Many studies involving large Markov chains require determining a smaller representative (aggregated) chain. Each superstate in the representative chain represents a group of related states in the original Markov chain. Typically, the choice of number of superstates in the aggregated chain is ambiguous, and based on the limited prior know-how. This paper presents a structured methodology of determining the best candidate for the number of superstates. It achieves this by comparing aggregated chains of different sizes. To facilitate this comparison a new quantity called heterogeneity of a superstate is developed, and subsequently it is used to establish the notion of marginal return of an aggregated chain. In particular, the notion of marginal return captures the decrease in the heterogeneity upon a unit increase in the number of superstates in the aggregated chain. Maximum Entropy Principle (MEP), from statistical mechanics, justifies the developed notion of marginal return, as well as the quantification of heterogeneity. Through simulations on synthetic Markov chains, where the number of superstates are known a priori, it is observed that the aggregated chain with the largest marginal return identifies this number. In case of Markov chains that model real-life scenarios it is shown that the aggregated model with the largest marginal return identifies an inherent structure unique to the scenario being modelled; thus, substantiating the efficacy of the proposed methodology.

许多涉及大型马尔可夫链的研究需要确定较小的代表性（聚合）链。代表链中的每个超状态代表原始马尔可夫链中的一组相关状态。通常，聚合链中超级状态数量的选择是模棱两可的，并且基于有限的先验知识。本文提出了一种结构化的方法来确定超级州数量的最佳候选者。它通过比较不同大小的聚合链来实现这一点。为了便于进行这种比较，开发了一个称为超级国家异质性的新量，随后它被用来建立边际收益的概念的聚合链。特别是，边际回报的概念反映了在聚合链中超级国家数量增加一个单位时异质性的减少。来自统计力学的最大熵原理 (MEP) 证明了边际收益的发展概念以及异质性的量化。通过对合成马尔可夫链的模拟，其中超级状态的数量是先验的，可以观察到具有最大边际回报的聚合链识别了这个数字。在对现实生活场景建模的马尔可夫链的情况下，表明具有最大边际回报的聚合模型识别了被建模场景独有的固有结构；因此，证实了拟议方法的有效性。