The Ordered Matrix Dirichlet for State-Space Models

TL;DR

This paper introduces the Ordered Matrix Dirichlet as a prior for ordered stochastic matrices in state-space models, enabling interpretable latent structures in dynamic systems like international relations without sacrificing predictive accuracy.

Contribution

It proposes the OMD prior for ordered matrices and demonstrates its effectiveness in state-space models, including a new dynamic Poisson Tucker decomposition for relational data.

Findings

OMD enables recovery of interpretable ordered latent structures

Models with OMD maintain strong predictive performance

Application to international relations data shows practical utility

Abstract

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.