Many dynamical systems in the real world are naturally described by latent states with intrinsic orderings, such as "ally", "neutral", and "enemy" relationships in international relations. These latent states manifest through countries' cooperative versus conflictual interactions over time. State-space models (SSMs) explicitly relate the dynamics of observed measurements to transitions in latent states. For discrete data, SSMs commonly do so through a state-to-action emission matrix and a state-to-state transition matrix. This paper introduces the Ordered Matrix Dirichlet (OMD) as a prior distribution over ordered stochastic matrices wherein the discrete distribution in the kth row stochastically dominates the (k+1)th, such that probability mass is shifted to the right when moving down rows. We illustrate the OMD prior within two SSMs: a hidden Markov model, and a novel dynamic Poisson Tucker decomposition model tailored to international relations data. We find that models built on the OMD recover interpretable ordered latent structure without forfeiting predictive performance. We suggest future applications to other domains where models with stochastic matrices are popular (e.g., topic modeling), and publish user-friendly code.

翻译：许多现实世界中的动态系统自然由具有内在顺序的潜在状态描述，例如国际关系中的"盟友"、"中立"和"敌对"关系。这些潜在状态通过各国随时间推移的合作与冲突互动表现出来。状态空间模型(SSM)将观测数据的动态过程与潜在状态的转移明确关联起来。对于离散数据，SSM通常通过状态-动作发射矩阵和状态-状态转移矩阵实现这一关联。本文提出有序矩阵狄利克雷(OMD)作为有序随机矩阵的先验分布，其中第k行的离散分布在随机意义上主导第(k+1)行的分布，使得向下移动行时概率质量向右偏移。我们在两个SSM中展示了OMD先验的应用：隐马尔可夫模型和针对国际关系数据定制的新型动态泊松塔克分解模型。研究发现，基于OMD构建的模型能够恢复可解释的有序潜在结构，同时不牺牲预测性能。我们建议将该方法推广到其他广泛应用随机矩阵模型的领域(如主题建模)，并公开发布了用户友好的代码。