A Novel Hyper-Heuristic Algorithm for Bayesian Network Structure Learning Based on Feature Selection

Abstract

Bayesian networks (BNs) are effective and universal tools for addressing uncertain knowledge. BN learning includes structure learning and parameter learning, and structure learning is its core. The topology of a BN can be determined by expert domain knowledge or obtained through data analysis. However, when many variables exist in a BN, relying only on expert knowledge is difficult and infeasible. Therefore, the current research focus is to build a BN via data analysis. However, current data learning methods have certain limitations. In this work, we consider a combination of expert knowledge and data learning methods. In our algorithm, the hard constraints are derived from highly reliable expert knowledge, and some conditional independent information is mined by feature selection as a soft constraint. These structural constraints are reasonably integrated into an exponential Monte Carlo with counter (EMCQ) hyper-heuristic algorithm. A comprehensive experimental study demonstrates that our proposed method exhibits more robustness and accuracy compared to alternative algorithms.

1. Introduction

Bayesian networks are among the most effective methods for studying high-dimensional uncertainty problems and have been widely used in medicine [1,2], fault diagnosis [3], environmental protection [4], and other fields. When BN theory is used to solve practical problems, the graph model of causal relationships between variables must be established first.

Global and local structure learning are the two current classifications for Bayesian network structure learning methods. The global learning method is further subdivided into three primary categories: constraint-based methods, score-based methods, and hybrid methods. Conditional independence (CI) tests are the foundation of constraint-based methods, which are used to determine causal links. Both SGS and TPDA [5] determine the network structure by evaluating the conditional independence among nodes. However, the computational complexity of these two algorithms increases exponentially with the number of nodes, so they are not suitable for dealing with large networks. To mitigate the computing cost, the PC and PC-Stable algorithms [6,7] take advantage of the characteristic that nodes in sparse networks do not require high-order independence tests. They employ a reduction strategy to efficiently establish Bayesian networks from the sparse model. However, the small sample size leads to a decrease in the statistical power of the CI test and an increase in the estimation error. The score-based method regards structure learning as a combinatorial optimization problem. Its core lies in searching for the optimal solution in the possible structure space through the score function and the search algorithm. This method consists of two key components: the design of the scoring function and the optimization of the search strategy. Prevalent scoring functions encompass K2, the BIC, and the MDL, whereas typical search spaces consist of three types: the directed acyclic graph (DAG) space [8], which is composed of all possible directed acyclic graphs; the equivalence class space [9], which groups Markov equivalent DAGs into the same equivalence class; and the ordering space [10], which is based on the topological ordering of variables. There are two primary types of search strategies: approximate learning and exact learning. Exact learning [11,12,13,14] theoretically ensures strict convergence to the theoretical optimal solution, but the computational cost increases exponentially with the number of variables. Therefore, this method is practically unfeasible for large-scale datasets. In contrast, approximate learning algorithms sacrifice controllable accuracy in exchange for computational feasibility and are the preferred choice for high-dimensional and real-time scenarios. Approximate learning algorithms converge probabilistically. The quality of the solution is affected by the sampling quantity and the design of the algorithm. The earliest proposed algorithms, hill-climbing algorithm [5] and K2 [15], are based on the greedy search strategy and are prone to fall into local optima. Subsequently, a series of metaheuristic algorithms, such as the genetic algorithm (GA) [16], evolutionary programming [17], ant colony optimization (ACO) [18], cuckoo optimization (CO) [19], water cycle optimization (WCO) [20], particle swarm optimization (PSO) [21,22], artificial bee colony algorithm (ABC) [23], bacterial foraging optimization (BFO) [24], and firefly algorithm (FA) [25], have been proposed to improve search efficiency and escape local optima. Although these metaheuristic algorithms demonstrate exceptional efficacy in addressing real-world issues, two hurdles persist when confronted with complex and extensive networks:

• A single metaheuristic algorithm cannot assure optimal performance while learning various Bayesian networks.
• In the case of large-scale Bayesian networks, these metaheuristic algorithms often succumb to local optima, leading to diminished accuracy of the resulting graph.

Hybrid methods combine two classes of algorithms in an attempt to integrate the advantages of both. The search space of the score-based method is typically constrained by the constraint-based algorithm, with the most commonly used hybrid algorithm being the MMHC algorithm [26].

The core objective of the local learning method is to identify the local dependency structure of the target variable. This is specifically manifested in the discovery of two key sets of nodes: the parent–child set (PC) and the Markov blanket (MB). The GS [27] algorithm, through a two-stage framework of the growing stage (full coverage) and the shrinking phase (minimization refinement), has for the first time provided a theoretically complete guarantee for Markov boundary learning, becoming a landmark method in the field of causal feature selection. However, the local structure learning approach is dependent on CI tests, similar to the constrained global structure learning method, and its accuracy significantly decreases as the sample size decreases. The BAMB [28] method and the EEMB [29] algorithm achieve an optimal balance between data utility (covering real dependencies) and time cost (controlling the computational scale) through the dynamic alternation mechanism of PC learning and spouse learning. The F2SL [30] algorithm achieves the coordinated optimization of causal discovery efficiency and accuracy through the process of feature selection, skeleton construction, and direction determination. LCS-FS [31] uses mutual information as a proxy for causal correlation measurement. By eliminating the condition set and incorporating a dynamic pruning mechanism, it significantly improves efficiency.

In conclusion, each method has distinct limitations and advantages. The core challenge currently faced in Bayesian network structure learning lies in how to achieve a deep synergy among different methodologies. Such as the integration strategy of constraints and scoring methods, the collaborative mechanism of global and local learning, and the enhancement of theoretical complementarity. This is the focus of this paper’s research. Hyper-heuristics have been employed across various domains to supplant metaheuristics because of their superior generalization capability and precision [32,33,34,35]. We explore the application of hyper-heuristics in lieu of metaheuristics for global structure learning.

The main contributions of this article are as follows:

• A feature selection-based method for MB discovery was employed to construct the search space and direction determination, and this knowledge served as soft constraints.
• We develop a library of low-level operations for BN structure learning.
• We propose an Exponential Monte Carlo with counter hyper-heuristic (EMCQ-HH) algorithm that integrates the soft and hard constraints derived from local structure learning techniques.

The subsequent sections of this article are structured as follows. Background information and associated concepts are introduced in Section 2. The study’s research design is described in Section 3. The performance evaluation and experimental findings are covered in Section 4. This work is concluded with some observations and recommendations for further research in Section 5.

2. Background

This section first introduces the DAG structure for discrete data through a BN and presents some basic properties and theorems of the DAG structure. This introduces the information theory basis for feature selection.

2.1. BN

Bayesian networks are a type of probabilistic graphical model. They represent the conditional dependencies among random variables through a directed acyclic graph and perform quantitative modeling on the basis of conditional probability distributions. A Bayesian network can be denoted as a two-tuple $(G,\theta )$, where G denotes a DAG and $\theta$ signifies the set of conditional probability distributions. As a DAG, G can be described as a two-tuple $(V,E)$, where $V=\left\{X_1,\cdots ,X_n\right\}$ represents the set of nodes, and each node corresponds to a variable. E represents the set of directed edges, and each edge denotes the dependency relationship between two nodes. The BN parameter $\theta$ is a distribution, which is manifested as a series of factorizations on the graph G.

$$P(X_1,\cdots ,X_n)=\prod _{i=1}^{n}P\left(X_i\right|Pa\left(X_i\right))$$

(1)

where $P\left(X_i\right|Pa\left(X_i\right))$ represents the conditional probability of $X_i$ when its parent node $Pa\left(X_i\right)$ is given. Each node in $Pa\left(X_i\right)$ has directed edges pointing to $X_i$. Figure 1 shows a simple BN structure in which nodes X and Y are both parent nodes of node Z. These three variables form a simple V-structure, and their conditional probability tables are shown in the figure.

For discrete variables X, Y, and Z, when and only when the joint probability distribution can be expressed as the product of the conditional marginal distributions:

$$X\perp Y|Z\iff P(X,Y\left|Z\right)=P(X,Y)P(Y,Z)$$

(2)

X and Y are conditionally independent given Z. The independence test can be characterized by the ${\chi }^2$ statistic and mutual information to depict the dependence and independence relationships between variables. For the two discrete variables X and Y, if the independence condition $X\perp Y$ holds, then the statistic

$$U^2=-2\sum _{x,y}p\left(xy\right)log{\displaystyle \frac{p\left(xy\right)}{p\left(x\right)p\left(y\right)}}$$

(3)

approximately follows a ${\chi }^2$-distribution with $(r_x-1)(r_y-1)$ degree of freedom. For the three discrete variables X, Y and Z, if the independence condition $X\perp Y|Z$ holds, then the statistic

$$U^2=-2\sum _{x,y,z}p(x,y,z)log{\displaystyle \frac{p(x,y|z)}{p\left(x\right|z\left)p\right(y\left|z\right)}}$$

(4)

approximately follows a ${\chi }^2$-distribution with $(r_x-1)(r_y-1)r_z$ degrees of freedom. In the above definition, $r_x$, $r_y$, and $r_z$ represent the number of possible values for X, Y, and Z, respectively.

2.2. Information Theory

Entropy $H\left(X\right)$ is the core metric in information theory used to quantify the uncertainty of a random variable X. Its calculation formula is

$$H\left(X\right)=-\sum _{x}p\left(x\right)logp\left(x\right)$$

(5)

The mutual information is a key metric in information theory for quantifying the shared information between two random variables X and Y. It measures the statistical dependence between the variables through the difference in probability distributions, and the calculation formula is

$$I(X;Y)=\sum _{x\in X}\sum _{y\in Y}p(x,y)log{\displaystyle \frac{p(x,y)}{p\left(x\right)p\left(y\right)}}$$

(6)

Conditional mutual information quantifies the remaining dependence or shared information between two variables X and Y, given a third random variable Z. The calculation formula is

$$I(X;Y|Z)=\sum _{z\in Z}p\left(z\right)\sum _{x\in X}\sum _{y\in Y}p(x,y|z)log{\displaystyle \frac{p(x,y|z)}{p\left(x\right|z\left)p\right(y\left|z\right)}}$$

(7)

2.3. Scoring Function

The BIC scoring function in Bayesian networks is the core indicator for evaluating the quality of the network structure. Its goal is to strike a balance between goodness of fit and model complexity. The formula is defined as

$$BIC(g,D)=logp(D;\widehat{\theta },g)-{\displaystyle \frac{logm}{2}}{d}_{\theta }$$

(8)

where g represents the network that requires scoring calculation, D denotes the training set, $\widehat{\theta }$ indicates the maximum likelihood estimator, m represents the sample size, and $d_{\theta }$ is equal to the dimension value of $\theta$.

3. Methodology

The proposed method is a combined strategy based on constraint-based and score-based algorithms. With respect to the constraint-based part, we employed the WLBRC [36] algorithm to extract conditional independence information, thereby establishing three restricted search spaces ($S{S}_1$,$S{S}_2$,$S{S}_3$) and identifying partial edges. The identified edges and search spaces act as soft constraints, whereas hard constraints are provided by expert knowledge. With respect to the score-based part, soft and hard constraints are incorporated into our proposed hyper-heuristic algorithm as structural constraints.

3.1. Proposed Hyper-Heuristic Algorithm

The hyper-heuristic algorithm can be divided into two main parts: a high-level selection strategy and a low-level heuristic operator library. For the Bayesian network structure learning problem, we assemble a set of low-level heuristic operators with different functions by referring to the proposed metaheuristic algorithm and consider a simple parameter-free EMCQ method [35] for the high-level strategy. Figure 2 shows the framework of the proposed EMCQ-HH algorithm.

In our work, EMCQ is similar to the simulated annealing method, where the probability density is defined as

$$\Psi =e^{-T/q}$$

(9)

where T is the number of current iterations and q is the control parameter for the diversity of the selection operator. The pseudocode of the EMCQ-HH algorithm is illustrated in Algorithm 1. The initial population starts with all edges identified by the hard and soft constraints, followed by multiple hill-climbing operations on $S{S}_3$ to optimize the initial structure and generate the initial population. At the beginning of the iteration, an operator is randomly selected to update the population, and then the operator continues to be selected if the global optimal score is improved. If the global optimal score does not increase, the probability density function is computed, and a random number between 0 and 1 is generated. If the random number is less than $\Psi$, the operator is retained; otherwise, a new operator is randomly selected. Finally, the algorithm stops when a local optimal solution is reached or the maximum number of iterations is reached.

Algorithm 1: EMCQHH

3.2. Low-Level Heuristics

By analyzing various heuristic algorithms for Bayesian network structure learning, we extract two mutation operators, three neighborhood hill-climbing operators, four learning operators, two restart operators, and expert knowledge operators to construct the underlying operator library. In addition, to better integrate expert knowledge and structural priors, we have made appropriate modifications to the underlying operators.

3.2.1. Mutation Operators

The two mutation operators extracted are the mutation operator and the neighborhood search operator. The former is extracted from a classical Bayesian network structure learning particle swarm algorithm [22], whereas the latter is a perturbation operator commonly used in combinatorial optimization problems. The former contains only one mutation operator, whereas the latter contains three operators: add, delete, and reverse.

3.2.2. Neighborhood Hill-Climbing Operators

The three neighborhood hill-climbing operators extracted are the chemotactic operator, employed bees, and onlooker bees. The chemotactic operator is extracted from a bacterial foraging optimization algorithm [24] applied to Bayesian network structure learning but does not consider the exchange operation. The operator operates on only one node at a time, greedily choosing to add, remove, or reverse edges for continuous operation. The last two operators are extracted from an artificial bee colony optimization algorithm [23] applied to BN structure learning, and they use different strategies to select an individual for climbing operations.

3.2.3. Learning Operators

The four learning operators extracted are the cognitive personal operator, cooperative global operator, moth-flame optimization, and teaching-learning-based optimization. The cognitive personal operator and cooperative global operator are extracted from the same particle swarm optimization algorithm [22] applied to Bayesian network structure learning, whose learning objects are individual historical optimal and global optimal, respectively. Moth-flame optimization randomly selects the historically optimal structure of an individual as a flame and learns from it. The teaching-learning-based optimization operator randomly selects an individual as its collaborator and learns from it if it is better than itself.

3.2.4. Restart Operators

The elimination and dispersal operator, derived from the bacterial foraging optimization method [24], is one component. Another component is scout bees [23], derived from an artificial bee colony optimization method. For large Bayesian networks, blindly rebooting new structures may be inefficient for escaping local optima. Therefore, we consider local restarts on a globally optimal structure, which only changes the local structure of one node at a time.

The elimination and dispersal operator works as follows:

• Remove all parents of the selected node X.
• Add the nodes that can increase the BIC score to the set of parent variables of X.
• Delete one by one the nodes in the parent node set of X that can increase the BIC score.
• Reverse the nodes in the parent node set of X one by one that can increase the BIC score.

Whether the operator is executed is dynamically controlled by whether the parameter $c_3$ is greater than the random number. Its adjustment formula is as follows:

$$c_3=0.1+0.9\left(L/L_{max}\right)$$

(10)

where L indicates that the global highest BIC score has not increased for L generations and where $L_{max}$ is used as the stop condition of our algorithm. The algorithm stops if the global maximum BIC score has not been increased for $L_{max}$ generations.

The scout bees work as follows:

• Randomly select a parent of node X and perform a parent–child conversion.
• Add the nodes that can increase the BIC score to the set of parent variables of X.
• Delete one by one the nodes in the parent node set of X that can increase the BIC score.
• Continue with Step 1 until the original parent of X has been executed.

Whether the operator is executed is controlled by the parameters $l_m$. If the highest BIC score in the history of an individual over successive $l_m$ generations is not improved, it is executed; otherwise, it is not executed.

3.2.5. Expert Knowledge Operator

The function of the expert knowledge operator is to guide the search by assigning hard and soft constrained knowledge to a partial proportion of individuals. In our work, expert knowledge consists of two parts: the edges that must exist and the edges that must not exist, the ratio of which we give directly in the experiment.

4. Experiments

In our experiment, $S{S}_2$ serves as the search space for the foundational operator of the hyper-heuristic algorithm. We evaluate the algorithm against other classical constraint-based approaches and score-based algorithms, with all subsequent experiments conducted on an AMD 1.7 GHz CPU equipped with 16 GB of RAM.

4.1. Datasets and Evaluation Metrics

The performance of the EMCQ-HH algorithm is evaluated by applying it to six standard Bayesian networks under four different sample sets. These six different-sized networks originate from the BNLEARN repository (https://www.bnlearn.com/bnrepository/, accessed on 9 March 2025), each collecting 1000, 3000, 5000, and 10,000 samples, respectively.

Table 1. Summary of networks.

Network	Nodes	Edges	Max.indeg	Max.outdeg	Domain Range
Alarm	37	46	4	5	2–4
Hepar2	70	123	6	17	2–4
Win95pts	76	112	7	10	2–2
Munin	189	282	3	15	1–21
Andes	223	338	6	12	2–2
Pigs	441	592	2	39	3–3

shows the basic situation of the selected Bayesian networks. The domain range denotes the minimum and maximum number of discrete values across all nodes.

In our experiment, the performance indicators of the algorithm are as follows:

• BIC: The mean and standard deviation of the BIC scores for multiple runs of the learning network.
• AE: The quantity of incorrect edges contained in the learned network compared with the original network.
• DE: The quantity of wrongly deleted edges in the learned network that were incorrectly deleted in comparison to the original network.
• RE: The quantity of edges with incorrect reversals in the learned network that were incorrectly reversed in comparison to the original network.
• RT: The average running time of the algorithm for multiple runs.
• F1: We use F1 to measure the accuracy of the algorithm. F1 is the harmonic average of precision and recall, where $F1=2\ast Precision\ast Recall/(Precision+Recall)$. Precision is defined as the ratio of the correct number of edges in the output to the total number of edges in the algorithm’s output, whereas recall is the ratio of the correct number of edges in the output to the actual number of edges in the original DAG. Therefore, F1 = 1 is the best case, and F1 = 0 is the worst case. The higher the F1 score is, the better.

4.2. Performance Evaluation of EMCQ-HH

Table 2. Parameters of the EMCQ-HH.

Param.	Value	Descriptions
p	0.1	The constraint rate of the edge that is sure to exist
q	0.5	The constraint rate of the nonexistent edge
$nPop$	50	The population size
$L_{max}$	$2·n$	The maximum number of unpromoted iterations allowed
$MaxIt$	5000	The maximum number of iterations allowed
$l_m$	20	Control parameters of the elimination and dispersal operator
$zj$	0.5	Percentage receiving expert guidance

lists the parameters of the EMCQ-HH.

Table 3. Performance of EMCQ-HH algorithm on different datasets.

Network	Dataset	BIC	AE	DE	RE	RT(s)	F1
Alarm	1000	$-1.2493\times {10}^4$ ± 53.83	0.2 ± 0.45	4.40 ± 0.55	2.20 ± 0.45	30.00	0.8974 ± 0.0166
	3000	$-3.4996\times {10}^4$ ± 222.41	0.40 ± 0.55	3.20 ± 1.10	1.80 ± 0.84	46.86	0.9192 ± 0.0292
	5000	$-5.6922\times {10}^4$ ± 8.51	0.80 ± 0.45	2 ± 0	0 ± 0	39.72	0.9692 ± 0.0048
	10,000	$-1.1211\times {10}^5$ ± 4.92	1.20 ± 0.45	2.20 ± 0.45	1.80 ± 0.45	62.72	0.9231 ± 0
Hepar2	1000	$-3.4081\times {10}^4$ ± 29.67	0 ± 0	71.60 ± 0.89	1.60 ± 0.55	58.92	0.5711 ± 0.0074
	3000	$-1.0037\times {10}^5$ ± 11.44	0 ± 0	56.20 ± 0.45	1.60 ± 0.55	96.23	0.6870 ± 0.0076
	5000	$-1.6643\times {10}^5$ ± 264.73	0 ± 0	47.20 ± 1.30	1.80 ± 0.45	126.51	0.7444 ± 0.0098
	10,000	$-3.3176\times {10}^5$ ± 680.34	0 ± 0	36 ± 0.71	3.60 ± 1.34	218.80	0.7943 ± 0.0099
Win95pts	1000	$-1.0767\times {10}^4$ ± 192.53	5.80 ± 1.92	33.80 ± 4.66	4.60 ± 1.67	158.56	0.7504 ± 0.0496
	3000	$-2.9859\times {10}^4$ ± 241.03	5 ± 1.22	25.80 ± 1.64	4.60 ± 2.51	183.29	0.8031 ± 0.0370
	5000	$-4.8323\times {10}^4$ ± 372.60	4.80 ± 2.39	21.20 ± 2.77	3.60 ± 1.52	236.90	0.8400 ± 0.0391
	10,000	$-9.4670\times {10}^4$ ± 2224.53	5.60 ± 1.82	13.60 ± 2.30	3.20 ± 1.64	427.16	0.8814 ± 0.0325
Munin	1000	−6.0507 $\times {10}^4$ ± 1744.05	43.40 ± 4.39	154.80 ± 2.59	7.80 ± 2.17	3676.36	0.5080 ± 0.0086
	3000	−1.5007 $\times {10}^5$ ± 1816.09	43 ± 1	129 ± 2	13.67 ± 2.08	4510.92	0.5666 ± 0.0148
	5000	−2.3602 $\times {10}^5$ ± 3403.33	46.67 ± 2.31	112.67 ± 3.51	12.33 ± 3.06	6014.27	0.6166 ± 0.0225
	10,000	−4.4662 $\times {10}^5$ ± 10,871.15	43 ± 1.73	100.67 ± 3.21	10.33 ± 2.89	8753.79	0.6635 ± 0.0079
Andes	1000	$-9.5801\times {10}^4$ ± 45.41	1.33 ± 0.58	80.33 ± 1.53	2 ± 1	1004.97	0.8565 ± 0.0020
	3000	$-2.8213\times {10}^5$ ± 64.71	1 ± 1	54.67 ± 1.53	2.33 ± 0.58	1344.88	0.9031 ± 0.0025
	5000	$-4.6752\times {10}^5$ ± 0.96	1.33 ± 0.58	39.33 ± 0.58	3.67 ± 0.58	1843.07	0.9248 ± 0
	10,000	$-9.3386\times {10}^5$ ± 2119.20	1 ± 1	28 ± 1	1.33 ± 0.58	3352.94	0.9512 ± 0.0038
Pigs	1000	−3.4827 $\times {10}^5$ ± 0	0 ± 0	0 ± 0	0 ± 0	2850.50	1 ± 0
	3000	−1.0120 $\times {10}^6$ ± 0	0 ± 0	0 ± 0	0 ± 0	2635.21	1 ± 0
	5000	−1.6760 $\times {10}^6$ ± 0	0 ± 0	0 ± 0	0 ± 0	2914.08	1 ± 0
	10,000	−3.3268 $\times {10}^6$ ± 0	0 ± 0	0 ± 0	0 ± 0	3388.36	1 ± 0

lists the learning results of our method across different training sets of Bayesian network structures. We computed the mean and standard deviation across 10 iterations.

The standard deviation of the BIC score over the four datasets of the pig network is 0, indicating that the algorithm consistently learns the network with an identical BIC score for these datasets. For the other five networks, except for Alarm3000, Hepar5000, Hepar10000, Win95pts10000, Munin10000, and Andes10000, the standard deviation of the learned structure on other datasets is similarly small. A significant reason for analyzing the variation in the BIC score is that the final output structure changes due to hard constraints. Overall, our algorithm has stable convergence performance.

According to the structural errors, learning from the four datasets of the Pigs network results in a structure that is the same as the Pigs network. In the other five networks, these structural errors are concentrated in the DE. With increasing sample size, DE tended to decrease, whereas AE and RE did not significantly change. This discovery shows that appropriately increasing the data helps to recover mistakenly removed edges in the learned structure. Moreover, when the amount of data increases, the F1 score of the learned structure evidently increases, suggesting that a reasonable increase in the sample size enhances the learning precision of our method. As the amount of data increased, the average execution time did not increase dramatically but rose gradually, indicating that our method is adept at handling large-scale datasets.

4.3. Comparison with Some Other Algorithms

This section compares EMCQ-HH with five existing Bayesian network structure methods. PC-stable is the classic method of conditional independent testing, whereas the GS algorithm is the inaugural theoretically sound Markov boundary algorithm; both methods have been categorized as constraint-based methods. F2SL is a score-based method of structural learning that relies on feature selection. BNC-PSO is a scoring method based on a swarm intelligence search, and the MMHC is the most commonly used hybrid algorithm for learning the structure of Bayesian networks.

For fair comparison, the EMCQ-HH algorithm does not consider expert knowledge. In addition, to compare the search performance of the BNC-PSO algorithm and the EMCQ-HH algorithm, we additionally include soft constraint information in BNC-PSO, specifically maintaining a consistent search starting point and search space.

Table 4. BIC scores of the algorithms for different datasets. Bold denotes the BIC score that was the best found amongst all methods.

Sample	Network	PC-Stable	GS	F2SL	MMHC	BNC-PSO	EMCQHH
1000	Alarm	$-1.2466\times {10}^4$	$-1.7902\times {10}^4$	$-1.2475\times {10}^4$	$-1.2561\times {10}^4$	$-1.2367\times {10}^4$	−1.2339 × ${10}^4$
	Hepar2	$-3.5318\times {10}^4$	$-3.4834\times {10}^4$	$-3.4261\times {10}^4$	$-3.4048\times {10}^4$	$-3.3994\times {10}^4$	−3.3992 × ${10}^4$
	Win95pts	$-1.3271\times {10}^4$	$-1.5375\times {10}^4$	$-1.0987\times {10}^4$	$-1.0917\times {10}^4$	$-1.0534\times {10}^4$	−1.0485 × ${10}^4$
	Munin	−	$-9.8582\times {10}^4$	$-7.3424\times {10}^4$	$-8.7319\times {10}^4$	$-5.8794\times {10}^4$	−5.6270 × ${10}^4$
	Andes	$-9.9080\times {10}^4$	$-1.0625\times {10}^5$	$-1.0074\times {10}^5$	$-9.8045\times {10}^4$	$-9.6344\times {10}^4$	−9.5844 × ${10}^4$
	Pigs	−	$-4.5682\times {10}^5$	$-3.5210\times {10}^5$	$-3.4827\times {10}^5$	$-3.5032\times {10}^5$	−3.4827 × ${10}^5$
3000	Alarm	$-3.5083\times {10}^4$	$-5.3065\times {10}^4$	$-3.5693\times {10}^4$	$-3.6877\times {10}^4$	$-3.4722\times {10}^4$	−3.4715 × ${10}^4$
	Hepar2	$-1.2180\times {10}^5$	$-1.0207\times {10}^5$	$-1.0124\times {10}^5$	$-1.0030\times {10}^5$	$-1.0007\times {10}^5$	−1.0000 × ${10}^5$
	Win95pts	$-3.5202\times {10}^4$	$-4.4373\times {10}^4$	$-3.1103\times {10}^4$	$-3.0680\times {10}^4$	$-2.9436\times {10}^4$	−2.9321 × ${10}^4$
	Munin	−	$-2.9331\times {10}^5$	$-1.7190\times {10}^5$	$-2.0244\times {10}^5$	$-1.4934\times {10}^5$	−1.4397 × ${10}^5$
	Andes	$-2.9066\times {10}^5$	$-3.1683\times {10}^5$	$-2.9970\times {10}^5$	$-2.8752\times {10}^5$	$-2.8284\times {10}^5$	−2.8213 × ${10}^5$
	Pigs	−1.0120 $\times {10}^6$	$-1.3594\times {10}^6$	$-1.0122\times {10}^6$	−1.0120 $\times {10}^6$	$-1.0155\times {10}^6$	−1.0120 × ${10}^6$
5000	Alarm	$-5.7240\times {10}^4$	$-8.8094\times {10}^4$	$-5.8871\times {10}^4$	$-5.9328\times {10}^4$	$-5.7181\times {10}^4$	−5.6858 × ${10}^4$
	Hepar2	$-1.7224\times {10}^5$	$-1.7118\times {10}^5$	$-1.6849\times {10}^5$	$-1.6635\times {10}^5$	$-1.6592\times {10}^5$	−1.6590 × ${10}^5$
	Win95pts	$-5.6061\times {10}^4$	$-7.0745\times {10}^4$	$-5.0993\times {10}^4$	$-4.8629\times {10}^4$	$-4.7595\times {10}^4$	−4.7410 × ${10}^4$
	Munin	−	$-4.9315\times {10}^5$	$-2.6622\times {10}^5$	$-3.1552\times {10}^5$	$-2.3419\times {10}^5$	−2.2662 × ${10}^5$
	Andes	$-4.8078\times {10}^5$	$-5.3222\times {10}^5$	$-4.9561\times {10}^5$	$-4.7403\times {10}^5$	$-4.6939\times {10}^5$	−4.6763 × ${10}^5$
	Pigs	$-1.6766\times {10}^6$	$-2.2608\times {10}^6$	$-1.6768\times {10}^6$	$-1.6764\times {10}^6$	$-1.6768\times {10}^6$	−1.6760 × ${10}^6$
10,000	Alarm	$-1.1383\times {10}^5$	$-1.7479\times {10}^5$	$-1.1623\times {10}^5$	$-1.1489\times {10}^5$	$-1.1200\times {10}^5$	−1.1197 × ${10}^5$
	Hepar2	$-3.3786\times {10}^5$	$-3.4119\times {10}^5$	$-3.3729\times {10}^5$	$-3.3182\times {10}^5$	$-3.3125\times {10}^5$	−3.3125 × ${10}^5$
	Win95pts	$-1.0726\times {10}^5$	$-1.3627\times {10}^5$	$-1.0393\times {10}^5$	$-9.8666\times {10}^4$	$-9.3616\times {10}^4$	−9.3509 ×${10}^4$
	Munin	−	$-9.8632\times {10}^5$	$-5.0656\times {10}^5$	$-9.7306\times {10}^5$	$-4.5065\times {10}^5$	−4.2882 × ${10}^5$
	Andes	$-9.5765\times {10}^5$	$-1.0700\times {10}^6$	$-1.0036\times {10}^6$	$-9.5184\times {10}^5$	$-9.3407\times {10}^5$	−9.3268 × ${10}^5$
	Pigs	$-3.3270\times {10}^6$	$-4.5233\times {10}^6$	−3.3268 $\times {10}^6$	−3.3268 $\times {10}^6$	$-3.3268\times {10}^6$	−3.3268 × ${10}^6$

and

Table 5. F1 scores of the algorithms for different datasets. Bold denotes the F1 score that was the best found amongst all methods.

Sample	Network	PC-Stable	GS	F2SL	MMHC	BNC-PSO	EMCQHH
1000	Alarm	0.8810	0.4516	0.8434	0.7912	0.8667	0.8315
	Hepar2	0.2927	0.3529	0.2953	0.5464	0.5385	0.5054
	Win95pts	0.4654	0.4459	0.5297	0.5455	0.6146	0.6597
	Munin	0.1180	0.0414	0.3129	0.2722	0.2850	0.3333
	Andes	0.7179	0.5217	0.5820	0.5742	0.7064	0.8152
	Pigs	0.7555	0.0707	0.9735	0.9975	0.9486	1
3000	Alarm	0.9545	0.4179	0.8537	0.7333	0.8764	0.8989
	Hepar2	0.3094	0.3827	0.3067	0.5670	0.6154	0.6392
	Win95pts	0.6932	0.5375	0.6349	0.6884	0.7273	0.7549
	Munin	0.2500	0.0621	0.3000	0.2659	0.3991	0.4169
	Andes	0.8068	0.5594	0.5667	0.6941	0.8066	0.8660
	Pigs	1	0.0943	0.9983	1	0.9713	1
5000	Alarm	0.9545	0.4412	0.8571	0.7957	0.9333	0.9111
	Hepar2	0.4255	0.4294	0.2267	0.5941	0.7122	0.6765
	Win95pts	0.7701	0.5697	0.6526	0.7544	0.7558	0.8462
	Munin	0.3769	0.0403	0.3581	0.3211	0.4915	0.5096
	Andes	0.8485	0.5217	0.5882	0.7797	0.8223	0.8875
	Pigs	0.9983	0.1178	0.9983	0.9958	0.9941	1
10,000	Alarm	0.9318	0.4478	0.8434	0.8602	0.9556	0.9011
	Hepar2	0.5572	0.4881	0.2267	0.7204	0.7793	0.7700
	Win95pts	0.8061	0.6550	0.6237	0.6756	0.8778	0.8357
	Munin	0.5380	0.0604	0.3113	0.3784	0.5031	0.5947
	Andes	0.8736	0.5021	0.3286	0.7101	0.8703	0.9252
	Pigs	0.9992	0.1031	1	0.9992	1	1

report the BIC scores and F1 scores for each algorithm’s output structure under different datasets. Notably, PC-stable and GS do not initially involve score values. However, for uniform comparison, we artificially calculated the scores of the structures they obtained.

As shown in Table 4, EMCQ-HH achieves the highest BIC scores on all datasets except the Munin3000 dataset, which clearly demonstrates the superior performance of the hyper-heuristic BN structure learning algorithm in score search. On 21/24 datasets, the constrained algorithm GS has a significantly lower BIC score than the other five algorithms do, and the difference becomes more pronounced. On the Hepar1000, Hepar3000, and Hepar5000 datasets, the BIC scores of the constrained algorithm PC-stable were markedly inferior to those of the other algorithms. For the three score search algorithms F2SL, MMHC, and BNC-PSO, they won 1, 4, and 19 times, respectively, in the competition of BIC scores. The above analysis shows that the score of the network learned by EMCQ-HH is significantly better than that of PC-stable and GS. For the four scoring algorithms, the ability to search high-scoring networks is ranked as follows: EMCQ-HH > BNC-PSO > MMHC > F2SL. Given that BNC-PSO and EMCQ-HH utilize identical initial structures and search spaces, we assert that the hyper-heuristic exhibits superior generalization ability and accuracy compared with the single metaheuristic.

As shown in Table 5, EMCQ-HH achieves the highest F1 score on 16/24 datasets. An important reason why the BIC and F1 scores do not win simultaneously on multiple datasets is that the data may not be faithful to the underlying Bayesian network. Moreover, akin to the constrained methods PC-stable and GS, the F1 scores of the EMCQ-HH algorithm exhibit an upward trend with larger sample sizes, suggesting that an appropriate increase in sample size enhances the graph precision of the EMCQ-HH algorithm. The F1 scores of the BNC-PSO algorithm are markedly inferior to those of the EMCQ-HH method. This shows that the EMCQ-HH algorithm has better search performance on large-scale networks than does the BNC-PSO algorithm. The GS algorithm outputs significantly lower F1 scores than the other algorithms do, especially on the Munin and Pigs networks. When the sample size is 1000, the F1 scores of the PC-stable algorithm and F2SL algorithm are markedly inferior to those of the other score-based algorithms for the Hepar2 network, and the F1 scores of the PC-stable algorithm are markedly inferior to those of the other scoring algorithms for the Munin network and Pigs network. In addition, when the F1 scores of the scoring algorithms on the six datasets with a sample size of 1000 were compared, the MMHC algorithm won 1 time, the BNC-PSO won 1 time, and the EMCQ-HH won 4 times. The above results show that the EMCQ-HH algorithm is more effective in learning from small sample datasets.

The above experimental results clearly demonstrate that EMCQ-HH is superior to other comparison algorithms in terms of score-searching ability and output graph accuracy and has stronger robustness in solving inadequate data and large-scale BN learning problems.

5. Conclusions and Future Research

This work employs a local learning method grounded in a feature selection technique and CI test to derive soft constraints; expert knowledge is established as the hard constraints, and both soft and hard constraints are utilized as structural constraints to increase the efficacy of our suggested hyper-heuristic algorithm. The experimental results demonstrate that, compared with other methods, our suggested hyper-heuristic algorithm has superior global search capability and graph precision. In dealing with challenges such as inadequate data and large-scale BN learning, simply searching for high-score structures may reduce the graph accuracy of the learning network. The hard constraints can mitigate this issue, producing high-scoring structures with elevated graph precision and ensuring robust performance. In conclusion, both soft and hard constraints substantially impact the performance of our proposed algorithm. We need to further optimize the low-level operator library and design ways to choose the best low-level operator library in the future.