & Friedman 2009 A
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P P-map learner is algorithm, such as PC, that (given a set of instances of the random variables X, drawn from a distribution P) attempts to outputs a P-map for that P. Should the distribution P not admit P-map, then the output will be DAG G, that either violates the faithfulness Markovness assumption. Two following Lemmas indicate a relation between I(g) and I(P).
Lemma 1 (Based (Spirtes et al. 2000 Consider random variables X with joint distribution P that admits a P-map G. Vertices X and are not adjacent in G if and only if X 1 LY for some (possibly empty) U X.
Lemma [Lemma 3.2 in (Koller & Friedman 2009 Consider random variahles X with joint distribution P that admits P-map G. Vertices X and Y are not adjacent in if and only if X MYY Pax or X Y Pay-
EXTENDABLE LEARNING
(X1, (XI,.... [NN } be the set of random variables with the joint probability distribution pr and 9 be an output of a P-map learner algorithm over X. Now, suppose a new variable Y is added, expanding the variable set to X
Let x=
[Y} whose joint distribution is denoted by P. We refer to X' (resp., P) as the extended variable set (resp., distribution). Following the common practice in the literature, we assume that there a P-map for the joint distribution P the extended variables X. but that is not necessarily the case with the joint distribution I' of the original variables X' as explained below. The goal is to efficiently learn a P-map & for X, leveraging the information already encoded in G.
Problem 1 Consider a set of random variables X and let G be the output of a P-map learner applied X. Consider random variable Y and the extended variable set X (Y with joint distribution P. Find a P-map G for P.
challenge arises because adding Y may alter the dependencies among the variables in X. Specif- ically, (X)) is the marginal distribution of P(X' over X. However, since Y was unobserved when was learned, 9 may not accurately capture the dependencies in P' (x)). particular, g may not be P-map for P'(X') due to hidden confounding introduced by Y. For example, when Y a confounding variable (hidden common cause) between two collider nodes in over X. DAG G cannot represent all independencies in P', violating the faithfulness assumption (Spirtes 1995). Consider = (X,, X2, X3, Xa} and as X1 + X2 t Y + X3 < Xa. Marginalization over Y results in two adjacent collider nodes X₂ and Xg leading to two immoralities X₁ - X2 + X3 and X2 + X3 + X4 in G. Since a valid DAG cannot contain adjacent collider nodes forming multiple immoralities, fails to satisfy the P-map condition.
We investigate how adding Y affects the dependencies among the variables in X. Consider two variables X, and X₂ in X. Three possible scenarios occur when Y is added:
Non-adjacent variables remain non non-adjacent: X2 and X₂ are not adjacent in G. they remain non-adjacent in G. by faithfulness, the absence of an edge implies a conditional independence given some subset C X {X1, X2) (Lemma which remains valid even after the inclusion of Y.
2. Spurious adjacencies may be removed: If X, and X₂ are adjacent in & but become conditionally independent given Y and some subset 24 C X') (X,, X2). the edge between them may be removed in G.
3. True adjacencies remain: If X, and X2 are adjacent in g and remain dependent given Y and any subset 24 S X' (X1,X2) the edge between them is preserved in G.
According to the first scenario, we show that following Proposition fflloow from Lemma
Proposition 1 Consider & is a P-map over & and a graph over X. If X, adjacent to X₃ in G. then X; is adjacent of X; in G.
Proof. If X, and X; are not adjacent in 9, Lemma proves there a subset LA S X'') (X,, X,) such that X, X, U. Because is a P-map then the faithfulness assumption means X; and X, are d-separated by 24 in G. Thus there can be edge between X; and X; in G.
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W
X1
X2
X1
X2
X1
X2
X²
(a)
(b)
(c)
X2
X1
X=
Xi
(d)
(e)
Figure I: Structures where not observing can lead spurious edge between X1 and X2. The dashed edge represents the possible direction of the spurious edge when the structure is learned by a P-map learner algorithm and is an unobserved node. In (a) either X1 - X2 or X1 X2 can occur, and in the other structures, only one direction occur.
An important result from Proposition i that adding Y does not introduce new edges between variables in X' that were not already connected in G. Therefore, edges in G and consider potential new edges between Y and the variables in X. We now present our main theoretical results, which characterize how the addition of Y affects the structure of the structure G.
only need to examine existing
Lemma 3 Consider variables & whose joint distribution admits P-map 9. Let Y E & and DAG g be the output of a P-map learner applied to (Y). Then every pair of non-adjacent nodes X, and X2 in & are adjacent in G only if
1. Y is common cause or mediator of X1 and X2 in or or
2. X, is linked to some node W which in turn linked to X2, and Y is linked to both W and
N2 (or the same statement but when Xı and X2 are exchanged).
Proof. X1 and X2 being adjacent in G implies that they remain dependent conditioned on any subset of X, i...
YU C X
L It..
(I)
On the other hand, X1 and X₂ being non-adjacent in 9 implies the existence of a subset of X' that together with Y drive X2 and X₂ independent, i.e.,
34Ex
XI 1
In view of equation [ equation 2 and & being P-map for X. it follows that there exists path T connecting X1 and X in & that is active if Y is not observed, and every path connecting X, and X₂ becomes inactive if Y and 2f are observed. The distance of Y X1 does not exceed two. Otherwise, for every path T. connecting Y to X1, let W; be the neighbor of X, on T, and Vú2 be the neighbor of W: on Tí If Y is not parent for X2, considering Lemma Y cannot impact the existence of an edge between X2 and X2. If Y is parent of X2, then two cases must be checked. (Z) There is no collider node V/ between X1 and X2 by observing W; Vú the path will be inactive and the other paths are blocked by parent nodes of X1 X2. (2) There is a collider node V, between X, and X2, with children W; and V. In this case, either V; a collider on the path between Y and X1, or there is a collider between V; and Y that blocks the path; otherwise, cycle would be formed in the graph. Therefore, two conditions are possible. (i) Y must an adjacent of both X, and X₂ such that Y not a collider between them, (ii) Y must be adjacent of one of them and has a trail with length two to the other. (i) is the same as case (I). For (ii). if there are other structures except what is indicated in Fig d.cc would be led to X2 1 X2 Ut for all' C X, which is contrary to (I)
Considering Lemma 3 only three cases exist where observing Y in & can remove the edge between X1 and X2: When Y confounding (Fig. [ (a)) mediator variable between them (Fig. (b.)) or Y is adjacent to X, and forms a collider with X₂ and W while W is a mediator node between X, and X2 (Fig. (d.))). Only in these cases is there active path between X, and Xa when Y is observed and where that path is blocked by observing Y in G.
Lemma Let & be a P-map for P. If L U for li C X (X1, X2). then cannot be a mediator common cause variable between them in G.
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X1
X2
X1
W
X3
X2
X
(b)
(a)
(c)
Figure 2: Three structures that show the effect of the unobserved node Y on an immorality
Proof. Consider Y as a mediator or common cause variable between X and X2. Then the path X1 = Y X2 is active when Y is hidden variable and X1 & X2/U for all c X X2). D
Lemma 5 Let G be P-map of P over X = X U 3 and G is the output of a P-map learner algorithm. If X X is a collider node in G, then it is a collider node in G.
Proof. Consider an immorality X, - X2 + Xa in G. There is a u C X Xx) so that
X2 & U and X1 J. Xz|U.. Also, for all u C X' {X1, X3] we have X1 / X3 X2 U. Then
X1 X X2,X2, U, Y. If X2 and X2, X3 are adjacent in g. then they form an immorality in G
and X1 + X2 t X3 appears in G. Now, consider the edge X2 < X3 is removed by observing Y. Therefore, X2, X3, and Y may form one of the structures shown in Fig. [ Of course, because X1 L X2 we cannot have a direct path as X2 + Y - X3. As a result, three types of structures might occur. If Y is a confounding variable for X2 and X3 (Fig. (a)) Lemma means there is no edge between and X1 and we have X1 / Xa which in turn means X2 must be a collider node between Y and X3 and the edge between X1, X2 orients as X₁ + X2 in G. In the second case, we have X3 + Y + X2 (Fig. 2 (b)). similar to the previous case, we have X1 + X2. Also, X1 and Y are adjacent, the edge between them orients as X1 + Y due to Lemma In third case (Fig. (c)) if X, + X2 then we have a direct path X3 - W - X2 -> X1 that can be blocked by observing X₂ or X3 1 X, X22 which is contradiction and similar to first cases X1 + X2. Since we have X3 L X2, 2/ it is impossible to have structure with X3 adjacent to Y and W a mediator node between X2 and X3 as X2 + W - X3. If we have an immorality X1 + X2 + X2 but the edges between X1, X2 and X3, X2 are removed by observing Y, so there are direct paths X3 + X2 or X3 W + X2, and X1 Y + X2 or X1 - + X2 in g for some W, V E X. Therefore, the direction between X1. X2, or X3. X2 does not change when Y is added to variables. As a result, the orientations of immoralities G will be unchanged in G, and so all orientations in & between nodes in X' are similar to the orientations in G.
have
3.1 CONSTRAINT-BASER APPROACH
Checking CI tests to detect independencies is the main idea in constraint- based algorithms. Two steps are required to add the new variable Y to the previous structure. The first step is checking the relation between Y and other nodes X, and the second step is investigating the effect of Y on the edges in the previous structure.
The PC algorithm is one of the most popular constraint-based algorithms to learn such structures. Algorithm| iss an extendable version of the PC algorithm. According to the PC algorithm, the quantity of CI tests required to verify the existence of an edge between two nodes is directly proportional to the number of adjacent nodes in the true DAG. Hence, to identify the existence of edge between Y and X E X, it is necessary to perform CI tests between Y and X while conditioning on all subsets of both adjacents of X and Y in ÿ. which are indicated by Adj(Ř, X) and Adj(G, Y), respectively. Moreover, based on the PC algorithm, by adding the node of Y into the graph G. all nodes in X' must be connected to Y, forming an initial graph G. Subsequently, the process involves refining the graph by eliminating any surplus or spurious edges. The count of adjacent nodes to Y is Adj(G, Y) |= N, whereas Adj X) I N for any X E X. Consequently, to determine the existence of edges between Y and each X E X. it is appropriate to initially conduct CI tests on l C. Adj(G, X) and subsequently on Vi c Adj Y). Once the true edges between Y and all X E X' are detected, we can then identify the spurious edges between X, Z E X
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If represents the maximum degree of nodes in 9, and m is the degree of node Y in true DAG, employing the PC algorithm for all nodes in X' X {Y} imposes a bound on the number of CI tests, which is (N + 1) where M max(d, m. This bound is established because the PC algorithm does not leverage information from the prior graph. However, applying the Extendable PC variable the variable set mitigate the number of required CI tests. Table 1 iillustrates the count of CI tests at each step in the Extendable PC algorithm. N2 (resp..mdd2 constrain the number of CI tests in steps (resp. 4), and step may require up KT CI tests, where K E {m, m + 1, N) denotes the number of adjacents of Y after step 2. Nevertheless, in step certain edges between Y and other nodes may be eliminated. If the number nodes adjacent to Y decreases, the number of conditional independence tests will accordingly decrease in step 3. As a result, we have proved that the number of CI tests for the Extendable PC algorithm is always fewer than the PC one; see Proposition 2
algorithm when adding
In addition, Theorem Proves the output of Algorithm Tis P-map.
Algorithm 1: The Extendable PC Algorithm
Input: new variable Y and graph obtained from the PC algorithm over X;; Output: New graph & over the set of variables X' = X' (Y):
Connect Y to all nodes in & and construct the graph 9; Adj(G, Y) = X; // Step 1: Initializing & and the adjacent sets
Adj(G, X) X) for all X X: Sepsel(X,))) 0 for X E:;; m =
while maximum degree of nodes X in & is greater than m do
// Step 2: Checking edges between the new variable and other nodes by conditioning on the neighbors of nodes in X. for X E X
for S Adj(š, and i LYY U
Remove the edge X Y from 9; Scpset(X, Y) t U;
+ 1;
12 m O;
13
while degree of Y in & is greater than m do
// Step 3: Checking the edges between the new node and its the neighbors of the new node.
neighbors by conditioning for X E Adj(G. Y) for U S Adj(G, Y) (X} and U m=
14
15
if 1 U Remove the edge X - from 9; Sepset (X,Y) t U;
18
19
m m
0;
while maximum node degree in is greater than do
// Step 4: Check i ng edges between nodes in X' with observing
new variable Y
for X E Y)
22
for Z E Adj(G, X) (Y)
2 E Adj(G, Y) or Adj(E, X Adj(G, AdiiC,, Y) +
24
for C Adj(C, X) (Z) and U |=
if X 1 (Y) UU
Remove the edge X from g; Sepset(X (X, Z) U:
27
+
if X, Z E Adj(G, Y), and X & Adj Z) // Step 5 Immorality detection
x / 7 Y and Y & Sepset (X, Z) 31
Orient X = = Z X + Z. Orient the other edges by orientation rules in Spirtes et al. 2000).
// Step 6

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Table 1: The number of CI tests for each step of the Extendable PC Algorithm
Step
2
3
4
Number of tests
(N22))
((Km)
(mmd24)
Proposition 2 The number of CI tests of Algorishm 1 fewer than the PC algorithm.
Theorem The output of Algorithms anddd 4 is a P-map.
Proof. The proof is straightforward conclusion using Lemma 1, Lemma and Lemma 4 (in Appendix). Lemma 4 shows that adding new variable cannot add an edge between two nodes. Hence, according Lemma 1, the output skeleton of the proposed extendable algorithm finds the skeleton of the true DAG. Then, Lemma 3 shows that all collider nodes were found correctly by the proposed algorithm. So the output PDAG for constraint-han algorithms, such as Algorithm 1, is P-map structure.
In addition, we use a straightforward modification of the PC algorithm using Proposition ! As discussed above, adding new variable cannot add any edge to the previous structure. Therefore, we can use the previous skeleton G as the input graph of the PC algorithm and check the other CI tests to obtain G. This algorithm is called the Initialized PC algorithm (IPC).
3.2 SCORE-BASED APPROACH
In the score-based approach, score function is used to find optimal structure over all possible DAGs, or perhaps a sub-optimal solution over a subset of possible DAGs. Therefore, the number of DAGs in the search space has key role in the complexity of the structure learning algorithm. If DAG was obtained by score-based algorithm over X', the search space for learning new structure that includes Y could be estimated by Proposition [ and Lemmas 35 This means the number of DAGs in this search space will be lower than all possible DAGs on X. Let Sx be search space on X. The DAGs ς in Sx must satisfy following conditions:
1. If X, E are not adjacent in G. then they are not G.
2. If X, 2' is collider node in G, then it is collider node in g.
3. If X, are adjacent to each other in G, if X,, X, and form a structure similar to one of the structures in Fig. then the edge between them be deleted in g.
4. If X,, X, E are not adjacent to each other in 9, and both them are adjacent to in G, then must be collider (i.e., X; Y < Xj 9).
5. If X,, Xj X' are adjacent to each other G, and both of them are collider nodes G, then must be confounding variable X; + X, in G.
Algorithms and are developed for extendable score- score-based structure learning approach. Algorithm 2 represents general extendable score-based algorithm that includes:(1) search space trimming Tunction (T-function in Algorithm ( that restricts the graph search space, based on the analysis from Lemmas 5 and (2) score-based P-map learner (for example global minimization of the BIC score), that finds the best graph within the restricted search space.
Algorithm 2: The Extendable Score-based Algorithm
Input: A new variable Y and a structure & over X Output: A P-map & over X X u (Y}
SR - T(G, Y,S
// By r-function in algorithm // PF is score-based p-map learner
g (F
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3.3 ITERATIVE STRUCTURE LEARNING APPROACH
We developed a new structure learning paradigm using the extendable approach, allowing standard algorithms to be modified to reduce the run-time. This is achieved through an iterative process where the extendable structure learning algorithm is applied at each step. As shown in Algorithm B] starting with two randomly selected variables from X. denoted as X1 and X2, a structure G1 is learned. Then, a third variable X3 is selected from X 1X1, }. and new structure 92 is formed by incorporating X3 using the extendable algorithm. This process is repeated iteratively, with each new variable, such as X, E X \ {X,, X2, X3). being added to the current set to form the next structure. The procedure continues until all N variables are included, resulting in a P-map graph over X'. Using iterative approach, at each step, we leverage information about the relationships between nodes from the previous graph to determine the current graph. Since the number of nodes impacts the number of CI tests, fewer nodes result in fewer CI tests when applying Lemma] [ which restricts the space of possible graphs. Also, the performance of the iterative algorithms depends variables are selected. According to Lemma and Figure I if the ordering is close to topological causal ordering the performance of the iterative will be closer to the standard algorithm. For example. consider a naive Bayes structure with children {X1,-.. 3 and a parent node Y. With ordering like (X,,... X,, Y), before dealing with Y, the iterative algorithm will first produce the complete graph over {X1. ... ..,). However, with this ordering (Y, X1, Xn)) the number of CI tests for Algorithm will be fewer than the previous ordering.
the order in which
Algorithm 3:
The Iterative P-map learner Algorithm Input: A set of variables X' and their joint probability distribution P Output: A partially directed acyclic graph
X = Xx1
G PPmap lea)ner(A')
while X' \ 2 + 0 do Select X E X \ &
g - Extendable P-map learner(G, X)
p t pu{}} grg 9
4 NUMERICAL RESULTS
We now compare the results of our Extendable PC algorithm with the PC, and Initialized PC
algorithms on the data sets ASIA (Lauritzen Spiegelhalter 1988), CANCER (Korb & Nicholson
2010), SURVEY (Scutari & Denis (202)) EARTHQUAKE (Korb & Nicholson (2000). (0010) AL.RRM
(Beinlich et al. 1989). INSURANCE (Binder et al. 1997). CHILD (Spicgelhalter & Cowell 1992).
WATER (Jensen et al. 1989) SACHS (Jensen & Jensen 2013), MILDEW (Jensen & Jensen 2013).
WIN95PTS (Jensen & Jensen 2013), HEPAR2 (Onisko 2003), and ANDES (Conati et al.) 1997). For
each task, we first draw 10000 instances from distributions for use in structure learning algorithms. For each data set a variable is chosen randomly and a structure is learned over the other variables by the PC algorithm. Then the chosen variable is added to the data set, and the learned structure is considered as the input of the Extendable PC algorithm and Initialized PC to learn the new structure. For iterative PC, the first two variables are chosen randomly, and the iterative PC is used estimate the structure over the whole of variables. The number of CI tests for the PC, Initialized PC, and Extendable PC algorithms are shown in Table the runtime in Table In addition, by considering the structural Hamming distance, we recorded the number of incorrect edges either missing or extra compared to the true graph and divided by the total number of edges in the true DAG (Table These results suggest that the extendable approach can significantly reduce both the number of required CI tests and the runtime, particularly in large networks. Additionally, Table 5 shows the number of CI tests for iterative PC and PC algorithms, and Tables [6] and illusttrte the runtime and error of them. The iterative approach applied to the PC algorithm demonstrates the runtime across most datasets is less than the standard PC algorithm, and the error did not change.
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Table 2: Number of CI tests
DATASET
EXTENDABLE PC
INITIALIZED PC
PC
EARTHQUAKE
15
48
CANCER
12
39
45
SURVEY
11
49
55
ASIA
26
87
124
CHILD
1242
2124
SACHS
618
682
971
ALARM
103
745
3283
MILDEW
200
670
3629
WIN95PTS
86
1975
12501
INSURANCE
147
1571
5078
WATER
71
278
1346
HEPAR2
536
5108
23202
ANDES
277
11426
68375
Table 3: Run-Time (sec)
DATASET
EXTENDABLE PC
INITIALIZED PC
EARTHQUAKE
0.033
0.116
0.283
CANCER
0.029
0.115
0.294
SURVEY
0.022
0.186
0.425
ASIA
0.071
0,244
0.744
CHILD
6.11
33.76
65.71
SACHS
15.39
15.86
34.16
ALARM
1.45
17.95
22.10
MILDEW
28.13
31.01
316
WIN9SPTS
0.688
77.81
111
INSURANCE
2.36
36.9
58.28
WATER
0.289
1.38
5.77
HEPAR2
47.57
474
1832
ANDES
1.97
532
2652
Table 4: Structural Hamming Distance divided by the total number of true edges (%)
DATASET
EXTENDABLE PC
INITIALIZED PC
PC
EARTHQUAKE
0
CANCER
0
0
SURVEY
ASIA
12.5
12.5
CHILD
4
SACHS
0
0
0
ALARM
8.7
8.7
8.7
MILDEW
13
17.4
17.4
WIN9SPTS
38.4
38.4
38.4
INSURANCE
30.8
30.8
30.8
WATER
57.6
59.1
59.1
HEPAR2
51.2
51.2
51.2
ANDES
19.5
19.5
19.5
CONCLUSION
The proposed extendable structure learning approach allows to add new variables to the given (correct) Bayesian network, with a significantly lower computational burden compared with learning structure from scratch. The proposed approach be applied all constraint-based and score-based algorithms. A key challenge is using P-map learner algorithms in the presence of hidden variable. In this case, the output of the algorithms is P-map and in some situations the

---

Published as a conference paper at ICLR 2025
faithfulness assumption is violated. We proposed Lemmas to detect situations in which unfaithfulness can occur while there an unobserved variable. Then, we proposed an extendable strategy for variable is added the set of variables. We applied the extendable
constructing P-map when approach to the PC algorithm, then shows that this algorithm could reduce the runtime up to 1300 times compared with the PC when compared with the Initialized PC algorithm. In addition, the iterative paradigm for structure learning based on the extendable approach, and demonstrated that this approach can be used effectively for all types of structure learning algorithms. The structure learning starts with two variables and then incrementally adds one variable at a time, to the previous structure using the extendable approach. until all variables are added to the graph. The iterative PC algorithm can reduce the number of CI tests and the runtime for most datasets, while also increasing accuracy in some cases. future work, optimizing variable ordering to reduce the number of CI tests in the iterative approach could enhance efficiency.
variable is added the set of variables and up to 270 times
Table 5: Number of CI tests
Table 6: Run-Time (sec)
DATASET
ITERATIVE PC
PC
DATASET
ITERATIVE PC
PC
EARTHQUAKE
31
57
EARTHQUAKE
0.09
0.283
CANCER
27
45
CANCER
0.06
0,294
SURVEY
37
55
SURVEY
0.09
0.425
ASIA
66
124
ASIA
0.15
0.744
CHILD
3344
2124
CHILD
124
65.71
SACHS
1276
971
SACHS
35.5
34.16
ALARM
4847
3283
ALARM
81
22.1
MILDEW
2597
3629
MILDEW
715
316
WIN95PTS
10412
12501
WIN95PTS
144
111
INSURANCE
2589
5078
INSURANCE
39.5
58.28
WATER
872
1346
WATER
3.11
5.77
HEPAR2
8371
23202
HEPAR2
597
1832
ANDES
35327
68375
ANDES
307
2652
Table 7: Structural Hamming Distance divided by the total number of true edges (%)
DATASET
ITERATIVE PC
PC
FARTHQUAKE
0
0
CANCER
0
SURVEY
0
ASIA
12.5
12.5
CHILD
4
4
SACHS
0
0
ALARM
17.4
8.7
MILDEW
54.3
17.4
WIN9SPTS
31.25
38.4
INSURANCE
26.9
30.8
WATER
47
59.1
HEPAR2
46.3
51.2
ANDES
23.4
19.5
REFERENCES
Juan R. Alcobe. Incremental methods for bayesian network structure learning. Artificial Intelligence Communications, 18(1):11-62, 2005.
L.A. Beinlich, HJJ. Suermondt, R.M. Chavez, and G.F. Cooper. The alarm monitoring system: A case study with two probabilistic inference techniques for belief networks. In Proceedings of the 2nd European Conference Artificial Intelligence in Medicine, pp. 247-256. Springer- Verlag, 1989.
Jeffrey Binder, Daphne Koller, Stuart Russell, and Keiji Kanazawa. Adaptive probabilistic networks with hidden variables. Machine Learning, 29(2-3):2132444 1997.
10

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Published as a conference paper at ICLR 2025
Johan Bollen, Huina Mao, and Xiaojun Zeng. Twitter mood predicts the stock market. Journal of Computational Science, 2(1):1-8, 2011.
David Card. The causal effect of education on earnings. In Handbooo of Labor Economics, volume 3, pp. 1801-1863. 1999.
C. Conati, A. S. Gertner, VanLehn, and M. Druzdzel. On-line student modeling for coached problem solving using bayesian networks. In Proceedings of the 6th International Conference on User Modeling, pp. 231-242, Springer- Verlag, 1997.
Ruocheng Guo, Li Cheng, Jundong Li, Peter R. Hahn, and Huan Liu. A survey of learning causality with data: Problems and methods. ACM Computing Surveys (CSUR), 53(4):1-37, 2020. 2020.
A.I. Jensen and F.V. Jensen. Midas-an influence diagram for management of mildew in winter wheat. arXiv preprint arXiv: 2013.
Finn V. Jensen, Uffe Kjærulff, Karl G. Olesen, and Jens Pedersen. Et forprojekt til et ekspertsystem for drift af spildevandsrenssnngg (an expert system for control of waste water treatment pilot project). Technical report, 1989. Technical Report, Judex Datasystemer A/S, Aalborg, In Danish.
N.K. Kitson, A.C. Constantinou, Z. Guo, Y. Liu, and K. Chobtham. A survey of bayesian network structure learning. 2021.
Damla Kocacoban and James Cussens. Online causal structure learning in the presence of latent variables. 2019 18th IEEE International Conference On Machine Learning And Applications (ICMLA), pp. 392-395, IEEE, December 2019.
Daphne Koller and Nir Friedman. Probabilistic Graphical Models: Principles and Techniques. MIT Press, 2009.
Kevin B. Korb and Ann E. Nicholson. Bayesian Artificial Intelligence. CRC Press, 2010.
Steffen L. Lauritzen and David J. Spiegelhalter: Local computations with probabilities on graphical structures and their application expert systems. Journal the Royal Statistical Society: Series B (Methodological),
1072))157-194, 1988.
Agnieszka Onisko. Probabilistic Causal Models in Medicine: Application to Diagnosis of Liver Disorders. PhD thesis, Institute Biocyberneties and Biomedical Engineering, Polish Academy of Sciences, Warsaw, Poland, 2003.
Brian A. Primack, Ariel Shensa, Jeanine E. Sidani, Emily O. Whaite, Lloyd Yi Lin, David Rosen, Jason B. Colditz, Ana Radovic, and Elizabeth Miller. Social media use and perceived social isolation among young adults in the us. American Journal of Preventive Medicine, 53(1):1-8, 2017.
Marco Scutari and lean-Baptiste Denis. Bayesian Networks: With Examples in R. Chapman and Hall/CRC, 2021.
David Spiegelhalter and Robert G. Cowell. Learning in probabilistic expert systems. In J.M. Bernaroo, J.O. Berger, A.P. Dawid, and A.F.M. Smith (eds.), Bayesian Statistics 4, pp. 447-466. Clarendon Press, Oxford, 1992.
Peter Spirtes. Building causal graphs from statistical data in the presence of latent variables. In Studies in Logic and the Foundations of Mathematics, volume 134, pp. 813-829. Elsevier, 1995.
Peter Spirtes, Clark N. Glymour, Richard Scheines, and David Heckerman. Causation, prediction. and search. MIT Press, 2000.
11

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Published as a conference paper at ICLR 2025
A
APPENDIX
Algorithm 4: The Extendable Constraint-based Algorithm
Input: A new variable Y and a structure 9 over X
Output: PDAG 9 over X X XU(Y)
Form the & over nodes X by connecting Y to all nodes & by undirected edge; for X E
// Step 1:
Check the edge between Y and X
for X E Adj(G, Y)
11 Step
for 7 E Adj(G. X)
Z E Adj(g, or Adj(G, n Adj(E, 2) Adj(G, Y) Check the edge between X and Z
Orient the new edges using the orientation rules in (Spirtes et al. 2000). // Orientation
Algorithm 5:
The Iterative PC Algorithm
Input:
A set of variables A' and their joint probability distribution P Output: partially directed acyelic graph
Sepset (X1,X2)
X'
Sepsot PC(() while X' &' o 0 do
X E X 1
9. Sepset + PCCl X, Sepset.)
&+X \u{x}
G-ğ
Orient the edges using the orientation rules in (Spirtes et al. 2000).
// Orientation
Algorithm 6: The PC Algorithm
Input: A set of variables X' and their joint probability distribution P
Output: partially directed acyelic graph
Form the complete undirected graph G over nodes X:; Sepset(X, Y) = @ for all X, Y E X:;
m
while maximum node degree in G is greater than m do
for X X
// CI tests
for Y E Adj(G, X)
for 14 C Adj(G, X) \ {Y} and u | m if X 1 Y u
Remove the edge X from 9; Scpset(X,X) < U;
1;
12 Orient the edges using the orientation rules in Spirtes et al. 2000).
// Orientation
12

---

Published as a conference paper at ICLR 2025
Algorithm 7: The Extendable exhaustive search structure learning algorithm
Input:
A new variable Y and a structure 9 over X'
Output: A P-map & over X' = X U {Y}
Form G the union of 9 and one-poni point graph Y
i i=1
while < do
Ng DAG Finder(g)
Ng + Y,g)
// By T-function in algorithm
g t arg maxgeNg ScoreBic(9)
i+i+1
Algorithm 8: T-function (search space trimming)
Input: Y, graph structure & over X, and set of initial DAGs over X denoted as Sxx Output: Sx for X' = X (U
for G Sx
for Xi, E X if x; & Adi(š, and X; X,) Delete & from Sx
(X, + X,+ + Xx) & and ((X; t- X3 t- xx) (X, + X, - Xx) E Delete from Sx
X; E Adi(š, Xj) and X, & Adj(g.X,)
if Edges between X,, Xj and Y do not form structure similar any of the
structures in Fig.
Delete from Sx
if X, & Adj( and X, X, E Adj(G, Y) and (X, - + X,) g Delete from Si
10
11
if X; E Adj(C, Xj) and Xi, X, are collider nodes in & and (X; +- Y - Xj) EG
13
Delete from Sx
14 Return Sx
13


Reviews:
[{"review_id": "Bo8MlhFFTn", "reviewer": "Reviewer_RR2Y", "summary": "This paper proposes an iterative/incremental algorithm to learn the structure of a Bayeisan network from observed data.\nThe algorithm added one variable at a time, and modify the previously learn structure accordingly.\nThe novelty is to prove that adding one variable only leads to deletion of previous edges, and therefore trim the search space of possible networks.\n\nIn particular, Lemma 1 shows that if the new variable Y satisfies some properties, a certain kind of edges in the old graph can be safely deleted. The paper introduced constrained-based and score-based approaches to utilize this lemma. The results is a reduction of number of CI tests.\n\nExperiments on 13 datasets shows that the running time is significantly reduced without compromising accuracy of the learned structure.", "questions": "- How the score-based and constraint-based methods are used in the extendable PC algorithm?\n- Are the more existing work that incrementally learn the structure? What are their strengths and weaknesses?", "limitations": "", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "+ structure learning is a challenging problem due to exponentially large search space, and the paper makes progress in speeding up the search without sacrificing accuracy.\n+ the proof is solid and the presentation is clear.\n+ strong experimental results in accuracy and running time.", "weaknesses": "- The scope of the paper (symbolic Baysian network) may not fit ICLR conference (representation learning).\n- Insufficient discussion of related work. There shall be a large number of related work on incremental structure learning, while the submission only cite two most relevant ones ((Kocacoban & Cussens, 2019) and (Alcobe, 2005)). This makes the contribution of the submission less clear.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "fCL3f49roZ", "reviewer": "Reviewer_g7c8", "summary": "This paper provides a novel perspective to the Bayesian structure learning problem with an efficient mechanism to add new variables to an underlying graphical structure. Specifically, the learning strategy hinges on efficiently incorporating a new variable $Y$ into an existing Bayesian network ${\\cal G}$ over the set of variables ${\\cal X}$, which results in an. updated Bayesian network $\\bar {\\cal G}$ on the augmented set of variables ${\\cal X} \\cup Y$. This learning strategy is further extended to provide a novel learning paradigm for structure learning for Bayesian networks. Experiments demonstrate significant computational efficiency over existing state-of-the-art algorithms.", "questions": "1. Is the Extendable P-map learner in Algorithm 3 implemented using Algorithm 1?\n2. What is Sepset(X,Y)?\n3. An intuitive explanation for why the iterative learning algorithm outperforms PC algorithm while learning the complete graph is recommended.\n4. How is this work positioned relative to the prior works that study the information-theoretic limits of Bayesian structure learning?\n\n\n____________________________________________\nPOST AUTHOR REBUTTAL\nI thank the authors for their detailed response. I will maintain my rating.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The problem studied in the paper is well-motivated and will be of interest to ML community. Recycling the existing information when new variables are added or revealed provides a novel perspective to the structure learning problem. The iterative structure learning algorithm is a valuable contribution. Experiments convincingly demonstrate the computational efficiency of the proposed approach.", "weaknesses": "I found the discussions and organization of Section 3 to be more convoluted than necessary. In particular, it would be helpful to have the relevance of Algorithms 2-4 explictly elucidated.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "7zrKA7W6k0", "reviewer": "Reviewer_wmKg", "summary": "The paper studies the structure learning problem in Bayesian networks, i.e.. learning a directed acyclic graph (DAG) which defines the conditional probability distribution  over the given variables. Particularly, an extendable structure learning strategy is proposed to update an existing Bayesian network graph efficiently, when  a new variable is introduced. Two approaches (constraint-based and score-based) are discussed for extendable structure learning, which leverage the previous graph structure, and have   much lower computational cost compared to relearning the graph. It is then shown that these procedures can be used to design an iterative algorithm for structure learning. Numerical results on many graph datasets illustrate the performance of  the proposed method, and shows significant speedup over the approach where the graph is learned from scratch.", "questions": "I have the following comments about the paper:\n\n1. Performance guarantees such as error analysis, or on how to check whether the algorithm has converged to the correct graph structure are not discussed. \nA discussion on how to verify the correctness of  the P-map finder algorithm used should be added. If the iterative approach is used, how does the error accumulate?\nAre the conditions in Lemma 1 and 2 sufficient for  the P-map finder algorithm output the true structure?\n\n\n2. The key assumption that when a new variable $Y$ is added to the existing set $X$, that no new edges get assigned between the elements of $X$, this assumption needs further explanation. This seems to be a necessary condition of the algorithm to have low computational cost. In many situations, the introduction of a new variable might introduce new dependencies between existing nodes, e.g., in root-cause analysis, causal learning, molecular prediction, and others. Also, such situations could occur in time evolving DAGs. Further discussion on these will illustrate the applicability of the proposed method to different problems. \n\n\n3. Minor Comment:\ni. In abstract, algorithm named PC might not be known to general readers. Similarly, FCI. Some of these acronyms are not defined.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Strengths:\n1.  Extendable structure learning for Bayesian networks is studied and two new approaches are proposed. \n2. The new approaches achieve much lower runtime than relearning the graphs without prior structure. \n3. Numerical results are presented on  many graph datasets showing  significant speedup.", "weaknesses": "Weakness:\n1. Performance guarantees for the proposed method are not presented.\n2. Details about a key assumption can be further discussed.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "NYm2fdXtt6", "reviewer": "Reviewer_hH34", "summary": "The paper proposes an extendable stracture learning method for Bayesian networks, which updates an existing network by adding new variables. The authors also propose a iterative approach for structured learning by starting with a small set and adding the remaining variables to the P-map graph. The authors run the extendable PC algorithm on multiple datasets, and show that their approach requires fewer number of CI tests compared to the original approaches.", "questions": "- Is there any bound on the total number of CI tests required for running Algorithm 3?", "limitations": "", "rating": 3, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- The paper is well organized. The notations and definitions are mostly self contained.\n- The authors provide experiments on multiple datasets, and show that their approach save significant runtime because of the fewer number of CI tests required in the proposed iterative method. \n- The proposed method seems straightforward and easy to implement in practice.", "weaknesses": "I am not an expert on this line of literature. My main concern is that the paper seems very heuristic with no formal guarantees: \n- While the experimental results look good, I wonder if the proposed method of extendable PC has any consistency, faithfulness, or optimality guarantee. \n- The results in Table 2 - 3 suggest that extendable PC always has a better runtime with fewer number of CI tests compared to PC. Can that be proved?\n- The result in Table 5 shows that the proposed iterative PC does not always require fewer CI tests compared to PC. Under what statistical or topological conditions will that happen? In my opinion this is be the bigger risk, because without any theoretic characterization, this shows the possibility that the proposed approach does not generalize.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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": "yimMgWayVr", "reviewer": "Reviewer_dGLi", "summary": "This paper presents an efficient method for updating Bayesian network structures as new variables are introduced, eliminating the need for retraining from scratch. The approach reduces computational costs by up to 1300x without sacrificing accuracy. The authors also propose an iterative strategy that builds the network iteratively, thereby offering runtime benefits comparable to common algorithms like PC while maintaining accuracy. This scalable approach is well-suited for real-time and high-dimensional applications.", "questions": "n/a", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1) A simple iterative type algorithm for learning bayesian networks, which is computationally efficient. \n2) Insights on unidentifiability and its relationship to faithfulness of the graph. \n3) Backed up by theoretical claims", "weaknesses": "n/a", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "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:

Decision: Accept (Poster)