# The Ordered Matrix Dirichlet for State-Space Models

arXiv: https://arxiv.org/abs/2212.04130 · HTML: https://mlsanity.com/paper/2212.04130/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 45 references that are in the map.
- First posted on arXiv 8 Dec 2022.

## Builds on (most cited first, 10 of 10)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Pyro: Deep Universal Probabilistic Programming](https://mlsanity.com/paper/1810.09538/) (2018)
- [Fast Differentiable Sorting and Ranking](https://mlsanity.com/paper/2002.08871/) (2020)
- [Composable Effects for Flexible and Accelerated Probabilistic Programming in NumPyro](https://mlsanity.com/paper/1912.11554/) (2019)
- [Understanding predictive information criteria for Bayesian models](https://mlsanity.com/paper/1307.5928/) (2013)
- [SemAxis: A Lightweight Framework to Characterize Domain-Specific Word Semantics Beyond Sentiment](https://mlsanity.com/paper/1806.05521/) (2018)
- [Bayesian Poisson Tucker Decomposition for Learning the Structure of International Relations](https://mlsanity.com/paper/1606.01855/) (2016)
- [Bayesian Poisson Tensor Factorization for Inferring Multilateral Relations from Sparse Dyadic Event Counts](https://mlsanity.com/paper/1506.03493/) (2015)
- [Poisson--Gamma Dynamical Systems](https://mlsanity.com/paper/1701.05573/) (2017)
- [An Ordinal Latent Variable Model of Conflict Intensity](https://mlsanity.com/paper/2210.03971/) (2022)

## Built on by (most cited first, 2 of 2)

- [Unsupervised Contrast-Consistent Ranking with Language Models](https://mlsanity.com/paper/2309.06991/) (2023)
- [The ALℓ0CORE Tensor Decomposition for Sparse Count Data](https://mlsanity.com/paper/2403.06153/) (2024)

## Abstract (arXiv)

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.