Active inference selects actions by minimising an expected free energy functional over predicted futures. However, adding an expectation over yet-unobserved outcomes means the free energy functional no longer has a Kullback-Leibler structure, which hinders message passing treatments of inference procedures. We propose an alternative formulation based on a Bethe free energy functional, fully supporting inference by message passing. The epistemic drive is maintained by imposing an information constraint, next to normalisation, marginalisation and form constraints, insisting that the mutual information between future observations, states and parameters given actions must be at least as large as the entropy of the goal prior. For a specific value of the corresponding Karush-Kuhn-Tucker multiplier, the stationary point of this constrained Bethe Lagrangian recovers the expected free energy solution. We show that, as the information demand is varied, the solved multiplier moves through its inactive, interior, and saturated regimes. In the inactive regime the agent's epistemic drive switches off entirely, while in the saturated regime it is maximal. We compare the performance of the constrained Bethe agent on three tasks against EFE and Q-MDP.
Ajith Anil Meera, Pablo Lanillos, Wouter Kouwcs.RO cs.IT cs.LG
An autonomous robot efficiently exploring an unknown environment, such as looking for water sources on Mars, faces two simultaneous demands: building an accurate information map while quickly finding the regions of greatest value, and paying for every meter of travel and the cost of every measurement it takes. Classical information-seeking and reward-seeking criteria address only one of these objectives at a time. Here, we propose Expected Free Energy (EFE), the principled action-selection objective from active inference, as a unifying criterion for budgeted robotic informative path planning. Maintaining a Gaussian-process belief over the information field, our agent plans continuous trajectories that minimize expected free energy under hard path-length constraints. The results from multiple realizations show that EFE-based planning yields accurate posterior maps and locates the highest-value regions simultaneously, outperforming information-theoretic baselines under the same settings. In robotic exploration, these unified, easy-to-tune principled information-gathering strategies facilitate autonomous deployment while enforcing efficiency and resource constraints.
Karim Zaghw, Andrew Pashea, Marc Pritsch +3cs.AI cs.CV
Active inference offers a unified framework for perception, learning, and action, but scaling discrete active-inference models to rich spatial and temporal domains remains difficult. Renormalising generative models (RGMs) address this challenge by composing discrete generative models across spatial and temporal scales, coarse-graining lower-level states and paths into higher-level causes for objects, events, and action. However, fully reproducing and adapting the framework remains difficult: the mathematical exposition is compact, and the reference implementations are deeply integrated within specialized software environments, leaving many algorithmic details implicit. This paper addresses these challenges by providing a self-contained, derivation-oriented account of RGMs together with an open, verified implementation. We explain how the hierarchy is built, how beliefs and actions are updated within it, and how information is passed between levels. Where the published equations and implementation differ in emphasis, we make those choices explicit and explain their modelling consequences. By clarifying the theory and separating it from its original implementation context, this work lowers practical barriers to entry and makes RGMs more transparent, auditable, and reproducible, providing a foundation for future quantitative evaluation and development on machine-learning benchmarks.
Despite advances in artificial intelligence (AI) across multiple sectors, today's AI tools, including deep learning and generative AI, still fail when embedded into physical systems, such as robots and vehicles operating under real-world physical laws. This stems from their inability to maintain reliable world models for long-horizon planning under uncertainty and generalize to unseen scenarios. In this context, wireless networks, through pervasive sensing and communication, can orchestrate physical intelligence. However, current architectures optimize throughput, latency, and reliability and cannot support real-time physical AI coordination, requiring agents to maintain shared spatiotemporal context. To address these challenges, a network of holonic digital twins (HDT-Nets) framework is proposed to deliver real-time physical AI inference through holonic agents that actively reason about their environment rather than passively mirror physical assets. Each HDT is realized as a hierarchical structure spanning the physical agent and network edge, reasoning autonomously at the local level while cooperating with neighboring HDTs to form collectively intelligent units. In HDT-Net, causal Markov blankets spanning sensing, communication, and control determine which agents must coordinate and enable counterfactual reasoning over multi-domain interventions. Active inference within these boundaries unifies perception, action, and learning by minimizing expected free energy while deciding which beliefs to transmit based on their cognitive value to the receiver. Category theory ensures that transmitted beliefs preserve semantic structure across heterogeneous agents with incompatible representations. Finally, integrated information theory quantifies when collective intelligence exceeds independent operation and how network intelligence evolves through coordinated learning and information exchange.
St John Grimbly, Nicolas Kuske, Evert A. Boonstra +7cs.AI
Biological systems must regulate competing needs under limited perceptual bandwidth, where sharpening one estimate costs the capacity to sharpen the others. Any fixed-budget system therefore has to decide where to allocate its perceptual precision. We study this in a foraging agent that must keep several bodily needs satisfied to survive, modelled with active inference. At each step it reads its own body-state beliefs, identifies the most-needed channel, and reallocates a fixed budget of interoceptive precision toward it, so that the same precision-shaped likelihood feeds both belief update and planning. In AffectWorld, a four-channel foraging gridworld, this selective allocation more than doubles learning-phase survival at matched budget against a uniform-precision agent ($0.414$ vs $0.199$ across 11 layouts, $n{=}32$ seeds each, paired cluster-bootstrap $p \leq 10^{-4}$). Two further results sharpen the mechanism. The benefit runs through planning as well as perception, since denying the shaped likelihood to the planner alone removes about half of it. It is also need-aligned, since aiming precision at the least-needed channel does worse than spreading it evenly. The attended channel additionally learns its own dynamics about twice as fast, and stays ahead even at matched observation count, a behavioural trace of the same precision routing, visible in learning speed, not survival.
In noisy social dilemmas, intended actions are stochastically corrupted before execution, so an observed defection may reflect hostile intent or action error. Standard Markov Decision Process (MDP) formulations treat executed actions as states, structurally precluding this distinction and causing systematic over-retaliation. We introduce a Partially Observable MDP (POMDP) formulation encoding opponent intentions as latent states and executed actions as noisy observations, solved within the active inference (AIF) framework with a cost function that decomposes into epistemic and pragmatic components that jointly address inferring current intent and learning how intent evolves. In the Iterated Prisoner's Dilemma with symmetric noise, we derive a critical noise threshold governing cooperation collapse, connecting it to a fixed-point condition on learned priors. Experiments reveal that the value of intention inference is context-dependent: the POMDP provides consistent advantages against conditionally cooperative opponents, but mutual intention inference under sufficient noise produces correlated belief-driven collapse. The advantage is specific to games where intent attribution is decision-relevant.
Predictive coding offers a powerful theory of cortical computation, but corresponding scalable algorithmic implementations for artificial intelligence have remained elusive. This paper introduces the Bayesian reflex, a computational framework that directly instantiates predictive coding through three pillars: belief maintenance via hierarchical generative models, sequential Bayesian updating via prediction-error minimization, and uncertainty-driven action via active inference. We show that recent breakthroughs---ellipsoidal decomposition for exact $i.i.d.$ sampling, recursive Gaussian processes for deep hierarchical inference, and derivative-aware Bayesian optimization---provide the missing algorithmic ingredients. The resulting framework enables mathematically principled, scalable, and brain-inspired continual learning, perception, and decision-making. We illustrate its versatility through applications ranging from climate model evaluation to prime number discovery, offering a blueprint for truly adaptive artificial intelligence.
A recent line of work measures causal emergence in reinforcement learning agents through Integrated Information Decomposition, reporting that $Φ_r$ grows with training and tracks reward improvement. For active inference, this raises the question of how reward-free predictive organization relates to such information-theoretic signatures. I test this within an active inference agent whose architecture separates a fast perception latent $z$ from a slow global latent $g$, where $g$ is driven by prediction error and structurally decoupled from policy gradients. In a reward-free environmental regime-switching protocol, $Φ_r$ concentrates in $g$; its aggregate magnitude is largely architectural and decreases with training. The substantive effect of learning becomes legible only at the atom-compositional level: decoupling flips sign from negative to positive and becomes regime-invariant under environmental change, while downward causation carries the regime-dependent adjustment. These results identify $g$ as the architectural locus of $Φ_r$-relevant temporal organization in an active inference agent, and argue against reading scalar $Φ_r$ as a direct index of learned integration.
Sophisticated Inference is a variant of active inference often associated with recursive belief modeling and tree search. We argue that its central computational role is simpler: within a planning horizon, it makes active inference closed-loop by allowing future actions to depend on future states and observations. This closed-loop structure can be represented in the epistemic-prior variational free energy framework. Epistemic priors supply the active-inference objective, while a joint posterior over future states and actions supplies the state-contingent control structure. We evaluate this decomposition in the Reactivity Maze, a stochastic benchmark designed to separate epistemic incentive from inner-horizon closed-loop control. The comparison includes three variational objectives with the same state-action posterior family, an action-state factorized active inference objective, Sophisticated Inference, and standard Expected Free Energy planning. The results show that neither ingredient is sufficient on its own. Methods without an epistemic component do not seek information, while methods that prevent future actions from depending on future states cannot turn information into reliable goal-reaching. By contrast, both Sophisticated Inference and full-joint epistemic-prior active inference solve the environment by combining epistemic drive with closed-loop inference. These results show that the advantage associated with Sophisticated Inference need not be specific to tree search itself. It arises from the closed-loop form of active inference, and this form can be represented in epistemic-prior variational inference when the posterior keeps future actions dependent on future states.
An agent acting under partial observability must decide when to gather information and which observations are worth their cost. Standard POMDPs value information only through its eventual effect on reward. The $ρ$-POMDP framework instead rewards uncertainty reduction directly, through a belief-dependent utility $ρ$, but in practice both the choice of $ρ$ and the weight placed on it are tuned by hand for every task. We show that active inference removes this tuning entirely. Minimizing Expected Free Energy (EFE) is exactly equivalent to solving a $ρ$-POMDP whose utility is expected information gain, and the exploration weight is fixed at $w=1$ because the variational bound expresses pragmatic and epistemic value in the same units (nats). We prove this equivalence for observe-then-commit POMDPs and extend it to factored observation POMDPs, a broader class that covers interleaved observe-act problems such as non-destructive testing and mobile sensing, where gathering information leaves the hidden state unchanged. Experiments support the theory. Across environments ranging from the classic Tiger problem to RockSample and a new Structural Inspection benchmark with over 65,000 states, the untuned weight matches or outperforms reward-only planning at the same horizon, avoids the over-exploration of bonuses tuned per task, and sits near the reward-maximizing knee of the success-reward Pareto frontier. The practical payoff is an exploration objective that works out of the box. In applications such as fault detection and medical screening, where every test has a price and every missed fault has a cost, EFE supplies a belief-dependent utility that is derived rather than tuned.
Across environments with mixed sources of uncertainty, unsupervised reinforcement learning requires intrinsic motivation that does not precommit to a particular direction of surprise. Surprise minimization is scoped by design to ``unstable'' environments. Prediction-error curiosity rewards total expected surprise, including irreducible noise. Bandit or mixture switching between surprise-minimizing and surprise-maximizing rewards reintroduces non-stationarity by construction. We propose a single intrinsic reward, stationary within each window, derived from the novelty contribution of a preference-free Expected Free Energy objective, expressed in reward-maximization form. Our claim is that parameter information gain, the expected surprise of the next state minus its irreducible part, is the appropriate intrinsic signal in both high-entropy and low-entropy components of the state space. Maximizing it seeks exactly the surprise the model can explain away. In regions of unresolved dynamics, this epistemic term drives exploration. As dynamics become resolved, the epistemic term vanishes, while an aleatoric penalty favors lower-variance transitions, all without fitting an explicit next-state predictor. A pseudocount supplies epistemic value, a probe-based penalty captures aleatoric variance, and a short-horizon gate protects informative successors. A window-based freeze of all reward-defining objects yields a stationary Bellman operator, explicit bounds on learning targets, and a conditional uniform-concentration result for the nonparametric estimators under mixing, smoothness, bandwidth, and capacity assumptions. In active-inference terms, the agent is preference-free where novelty is retained, standard likelihood ambiguity vanishes under full observability, a nonstandard transition-entropy penalty is added, and surprise minimization emerges in resolved regions of the state space.
Joint-Embedding Predictive Architectures (JEPAs) are the dominant design for latent world models, yet they are usually justified by empirical performance rather than a normative principle. We show that the choice of anti-collapse regulariser determines whether a JEPA's training objective, a prediction loss plus a weighted embedding regulariser, is a valid Active Inference (AIF) variational free energy. We organise four non-contrastive regularisers (VICReg, LogDet, PairDist, and SIGReg) into an entropy-estimator hierarchy indexed by a prior-miscalibration gap, and show that the gap's sign, whether the estimator bounds the latent entropy from above or below, decides whether the AIF surprise bound survives: VICReg and LogDet are unsafe upper bounds, PairDist a safe lower bound, and SIGReg eliminates the gap. We then prove a correspondence theorem: under the standard constant-noise encoder model and successful SIGReg enforcement (isotropic-Gaussian embeddings), the gap vanishes, the objective becomes an exact information bottleneck, the surprise bound is preserved, and the latent goal cost becomes an exact proxy for AIF pragmatic value, whereas VICReg leaves an irreducible second-order anisotropy term. We extend the correspondence to multi-step expected free energy, ensemble epistemic value, and a learned-policy regime, and we identify the one AIF term no current JEPA world model computes: the state-epistemic value, a future-state coverage signal. The predictions differ in kind, not degree, and are stated here as theoretical consequences left for empirical test in separate work; full proofs are in Appendix A, and the algebraic core of every result is machine-verified in Lean 4 (Appendix D).
I discuss some quantitative representations of Promise Theory for processes involving autonomous agents. Agent models are common in software systems, machine learning, and biology, for example, but may also apply to physics and other forms of engineering. I describe how Bayesian probability and information theoretic optimization, including Active Inference, may be incorporated with promise semantics -- as well as how Promise Theory supplements solutions, helping to avoid probability's pitfalls, which include non-local coordination, calibrating, and normalizing probabilistic computations. The role of boundary conditions in constraining allowed states and selecting decision thresholds is a form of promise, and agent alignment provides a scalable definition of intent. Autonomous agents may congeal into swarms with superagent characteristics by trying to minimize their information, despite uncertainty that works to maximize it. The use of Promise Theory involves some research challenges as well as stylistic preferences.