arXiv subjects

Karl Friston

Publications and source records attributed to Karl Friston.

At least 19 recordsLinked to original sources

arXiv2026

Renormalising Generative Models for Active Inference: Foundations, Derivations, and Verification

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.

cs.AI↗

arXiv2026

Active Inference as the Test-Time Scaling Law for Physical AI Agents

In this paper, a novel test-time scaling law for physical artificial intelligence (AI) agents is introduced. This scaling law enables physical AI agents to reason with their world models to generalize in unforeseen scenarios at test time. The derived scaling law is grounded in the first principle of active inference, which equips agents with the general objective to survive in the real world, under which their specific task objectives are subsumed. Active inference achieves this by providing the reasoning to resolve prediction errors that arise when the agent encounters unforeseen situations outside its training distribution, enabling generalization in non-stationary environments. The proposed scaling law captures this by dynamically updating the agent's policy with this reasoning at test time. This policy update is modeled as a soft Bayesian inference process in which beliefs about the policy are updated using the reasoning that reduces expected prediction errors under allowable policies as a likelihood. The resulting posterior policy admits a biological interpretation, recovering the scaling mechanism that engages the brain's basal ganglia and prefrontal cortex at test time. To solve this analytically intractable problem, a variational inference solution minimizing free energy bounds is developed. This solution extends to enable learning beyond training by reinforcing new instances, resolved at test time, in both the policy and world model. Unlike existing scaling laws constrained by model size and training data, the derived solution scales with the continuous real-world experience of a physical AI agent. Simulation results on an autonomous driving task demonstrate that the proposed solution outperforms model-free Q-learning and model-based Bayesian reinforcement learning, achieving robust generalization to unforeseen scenarios while improving inference efficiency by over 36%.

cs.AI↗

arXiv2026

The Complex Brain Hypothesis: Resolving the Entropy-Content Conundrum in Minimal Phenomenal Experience

Minimal Phenomenal Experiences (MPEs) are states of consciousness in which wakefulness is preserved but phenomenal content is low or absent. The Entropic Brain Hypothesis (EBH) is a model of conscious processes that regards the entropy of spontaneous brain activity as a marker of 'phenomenal richness', exemplified by high-content psychedelic experiences (HCPEs). Yet recent human neuroimaging studies of MPEs induced by meditation -- and possibly 5-MeO-DMT -- suggest that these states, defined by their phenomenological simplicity, also show signs of increased neurophysiological entropy. This presents a conundrum for the EBH: brain entropy is elevated with increased and decreased richness of the phenomenal experience. Here, we put forward the Complex Brain Hypothesis (CBH), which proposes that the richness of experience differentiating MPEs from HCPEs is better indexed by complexity than by entropy. We argue that brain complexity is modulated by the grain of inference through which the brain resolves uncertainty: some HCPEs exemplify a fine-grained regime, in which loosened constraints amplify fluctuations into proliferating content, whereas some MPEs exemplify a coarse-grained regime, in which a simpler model dissolves variety into an experience of 'contentless' awareness. Both regimes can be associated with elevated brain entropy, but they diverge in phenomenology and perturbational signatures. By resolving the entropy-content conundrum, the CBH refines the EBH and highlights MPEs as an important test case for computational theories of consciousness.

q-bio.NC↗

arXiv2026

Online Generalised Predictive Coding

This paper introduces an extension of generalised filtering for online applications. Generalised filtering refers to data assimilation schemes that jointly infer latent states, learn unknown model parameters, and estimate uncertainty in an integrated framework -- e.g., estimate state and observation noise -- at the same time (i.e., triple estimation). This framework appears across disciplines under different names, including variational Kalman-Bucy filtering in engineering, generalised predictive coding in neuroscience, and Dynamic Expectation Maximisation (DEM) in time-series analysis. Here, we specialise DEM for ``online'' data assimilation, through a separation of temporal scales. We describe the variational principles and procedures that allow one to assimilate data in a way that allows for a slow updating of parameters and precisions, which contextualise fast Bayesian belief updating about the dynamic hidden states. Using numerical studies, we demonstrate the validity of online DEM (ODEM) using a non-linear -- and potentially chaotic -- generative model, to show that the ODEM scheme can track the latent states of the generative process, even when its functional form differs fundamentally from the dynamics of the generative model. Framed from a neuro-mimetic predictive coding perspective, ODEM offers a biologically inspired solution to online inference, learning, and uncertainty estimation in dynamic environments.

stat.ML↗

arXiv2026

Active Inference: A method for Phenotyping Agency in AI systems?

The proliferation of agentic artificial intelligence has outpaced the conceptual tools needed to characterize agency in computational systems. Prevailing definitions mainly rely on autonomy and goal-directedness. Here, we argue for a minimal notion open to principled inspection given three criteria: intentionality as action grounded in beliefs and desires, rationality as normatively coherent action entailed by a world model, and explainability as action causally traceable to internal states; we subsequently instantiate these as a partially observable Markov decision process under a variational framework wherein posterior beliefs, prior preferences, and the minimization of expected free energy jointly constitute an agentic action chain. Using a canonical T-maze paradigm, we evidence how empowerment, formulated as the channel capacity between actions and anticipated observations, serves as an operational metric that distinguishes zero-, intermediate-, and high-agency phenotypes through structural manipulations of the generative model. We conclude by arguing that as agents engage in epistemic foraging to resolve ambiguity, the governance controls that remain effective must shift systematically from external constraints to the internal modulation of prior preferences, offering a principled, variational bridge from computational phenotyping to AI governance strategy

cs.AI↗

arXiv2025

Active inference and artificial reasoning

This technical note considers the sampling of outcomes that provide the greatest amount of information about the structure of underlying world models. This generalisation furnishes a principled approach to structure learning under a plausible set of generative models or hypotheses. In active inference, policies - i.e., combinations of actions - are selected based on their expected free energy, which comprises expected information gain and value. Information gain corresponds to the KL divergence between predictive posteriors with, and without, the consequences of action. Posteriors over models can be evaluated quickly and efficiently using Bayesian Model Reduction, based upon accumulated posterior beliefs about model parameters. The ensuing information gain can then be used to select actions that disambiguate among alternative models, in the spirit of optimal experimental design. We illustrate this kind of active selection or reasoning using partially observed discrete models; namely, a 'three-ball' paradigm used previously to describe artificial insight and 'aha moments' via (synthetic) introspection or sleep. We focus on the sample efficiency afforded by seeking outcomes that resolve the greatest uncertainty about the world model, under which outcomes are generated.

q-bio.NC↗

arXiv2025

EcoNet: Multiagent Planning and Control Of Household Energy Resources Using Active Inference

Advances in automated systems afford new opportunities for intelligent management of energy at household, local area, and utility scales. Home Energy Management Systems (HEMS) can play a role by optimizing the schedule and use of household energy devices and resources. One challenge is that the goals of a household can be complex and conflicting. For example, a household might wish to reduce energy costs and grid-associated greenhouse gas emissions, yet keep room temperatures comfortable. Another challenge is that an intelligent HEMS agent must make decisions under uncertainty. An agent must plan actions into the future, but weather and solar generation forecasts, for example, provide inherently uncertain estimates of future conditions. This paper introduces EcoNet, a Bayesian approach to household and neighborhood energy management that is based on active inference. The aim is to improve energy management and coordination, while accommodating uncertainties and taking into account potentially conditional and conflicting goals and preferences. Simulation results are presented and discussed.

eess.SY↗

arXiv2025

Variational Representational Similarity Analysis (vRSA) for M/EEG

This paper introduces variational representational similarity analysis RSA (vRSA) for electromagnetic recordings of neural responses (e.g., EEG, MEG, ECoG or LFP). Variational RSA is a Bayesian approach for testing whether the similarity of stimuli or experimental conditions is expressed in univariate or multivariate neural recordings. Extending an approach previously introduced in the context of functional MRI, vRSA decomposes the condition-by-condition data covariance matrix into hypothesised effects and observation noise, thereby casting RSA as a covariance component estimation problem. In this context, peristimulus time may be treated as an experimental factor, enabling one to test for the probability that different experimental effects are expressed in data at different times. Variational Bayesian methods are used for model estimation and model comparison, which confer a number of advantages over classical approaches, including statistically efficient hypothesis testing, quantification of uncertainty using Bayesian credible intervals and computational efficiency. After introducing the theory, we provide a worked example using openly available EEG data. Software functions implementing vRSA for the SPM software package accompany this paper, together with exemplar analysis scripts.

q-bio.NC↗

arXiv2025

Distributed Intelligence in the Computing Continuum with Active Inference

The Computing Continuum (CC) is an emerging Internet-based computing paradigm that spans from local Internet of Things sensors and constrained edge devices to large-scale cloud data centers. Its goal is to orchestrate a vast array of diverse and distributed computing resources to support the next generation of Internet-based applications. However, the distributed, heterogeneous, and dynamic nature of CC platforms demands distributed intelligence for adaptive and resilient service management. This article introduces a distributed stream processing pipeline as a CC use case, where each service is managed by an Active Inference (AIF) agent. These agents collaborate to fulfill service needs specified by SLOiDs, a term we introduce to denote Service Level Objectives that are aware of its deployed devices, meaning that non-functional requirements must consider the characteristics of the hosting device. We demonstrate how AIF agents can be modeled and deployed alongside distributed services to manage them autonomously. Our experiments show that AIF agents achieve over 90% SLOiD fulfillment when using tested transition models, and around 80% when learning the models during deployment. We compare their performance to a multi-agent reinforcement learning algorithm, finding that while both approaches yield similar results, MARL requires extensive training, whereas AIF agents can operate effectively from the start. Additionally, we evaluate the behavior of AIF agents in offloading scenarios, observing a strong capacity for adaptation. Finally, we outline key research directions to advance AIF integration in CC platforms.

cs.DC↗

arXiv2025

AXIOM: Learning to Play Games in Minutes with Expanding Object-Centric Models

Current deep reinforcement learning (DRL) approaches achieve state-of-the-art performance in various domains, but struggle with data efficiency compared to human learning, which leverages core priors about objects and their interactions. Active inference offers a principled framework for integrating sensory information with prior knowledge to learn a world model and quantify the uncertainty of its own beliefs and predictions. However, active inference models are usually crafted for a single task with bespoke knowledge, so they lack the domain flexibility typical of DRL approaches. To bridge this gap, we propose a novel architecture that integrates a minimal yet expressive set of core priors about object-centric dynamics and interactions to accelerate learning in low-data regimes. The resulting approach, which we call AXIOM, combines the usual data efficiency and interpretability of Bayesian approaches with the across-task generalization usually associated with DRL. AXIOM represents scenes as compositions of objects, whose dynamics are modeled as piecewise linear trajectories that capture sparse object-object interactions. The structure of the generative model is expanded online by growing and learning mixture models from single events and periodically refined through Bayesian model reduction to induce generalization. AXIOM masters various games within only 10,000 interaction steps, with both a small number of parameters compared to DRL, and without the computational expense of gradient-based optimization.

cs.AI↗

arXiv2025

Self-orthogonalizing attractor neural networks emerging from the free energy principle

Attractor dynamics are a hallmark of many complex systems, including the brain. Understanding how such self-organizing dynamics emerge from first principles is crucial for advancing our understanding of neuronal computations and the design of artificial intelligence systems. Here we formalize how attractor networks emerge from the free energy principle applied to a universal partitioning of random dynamical systems. Our approach obviates the need for explicitly imposed learning and inference rules and identifies emergent, but efficient and biologically plausible inference and learning dynamics for such self-organizing systems. These result in a collective, multi-level Bayesian active inference process. Attractors on the free energy landscape encode prior beliefs; inference integrates sensory data into posterior beliefs; and learning fine-tunes couplings to minimize long-term surprise. Analytically and via simulations, we establish that the proposed networks favor approximately orthogonalized attractor representations, a consequence of simultaneously optimizing predictive accuracy and model complexity. These attractors efficiently span the input subspace, enhancing generalization and the mutual information between hidden causes and observable effects. Furthermore, while random data presentation leads to symmetric and sparse couplings, sequential data fosters asymmetric couplings and non-equilibrium steady-state dynamics, offering a natural generalization of conventional Boltzmann Machines. Our findings offer a unifying theory of self-organizing attractor networks, providing novel insights for AI and neuroscience.

q-bio.NC↗

arXiv2025

Meta-Representational Predictive Coding: Neuroscience-Informed Self-Supervised Learning

Self-supervised learning has become an increasingly important paradigm in the domain of machine intelligence. Furthermore, evidence for self-supervised adaptation, such as contrastive formulations, has emerged in recent computational neuroscience and brain-inspired research. Nevertheless, current work on self-supervised learning relies on biologically implausible credit assignment -- in the form of backpropagation of errors -- and feedforward inference, typically a forward-locked pass. Predictive coding, in its mechanistic form, offers a biologically plausible means to sidestep these backprop-specific limitations. However, unsupervised predictive coding rests on learning a generative model of raw input (akin to "generative AI" approaches), which entails predicting a potentially high dimensional input; on the other hand, supervised predictive coding, which learns a mapping between inputs to target labels, requires human annotation, and thus incurs the drawbacks of supervised learning. In this work, we present a scheme for self-supervised learning, specifically for an emerging research sub-domain that we label as neuroscience-informed self-supervised learning (NeuroSSL), within a neurobiologically plausible framework that appeals to the free energy principle, constructing a new form of predictive coding that we call meta-representational predictive coding (MPC). MPC sidesteps the need for learning a generative model of sensory input (e.g., pixel-level features) by learning to predict representations of the input across parallel streams, resulting in an encoder-only learning and inference scheme. This formulation notably rests on active inference (in the form of sensory glimpsing) to drive the learning of representations, i.e., the representational dynamics are driven by sequences of decisions made by the model to sample informative portions of its sensorium.

cs.NE↗

arXiv2025

Distributionally Robust Free Energy Principle for Decision-Making

Despite their groundbreaking performance, autonomous agents can misbehave when training and environmental conditions become inconsistent, with minor mismatches leading to undesirable behaviors or even catastrophic failures. Robustness towards these training-environment ambiguities is a core requirement for intelligent agents and its fulfillment is a long-standing challenge towards their real-world deployments. Here, we introduce a Distributionally Robust Free Energy model (DR-FREE) that instills this core property by design. Combining a robust extension of the free energy principle with a resolution engine, DR-FREE wires robustness into the agent decision-making mechanisms. Across benchmark experiments, DR-FREE enables the agents to complete the task even when, in contrast, state-of-the-art models fail. This milestone may inspire both deployments in multi-agent settings and, at a perhaps deeper level, the quest for an explanation of how natural agents -- with little or no training -- survive in capricious environments.

cs.AI↗

arXiv2025

Brain in the Dark: Design Principles for Neuromimetic Inference under the Free Energy Principle

Deep learning has revolutionised artificial intelligence (AI) by enabling automatic feature extraction and function approximation from raw data. However, it faces challenges such as a lack of out-of-distribution generalisation, catastrophic forgetting and poor interpretability. In contrast, biological neural networks, such as those in the human brain, do not suffer from these issues, inspiring AI researchers to explore neuromimetic deep learning, which aims to replicate brain mechanisms within AI models. A foundational theory for this approach is the Free Energy Principle (FEP), which despite its potential, is often considered too complex to understand and implement in AI as it requires an interdisciplinary understanding across a variety of fields. This paper seeks to demystify the FEP and provide a comprehensive framework for designing neuromimetic models with human-like perception capabilities. We present a roadmap for implementing these models and a Pytorch code repository for applying FEP in a predictive coding network.

cs.NE↗

arXiv2025

SPM 25: open source neuroimaging analysis software

Statistical Parametric Mapping (SPM) is an integrated set of methods for testing hypotheses about the brain's structure and function, using data from imaging devices. These methods are implemented in an open source software package, SPM, which has been in continuous development for more than 30 years by an international community of developers. This paper reports the release of SPM 25.01, a major new version of the software that incorporates novel analysis methods, optimisations of existing methods, as well as improved practices for open science and software development.

q-bio.QM↗

arXiv2024

A Rosetta Stone Hypothesis for Neurophenomenology: Mathematical Predictions from Predictive Processing

Consciousness science faces the challenge of bridging first-person experience with third-person empirical measurements. Neurophenomenology aims to build such `generative passages' connecting the content of experience with behavioural and neuroscientific data. However, the mathematical machinery for such bridges remains underdeveloped. Here we develop a Rosetta Stone hypothesis from predictive processing, where beliefs serve as a central hub connecting phenomenology, behaviour, and neural dynamics. This hinges on a central technical assumption that phenomenology is a function of beliefs. We pursue a conditional approach: if this assumption holds, then certain predictions mathematically follow. We derive predictions for subjective similarity judgements, cognitive metabolic cost, subjective cognitive effort, and time perception. We review the connection between beliefs and neural dynamics to complete the generative passage for neurophenomenology, omitting the connection between beliefs and behaviour as this is already well-documented elsewhere. Testing our predictions will inform the validity of the central assumption connecting beliefs and phenomenology, and advance the neurophenomenology research programme.

q-bio.NC↗

arXiv2024

Possible Principles for Aligned Structure Learning Agents

This paper offers a roadmap for the development of scalable aligned artificial intelligence (AI) from first principle descriptions of natural intelligence. In brief, a possible path toward scalable aligned AI rests upon enabling artificial agents to learn a good model of the world that includes a good model of our preferences. For this, the main objective is creating agents that learn to represent the world and other agents' world models; a problem that falls under structure learning (a.k.a. causal representation learning or model discovery). We expose the structure learning and alignment problems with this goal in mind, as well as principles to guide us forward, synthesizing various ideas across mathematics, statistics, and cognitive science. 1) We discuss the essential role of core knowledge, information geometry and model reduction in structure learning, and suggest core structural modules to learn a wide range of naturalistic worlds. 2) We outline a way toward aligned agents through structure learning and theory of mind. As an illustrative example, we mathematically sketch Asimov's Laws of Robotics, which prescribe agents to act cautiously to minimize the ill-being of other agents. We supplement this example by proposing refined approaches to alignment. These observations may guide the development of artificial intelligence in helping to scale existing -- or design new -- aligned structure learning systems.

cs.AI↗

arXiv2024

A theory of generalised coordinates for stochastic differential equations

Stochastic differential equations are ubiquitous modelling tools in physics and the sciences. In most modelling scenarios, random fluctuations driving dynamics or motion have some non-trivial temporal correlation structure, which renders the SDE non-Markovian; a phenomenon commonly known as ``colored'' noise. Thus, an important objective is to develop effective tools for mathematically and numerically studying (possibly non-Markovian) SDEs. In this report, we formalise a mathematical theory for analysing and numerically studying SDEs based on so-called `generalised coordinates of motion'. Like the theory of rough paths, we analyse SDEs pathwise for any given realisation of the noise, not solely probabilistically. Like the established theory of Markovian realisation, we realise non-Markovian SDEs as a Markov process in an extended space. Unlike the established theory of Markovian realisation however, the Markovian realisations here are accurate on short timescales and may be exact globally in time, when flows and fluctuations are analytic. This theory is exact for SDEs with analytic flows and fluctuations, and is approximate when flows and fluctuations are differentiable. It provides useful analysis tools, which we employ to solve linear SDEs with analytic fluctuations. It may also be useful for studying rougher SDEs, as these may be identified as the limit of smoother ones. This theory supplies effective, computationally straightforward methods for simulation, filtering and control of SDEs; amongst others, we re-derive generalised Bayesian filtering, a state-of-the-art method for time-series analysis. Looking forward, this report suggests that generalised coordinates have far-reaching applications throughout stochastic differential equations.

math.PR↗