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

Peter Amorese, Morteza Lahijanian
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:4591-4599, 2026.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-amorese26a, title = { Learning Markov Processes as Sum-of-Square Forms for Analytical Belief Propagation }, author = {Amorese, Peter and Lahijanian, Morteza}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {4591--4599}, year = {2026}, editor = {Khan, Emtiyaz and Li, Yingzhen and Solin, Arno and Ramdas, Aaditya}, volume = {300}, series = {Proceedings of Machine Learning Research}, month = {02--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v300/main/assets/amorese26a/amorese26a.pdf}, url = {https://proceedings.mlr.press/v300/amorese26a.html}, abstract = { 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. } }
Endnote
%0 Conference Paper %T Learning Markov Processes as Sum-of-Square Forms for Analytical Belief Propagation %A Peter Amorese %A Morteza Lahijanian %B Proceedings of The 29th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2026 %E Emtiyaz Khan %E Yingzhen Li %E Arno Solin %E Aaditya Ramdas %F pmlr-v300-amorese26a %I PMLR %P 4591--4599 %U https://proceedings.mlr.press/v300/amorese26a.html %V 300 %X 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.
APA
Amorese, P. & Lahijanian, M.. (2026). Learning Markov Processes as Sum-of-Square Forms for Analytical Belief Propagation . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:4591-4599 Available from https://proceedings.mlr.press/v300/amorese26a.html.

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