# Learning Markov Processes as Sum-of-Square Forms for Analytical Belief Propagation

arXiv: https://arxiv.org/abs/2604.07525 · HTML: https://mlsanity.com/paper/2604.07525/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 22 references that are in the map.
- First posted on arXiv 8 Apr 2026.

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

- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Differentiable Particle Filters: End-to-End Learning with Algorithmic Priors](https://mlsanity.com/paper/1805.11122/) (2018)
- [Non-parametric Models for Non-negative Functions](https://mlsanity.com/paper/2007.03926/) (2020)
- [Subtractive Mixture Models via Squaring: Representation and Learning](https://mlsanity.com/paper/2310.00724/) (2023)
- [PSD Representations for Effective Probability Models](https://mlsanity.com/paper/2106.16116/) (2021)
- [Sum of Squares Circuits](https://mlsanity.com/paper/2408.11778/) (2024)
- [Moment-Based Exact Uncertainty Propagation Through Nonlinear Stochastic Autonomous Systems](https://mlsanity.com/paper/2101.12490/) (2021)
- [Uncertainty Propagation in Stochastic Systems via Mixture Models with Error Quantification](https://mlsanity.com/paper/2403.15626/) (2024)
- [Nonlinearity and Uncertainty Informed Moment-Matching Gaussian Mixture Splitting](https://mlsanity.com/paper/2412.00343/) (2024)
- [Universal Learning of Stochastic Dynamics for Exact Belief Propagation using Bernstein Normalizing Flows](https://mlsanity.com/paper/2509.15533/) (2025)

## Abstract (arXiv)

Harnessing the predictive capability of Markov process models requires propagating probability density functions (beliefs) through the model. For many existing models however, belief propagation is analytically infeasible, requiring approximation or sampling to generate predictions. This paper proposes a functional modeling framework leveraging sparse Sum-of-Squares (SoS) forms for valid (conditional) density estimation. We study the theoretical restrictions of modeling conditional densities using the SoS form, and propose a novel functional form for addressing such limitations. The proposed architecture enables generalized simultaneous learning of basis functions and coefficients, while preserving analytical belief propagation. In addition, we propose a training method that allows for exact adherence to the normalization and non-negativity constraints. Our results show that the proposed method achieves accuracy comparable to state-of-the-art approaches while requiring significantly less memory in low-dimensional spaces, and it further scales to 12D systems when existing methods fail beyond 2D.