Bayesian networks are widely utilised in various fields, offering elegant representations of factorisations and causal relationships. We use surjective functions to reduce the dimensionality of the Bayesian networks by combining states and study the preservation of their factorisation structure. We introduce and define corresponding notions, analyse their properties, and provide examples of highly symmetric special cases, enhancing the understanding of the fundamental properties of such reductions for Bayesian networks. We also discuss the connection between this and reductions of homogeneous and non-homogeneous Markov chains.

翻译：贝叶斯网络广泛应用于各个领域，能优雅地表示因子分解和因果关系。我们通过状态合并的满射函数来降低贝叶斯网络的维度，并研究其因子分解结构的保持性。我们引入并定义了相应概念，分析了它们的性质，提供了高对称性特例的实例，从而加深对贝叶斯网络这类约简基本性质的理解。我们还讨论了这种约简与齐次及非齐次马尔可夫链约简之间的联系。