What Type of Inference is Active Inference?

Active inference casts decision-making as inference, with the Expected Free Energy (EFE) unifying goal-directed and information-seeking behavior. Recent work showed that EFE minimization can be written as Variational Free Energy (VFE) minimization on a generative model augmented with epistemic priors. We prove that the VFE of the augmented model can be rewritten as the VFE of the predictive model plus explicit entropy-correction terms, making the EFE contribution transparent. We then show that proper EFE-based planning requires combining these epistemic corrections with a planning correction that turns marginal inference into policy optimization, yielding a full variational characterization of EFE-based planning. This clarifies which corrections are needed for cross-entropy planning and for full EFE-based planning. The same entropy-corrected formulation leads to a detailed message-passing scheme for EFE-based planning together with simpler ablations. Experiments on three grid-world environments show that the planning correction already helps when observations are decisive, whereas the additional observation-side epistemic corrections matter most when observations are merely suggestive.

Composing Non-Conjugate Factor Graphs with Closed-Form Variational Inference

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.

Active Inference for Physical AI Agents -- An Engineering Perspective

Physical AI agents, such as robots and other embodied systems operating under tight and fluctuating resource constraints, remain far less capable than biological agents in open-ended real-world environments. This paper argues that Active Inference (AIF), grounded in the Free Energy Principle, offers a principled foundation for closing that gap. We develop this argument from first principles, following a chain from probability theory through Bayesian machine learning and variational inference to active inference and reactive message passing. From the FEP perspective, systems that maintain their structural and functional integrity over time can, under suitable assumptions, be described as minimizing variational free energy (VFE), and AIF operationalizes this by unifying perception, learning, planning, and control within a single computational objective. We show that VFE minimization is naturally realized by reactive message passing on factor graphs, where inference emerges from local, parallel computations. This realization is well matched to the constraints of physical operation, including hard deadlines, asynchronous data, fluctuating power budgets, and changing environments. Because reactive message passing is event-driven, interruptible, and locally adaptable, performance degrades gracefully under reduced resources while model structure can adjust online. We further show that, under suitable coupling and coarse-graining conditions, coupled AIF agents can be described as higher-level AIF agents, yielding a homogeneous architecture based on the same message-passing primitive across scales. Our contribution is not empirical benchmarking, but a clear theoretical and architectural case for the engineering community.

Spike-Timing-Dependent Plasticity for Bernoulli Message Passing

Bayesian inference provides a principled framework for understanding brain function, while neural activity in the brain is inherently spike-based. This paper bridges these two perspectives by designing spiking neural networks that simulate Bayesian inference through message passing for Bernoulli messages. To train the networks, we employ spike-timing-dependent plasticity, a biologically plausible mechanism for synaptic plasticity which is based on the Hebbian rule. Our results demonstrate that the network's performance closely matches the true numerical solution. We further demonstrate the versatility of our approach by implementing a factor graph example from coding theory, illustrating signal transmission over an unreliable channel.

A Spiking Neural Network Implementation of Gaussian Belief Propagation

Bayesian inference offers a principled account of information processing in natural agents. However, it remains an open question how neural mechanisms perform their abstract operations. We investigate a hypothesis where a distributed form of Bayesian inference, namely message passing on factor graphs, is performed by a simulated network of leaky-integrate-and-fire neurons. Specifically, we perform Gaussian belief propagation by encoding messages that come into factor nodes as spike-based signals, propagating these signals through a spiking neural network (SNN) and decoding the spike-based signal back to an outgoing message. Three core linear operations, equality (branching), addition, and multiplication, are realized in networks of leaky integrate-and-fire models. Validation against the standard sum-product algorithm shows accurate message updates, while applications to Kalman filtering and Bayesian linear regression demonstrate the framework's potential for both static and dynamic inference tasks. Our results provide a step toward biologically grounded, neuromorphic implementations of probabilistic reasoning.

Expected Free Energy-based Planning as Variational Inference

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, offers such a foundation by minimizing Expected Free Energy (EFE), a cost function that combines utility with epistemic drives like ambiguity resolution and novelty seeking. However, the computational burden of EFE minimization has remained a major 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 itself as variational inference. Our formulation yields optimal 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.

Improved Depth Estimation of Bayesian Neural Networks

This paper proposes improvements over earlier work by Nazareth and Blei (2022) for estimating the depth of Bayesian neural networks. Here, we propose a discrete truncated normal distribution over the network depth to independently learn its mean and variance. Posterior distributions are inferred by minimizing the variational free energy, which balances the model complexity and accuracy. Our method improves test accuracy on the spiral data set and reduces the variance in posterior depth estimates.

Toward Design of Synthetic Active Inference Agents by Mere Mortals

The theoretical properties of active inference agents are impressive, but how do we realize effective agents in working hardware and software on edge devices? This is an interesting problem because the computational load for policy exploration explodes exponentially, while the computational resources are very limited for edge devices. In this paper, we discuss the necessary features for a software toolbox that supports a competent non-expert engineer to develop working active inference agents. We introduce a toolbox-in-progress that aims to accelerate the democratization of active inference agents in a similar way as TensorFlow propelled applications of deep learning technology.

Realising Synthetic Active Inference Agents, Part I: Epistemic Objectives and Graphical Specification Language

The Free Energy Principle (FEP) is a theoretical framework for describing how (intelligent) systems self-organise into coherent, stable structures by minimising a free energy functional. Active Inference (AIF) is a corollary of the FEP that specifically details how systems that are able to plan for the future (agents) function by minimising particular free energy functionals that incorporate information seeking components. This paper is the first in a series of two where we derive a synthetic version of AIF on free form factor graphs. The present paper focuses on deriving a local version of the free energy functionals used for AIF. This enables us to construct a version of AIF which applies to arbitrary graphical models and interfaces with prior work on message passing algorithms. The resulting messages are derived in our companion paper. We also identify a gap in the graphical notation used for factor graphs. While factor graphs are great at expressing a generative model, they have so far been unable to specify the full optimisation problem including constraints. To solve this problem we develop Constrained Forney-style Factor Graph (CFFG) notation which permits a fully graphical description of variational inference objectives. We then proceed to show how CFFG's can be used to reconstruct prior algorithms for AIF as well as derive new ones. The latter is demonstrated by deriving an algorithm that permits direct policy inference for AIF agents, circumventing a long standing scaling issue that has so far hindered the application of AIF in industrial settings. We demonstrate our algorithm on the classic T-maze task and show that it reproduces the information seeking behaviour that is a hallmark feature of AIF.

Automating Model Comparison in Factor Graphs

Bayesian state and parameter estimation have been automated effectively in the literature, however, this has not yet been the case for model comparison, which therefore still requires error-prone and time-consuming manual derivations. As a result, model comparison is often overlooked and ignored, despite its importance. This paper efficiently automates Bayesian model averaging, selection, and combination by message passing on a Forney-style factor graph with a custom mixture node. Parameter and state inference, and model comparison can then be executed simultaneously using message passing with scale factors. This approach shortens the model design cycle and allows for the straightforward extension to hierarchical and temporal model priors to accommodate for modeling complicated time-varying processes.