The Ordered Matrix Dirichlet for State-Space Models
This work addresses the need for interpretable latent state modeling in domains like international relations, though it is incremental as it builds on existing state-space model frameworks.
The paper tackles the problem of modeling latent states with intrinsic orderings in state-space models by introducing the Ordered Matrix Dirichlet (OMD) prior, which enforces stochastic dominance in rows of transition and emission matrices, and demonstrates that models using OMD recover interpretable ordered structure without sacrificing predictive performance.
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.