Clustering of nodes in Bayesian Networks (BNs) and related graphical models such as Dynamic BNs (DBNs) has been demonstrated to enhance computational efficiency and improve model learning. Typically, it involves the partitioning of the underlying Directed Acyclic Graph (DAG) into cliques, or optimising for some cost or criteria. Computational cost is important since BN and DBN inference, such as estimating marginal distributions given evidence or updating model parameters, is NP-hard. The challenge is exacerbated by cost dependency, where inference outcomes and hence clustering cost depends on both nodes within a cluster and the mapping of clusters that are connected by at least one arc. We propose an algorithm called Dependent Cluster MAPping (DCMAP) which is shown analytically, given an arbitrarily defined, positive cost function, to find all optimal cluster mappings, and do so with no more iterations than an equally informed algorithm. DCMAP is demonstrated on a complex systems seagrass DBN, which has 25 nodes per time-slice, and captures biological, ecological and environmental dynamics and their interactions to predict the impact of dredging stressors on resilience and their cumulative effects over time. The algorithm is employed to find clusters to optimise the computational efficiency of inferring marginal distributions given evidence. For the 25 (one time-slice) and 50-node (two time-slices) DBN, the search space size was $9.91\times10^9$ and $1.51\times10^{21}$ possible cluster mappings, respectively, but the first optimal solution was found at iteration number 856 (95\% CI 852,866), and 1569 (1566,1581) with a cost that was 4\% and 0.2\% of the naive heuristic cost, respectively. Through optimal clustering, DCMAP opens up opportunities for further research beyond improving computational efficiency, such as using clustering to minimise entropy in BN learning.

翻译：贝叶斯网络及其相关图模型（如动态贝叶斯网络）中的节点聚类已被证明能提升计算效率并改进模型学习。该方法通常涉及将底层有向无环图划分为团，或针对某些成本或准则进行优化。由于贝叶斯网络与动态贝叶斯网络的推断（例如给定证据估计边缘分布或更新模型参数）属于NP难问题，计算成本至关重要。成本依赖性进一步加剧了这一挑战——推断结果以及相应的聚类成本既取决于聚类内的节点，也取决于通过至少一条弧连接的聚类之间的映射关系。我们提出了一种名为依赖聚类映射的算法，该算法在给定任意定义的正成本函数条件下，经分析证明能够找到所有最优聚类映射，且其迭代次数不超过同等信息水平的算法。DCMAP在一个复杂系统海草动态贝叶斯网络上得到验证，该网络每时间片包含25个节点，能够捕捉生物、生态和环境动力学及其相互作用，以预测疏浚压力对恢复力的影响及其随时间累积的效应。本算法通过寻找最优聚类来优化给定证据时推断边缘分布的计算效率。对于25节点（单时间片）和50节点（双时间片）的动态贝叶斯网络，其搜索空间规模分别为$9.91\times10^9$和$1.51\times10^{21}$种可能的聚类映射，但首个最优解分别在856次（95%置信区间852-866）和1569次（1566-1581）迭代时找到，其成本仅为朴素启发式成本的4%和0.2%。通过实现最优聚类，DCMAP为超越计算效率提升的进一步研究开辟了新途径，例如利用聚类最小化贝叶斯网络学习中的熵。