## 1 Introduction  
The Bayesian network (BN) [1], which is also called the probabilistic belief network or the causal network [2], is a kind of graphical model and knowledge representation tool. BNs can efficiently give the probabilistic description of the dependence or independence relationships between a set of random variables. A BN is composed of a directed acyclic graphical structure and a set of probability parameters. The directed acyclic graphical structure represents the dependence relationships between various variables, and the corresponding probability parameters specify their degree of dependence. Recently, learning the BN structure from a dataset has received increasing attention [3], and researchers have introduced various learning algorithms to obtain the structure for BNs. According to the modeling type [3; 4; 5], these structure learning algorithms can be classified into methods based on detecting conditional independencies [6; 7], also known as constraint-based methods, the \"score+search\" method [1; 2; 8; 9; 10; 11; 12; 13], and the algorithms that combine the above two methods [14; 15; 16; 17]. As to the score+search method, the BN structure learning is modelled as an optimization problem. A scoring metric is employed to evaluate how well the candidate network structure [3] matches the dataset. The better the candidate network structure matches the dataset, the higher the score. Thus, it can use the optimization technique to search for the network structure with the best score. However, searching for the optimum network for BNs is an NP-hard problem [5; 18], which means those exact methods become unfeasible. Therefore, the approximate algorithms are very useful for quickly obtaining a sufficiently highly qualified network structure. The greedy search methods, including the K2 algorithm and many meta-heuristic methods, such as genetic algorithm (GA) [12], evolutionary programming (EP) [12] and particle swarm optimization (PSO) [14], are frequently used in solving the BN learning problem. As a population-based meta-heuristic technique, ant colony optimization (ACO) also has been introduced successfully into the problem of BN structure learning from a dataset [1; 2; 8; 14; 15]. ACO, which was originally presented under the inspiration from the collective behavior of a real ant system [19; 20], has already been applied to a wide range of optimization problems. [PERSON] et al. [2] first introduced ACO to the problem of BN structure learning. They described all of the elements necessary to tackle the BN learning problem using ACO, and contrast experiments indicated that ACO shows better performance than the estimation of distribution algorithms (EDAs) as well as the greedy hill climbing (HC) algorithm. Based on the constraint-based local discovery algorithm, [PERSON] et al. [1] proposed a local discovery ACO by using the conditional dependence test max-min parents and children method. [PERSON] et al. [15] proposed a hybrid method that combines ACO, the conditional independence test, and the simulated annealing (SA) strategy, and their hybrid ACO method outperforms the basic ACO in terms of computational time and searching capability. However, there are two drawbacks in the above ACO-based hybrid algorithms. Firstly, the conditional independence test can make the computation more complex and unreliable [3]. Secondly, when the number of variables is very large, the ACO-based algorithm will easily fall into the local optimal solution and result in a premature stagnation. To avoid the drawback caused by the conditional independence test, it is necessary to improve the inherent searching mechanism of the ACO algorithm. In this paper, we propose a kind of improved ACO to enhance the convergence and accuracy of the basic ACO in solving the problem of BN structure learning. The improved ACO divides the entire ant colony into several small sub-colonies, denoted as groups.