  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.
