Composing Non-Conjugate Factor Graphs with Closed-Form Variational Inference
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| Main Authors: | , , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910269098164224 |
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| author | Lukashchuk, Mykola Yemets, Kyrylo Kouw, Wouter M. Bagaev, Dmitry Şenöz, İsmail Beck, Jeff de Vries, Bert |
| author_facet | Lukashchuk, Mykola Yemets, Kyrylo Kouw, Wouter M. Bagaev, Dmitry Şenöz, İsmail Beck, Jeff de Vries, Bert |
| contents | Stacking probabilistic building blocks into deeper architectures typically breaks closed-form inference. We show that closed-form inference can be preserved. We identify five factor-graph primitives: a bilinear factor, an exponential link, a Gamma prior, a Gaussian likelihood, and an equality node, and prove that any model composed from them admits closed-form variational message passing. The construction works because each primitive preserves a small set of message families: under mean-field factorization, messages on Gaussian variables remain Gaussian and messages on precision variables remain Gamma, while the only non-conjugate interface, the exponential link, remains tractable through the Gaussian moment-generating function and the sufficient statistics of the Gamma family. We demonstrate composition at increasing depth, from static ensembles through input-dependent gating to split-branch routing, and show that stacking routing layers encodes arbitrary decision trees, establishing universal function approximation with closed-form inference. Applied to ensemble time-series forecasting, the framework yields a Bayesian mixture of experts in which gating functions are inferred rather than learned, providing calibrated uncertainty over expert selection across five benchmark datasets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_29467 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Composing Non-Conjugate Factor Graphs with Closed-Form Variational Inference Lukashchuk, Mykola Yemets, Kyrylo Kouw, Wouter M. Bagaev, Dmitry Şenöz, İsmail Beck, Jeff de Vries, Bert Machine Learning Artificial Intelligence Stacking probabilistic building blocks into deeper architectures typically breaks closed-form inference. We show that closed-form inference can be preserved. We identify five factor-graph primitives: a bilinear factor, an exponential link, a Gamma prior, a Gaussian likelihood, and an equality node, and prove that any model composed from them admits closed-form variational message passing. The construction works because each primitive preserves a small set of message families: under mean-field factorization, messages on Gaussian variables remain Gaussian and messages on precision variables remain Gamma, while the only non-conjugate interface, the exponential link, remains tractable through the Gaussian moment-generating function and the sufficient statistics of the Gamma family. We demonstrate composition at increasing depth, from static ensembles through input-dependent gating to split-branch routing, and show that stacking routing layers encodes arbitrary decision trees, establishing universal function approximation with closed-form inference. Applied to ensemble time-series forecasting, the framework yields a Bayesian mixture of experts in which gating functions are inferred rather than learned, providing calibrated uncertainty over expert selection across five benchmark datasets. |
| title | Composing Non-Conjugate Factor Graphs with Closed-Form Variational Inference |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2605.29467 |