Dmitry Bagaev

Mykola Lukashchuk, Kyrylo Yemets, Wouter M. Kouw et al.

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.

Bert de Vries, Wouter Nuijten, Thijs van de Laar et al.

We address the problem of planning under uncertainty, where an agent must choose actions that not only achieve desired outcomes but also reduce uncertainty. Traditional methods often treat exploration and exploitation as separate objectives, lacking a unified inferential foundation. Active inference, grounded in the Free Energy Principle, provides such a foundation by minimizing Expected Free Energy (EFE), a cost function that combines utility with epistemic drives, such as ambiguity resolution and novelty seeking. However, the computational burden of EFE minimization had remained a significant obstacle to its scalability. In this paper, we show that EFE-based planning arises naturally from minimizing a variational free energy functional on a generative model augmented with preference and epistemic priors. This result reinforces theoretical consistency with the Free Energy Principle by casting planning under uncertainty itself as a form of variational inference. Our formulation yields policies that jointly support goal achievement and information gain, while incorporating a complexity term that accounts for bounded computational resources. This unifying framework connects and extends existing methods, enabling scalable, resource-aware implementations of active inference agents.

Dmitry Bagaev, Bert de Vries

We introduce Reactive Message Passing (RMP) as a framework for executing schedule-free, robust and scalable message passing-based inference in a factor graph representation of a probabilistic model. RMP is based on the reactive programming style that only describes how nodes in a factor graph react to changes in connected nodes. The absence of a fixed message passing schedule improves robustness, scalability and execution time of the inference procedure. We also present ReactiveMP.jl, which is a Julia package for realizing RMP through minimization of a constrained Bethe free energy. By user-defined specification of local form and factorization constraints on the variational posterior distribution, ReactiveMP.jl executes hybrid message passing algorithms including belief propagation, variational message passing, expectation propagation, and expectation maximisation update rules. Experimental results demonstrate the improved performance of ReactiveMP-based RMP in comparison to other Julia packages for Bayesian inference across a range of probabilistic models. In particular, we show that the RMP framework is able to run Bayesian inference for large-scale probabilistic state space models with hundreds of thousands of random variables on a standard laptop computer.