arXiv subjects

Lancelot Da Costa

Publications and source records attributed to Lancelot Da Costa.

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

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↗

arXiv2026

Learning POMDP World Models from Observations with Language-Model Priors

Whether navigating a building, operating a robot, or playing a game, an agent that acts effectively in an environment must first learn an internal model of how that environment works. Partially-observable Markov decision processes (POMDPs) provide a flexible modeling class for such internal world models, but learning them from observation-action trajectories alone is challenging and typically requires extensive environment interaction. We ask whether language-model priors can reduce costly interaction by leveraging prior knowledge, and introduce \emph{Pinductor} (POMDP-inductor): an LLM proposes candidate POMDP models from a few observation-action trajectories and iteratively refines them to optimize a belief-based likelihood score. Despite using strictly less information, \emph{Pinductor} matches the performance and sample efficiency of LLM-based POMDP learning methods that assume privileged access to the hidden state, while significantly surpassing the sample efficiency of tabular POMDP baselines. Further results show that performance scales with LLM capability and degrades gracefully as semantic information about the environment is withheld. Together, these results position language-model priors as a practical tool for sample-efficient world-model learning under partial observability, and a step toward generalist agents in real-world environments. Code is available at https://github.com/atomresearch/pinductor.

cs.LG↗

arXiv2026

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↗

arXiv2026

Sample Path Regularity of Gaussian Processes from the Covariance Kernel

Gaussian processes (GPs) are the most common formalism for defining probability distributions over spaces of functions. While applications of GPs are myriad, a comprehensive understanding of GP sample paths, i.e. the function spaces over which they define a probability measure, is lacking. In practice, GPs are not constructed through a probability measure, but instead through a mean function and a covariance kernel. In this paper we provide necessary and sufficient conditions on the covariance kernel for the sample paths of the corresponding GP to attain a given regularity. We focus primarily on Hölder regularity as it grants particularly straightforward conditions, which simplify further in the cases of stationary and isotropic GPs. We then demonstrate that our results allow for novel and unusually tight characterisations of the sample path regularities of the GPs commonly used in machine learning applications, such as the Matérn GPs.

cs.LG↗

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

Natural Building Blocks for Structured World Models: Theory, Evidence, and Scaling

The field of world modeling is fragmented, with researchers developing bespoke architectures that rarely build upon each other. We propose a framework that specifies the natural building blocks for structured world models based on the fundamental stochastic processes that any world model must capture: discrete processes (logic, symbols) and continuous processes (physics, dynamics); the world model is then defined by the hierarchical composition of these building blocks. We examine Hidden Markov Models (HMMs) and switching linear dynamical systems (sLDS) as natural building blocks for discrete and continuous modeling--which become partially-observable Markov decision processes (POMDPs) and controlled sLDS when augmented with actions. This modular approach supports both passive modeling (generation, forecasting) and active control (planning, decision-making) within the same architecture. We avoid the combinatorial explosion of traditional structure learning by largely fixing the causal architecture and searching over only four depth parameters. We review practical expressiveness through multimodal generative modeling (passive) and planning from pixels (active), with performance competitive to neural approaches while maintaining interpretability. The core outstanding challenge is scalable joint structure-parameter learning; current methods finesse this by cleverly growing structure and parameters incrementally, but are limited in their scalability. If solved, these natural building blocks could provide foundational infrastructure for world modeling, analogous to how standardized layers enabled progress in deep learning.

cs.LG↗

arXiv2025

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↗

arXiv2025

Bayesian Predictive Coding

Predictive coding (PC) is an influential theory of information processing in the brain, providing a biologically plausible alternative to backpropagation. It is motivated in terms of Bayesian inference, as hidden states and parameters are optimised via gradient descent on variational free energy. However, implementations of PC rely on maximum \textit{a posteriori} (MAP) estimates of hidden states and maximum likelihood (ML) estimates of parameters, limiting their ability to quantify epistemic uncertainty. In this work, we investigate a Bayesian extension to PC that estimates a posterior distribution over network parameters. This approach, termed Bayesian Predictive coding (BPC), preserves the locality of PC and results in closed-form Hebbian weight updates. Compared to PC, our BPC algorithm converges in fewer epochs in the full-batch setting and remains competitive in the mini-batch setting. Additionally, we demonstrate that BPC offers uncertainty quantification comparable to existing methods in Bayesian deep learning, while also improving convergence properties. Together, these results suggest that BPC provides a biologically plausible method for Bayesian learning in the brain, as well as an attractive approach to uncertainty quantification in deep learning.

cs.LG↗

arXiv2024

Toward Universal and Interpretable World Models for Open-ended Learning Agents

We introduce a generic, compositional and interpretable class of generative world models that supports open-ended learning agents. This is a sparse class of Bayesian networks capable of approximating a broad range of stochastic processes, which provide agents with the ability to learn world models in a manner that may be both interpretable and computationally scalable. This approach integrating Bayesian structure learning and intrinsically motivated (model-based) planning enables agents to actively develop and refine their world models, which may lead to developmental learning and more robust, adaptive behavior.

cs.AI↗

arXiv2024

Probabilistic Principles for Biophysics and Neuroscience: Entropy Production, Bayesian Mechanics & the Free-Energy Principle

This thesis focuses on three fundamental aspects of biological systems; namely, entropy production, Bayesian mechanics, and the free-energy principle. The contributions are threefold: 1) We compute the entropy production for a greater class of systems than before, including almost any stationary diffusion process, such as degenerate diffusions where the driving noise does not act on all coordinates of the system. Importantly, this class of systems encompasses Markovian approximations of stochastic differential equations driven by colored noise, which is significant since biological systems at the macro- and meso-scale are generally subject to colored fluctuations. 2) We develop a Bayesian mechanics for biological and physical entities that interact with their environment in which we give sufficient and necessary conditions for the internal states of something to infer its external states, consistently with variational Bayesian inference in statistics and theoretical neuroscience. 3) We refine the constraints on Bayesian mechanics to obtain a description that is more specific to biological systems, called the free-energy principle. This says that active and internal states of biological systems unfold as minimising a quantity known as free energy. The mathematical foundation to the free-energy principle, presented here, unlocks a first principles approach to modeling and simulating behavior in neurobiology and artificial intelligence, by minimising free energy given a generative model of external and sensory states.

math-ph↗

arXiv2024

From pixels to planning: scale-free active inference

This paper describes a discrete state-space model -- and accompanying methods -- for generative modelling. This model generalises partially observed Markov decision processes to include paths as latent variables, rendering it suitable for active inference and learning in a dynamic setting. Specifically, we consider deep or hierarchical forms using the renormalisation group. The ensuing renormalising generative models (RGM) can be regarded as discrete homologues of deep convolutional neural networks or continuous state-space models in generalised coordinates of motion. By construction, these scale-invariant models can be used to learn compositionality over space and time, furnishing models of paths or orbits; i.e., events of increasing temporal depth and itinerancy. This technical note illustrates the automatic discovery, learning and deployment of RGMs using a series of applications. We start with image classification and then consider the compression and generation of movies and music. Finally, we apply the same variational principles to the learning of Atari-like games.

cs.LG↗

arXiv2024

Metacognitive particles, mental action and the sense of agency

This paper articulates metacognition using the language of statistical physics and Bayesian mechanics. Metacognitive beliefs, defined as beliefs about beliefs, find a natural description within this formalism, which allows us to define the dynamics of 'metacognitive particles', i.e., systems possessing metacognitive beliefs. We further unpack this typology of metacognitive systems by distinguishing passive and active metacognitive particles, where active particles are endowed with the capacity for mental actions that update the parameters of other beliefs. We provide arguments for the necessity of this architecture in the emergence of a subjective sense of agency and the experience of being separate from the environment. The motivation is to pave the way towards a mathematical and physical understanding of cognition -- and higher forms thereof -- furthering the study and formalization of cognitive science in the language of mathematical physics.

q-bio.NC↗

arXiv2024

Active Inference as a Model of Agency

Is there a canonical way to think of agency beyond reward maximisation? In this paper, we show that any type of behaviour complying with physically sound assumptions about how macroscopic biological agents interact with the world canonically integrates exploration and exploitation in the sense of minimising risk and ambiguity about states of the world. This description, known as active inference, refines the free energy principle, a popular descriptive framework for action and perception originating in neuroscience. Active inference provides a normative Bayesian framework to simulate and model agency that is widely used in behavioural neuroscience, reinforcement learning (RL) and robotics. The usefulness of active inference for RL is three-fold. \emph{a}) Active inference provides a principled solution to the exploration-exploitation dilemma that usefully simulates biological agency. \emph{b}) It provides an explainable recipe to simulate behaviour, whence behaviour follows as an explainable mixture of exploration and exploitation under a generative world model, and all differences in behaviour are explicit in differences in world model. \emph{c}) This framework is universal in the sense that it is theoretically possible to rewrite any RL algorithm conforming to the descriptive assumptions of active inference as an active inference algorithm. Thus, active inference can be used as a tool to uncover and compare the commitments and assumptions of more specific models of agency.

cs.AI↗

arXiv2023

Towards a Bayesian mechanics of metacognitive particles: A commentary on "Path integrals, particular kinds, and strange things" by Friston, Da Costa, Sakthivadivel, Heins, Pavliotis, Ramstead, and Parr

What could metacognition look like in simple physical terms? We define metacognition as having beliefs about beliefs, which can be articulated very simply using the language of statistical physics and Bayesian mechanics. We introduce a typology between cognitive and metacognitive particles and develop an example of a metacognitive particle. This can be generalized to provide examples of higher forms of metacognition: i.e. particles having beliefs about beliefs about beliefs and so forth. We conclude by saying that the typology of particles laid down in the target article seems promising, for seemingly enabling a physics of cognition that builds upon and refines the free energy principle, toward a physical description of entities that specifically possess higher forms of cognition.

nlin.AO↗

arXiv2023

Supervised structure learning

This paper concerns structure learning or discovery of discrete generative models. It focuses on Bayesian model selection and the assimilation of training data or content, with a special emphasis on the order in which data are ingested. A key move - in the ensuing schemes - is to place priors on the selection of models, based upon expected free energy. In this setting, expected free energy reduces to a constrained mutual information, where the constraints inherit from priors over outcomes (i.e., preferred outcomes). The resulting scheme is first used to perform image classification on the MNIST dataset to illustrate the basic idea, and then tested on a more challenging problem of discovering models with dynamics, using a simple sprite-based visual disentanglement paradigm and the Tower of Hanoi (cf., blocks world) problem. In these examples, generative models are constructed autodidactically to recover (i.e., disentangle) the factorial structure of latent states - and their characteristic paths or dynamics.

cs.LG↗

arXiv2023

On efficient computation in active inference

Despite being recognized as neurobiologically plausible, active inference faces difficulties when employed to simulate intelligent behaviour in complex environments due to its computational cost and the difficulty of specifying an appropriate target distribution for the agent. This paper introduces two solutions that work in concert to address these limitations. First, we present a novel planning algorithm for finite temporal horizons with drastically lower computational complexity. Second, inspired by Z-learning from control theory literature, we simplify the process of setting an appropriate target distribution for new and existing active inference planning schemes. Our first approach leverages the dynamic programming algorithm, known for its computational efficiency, to minimize the cost function used in planning through the Bellman-optimality principle. Accordingly, our algorithm recursively assesses the expected free energy of actions in the reverse temporal order. This improves computational efficiency by orders of magnitude and allows precise model learning and planning, even under uncertain conditions. Our method simplifies the planning process and shows meaningful behaviour even when specifying only the agent's final goal state. The proposed solutions make defining a target distribution from a goal state straightforward compared to the more complicated task of defining a temporally informed target distribution. The effectiveness of these methods is tested and demonstrated through simulations in standard grid-world tasks. These advances create new opportunities for various applications.

cs.LG↗

arXiv2023

The entropy production of stationary diffusions

The entropy production rate is a central quantity in non-equilibrium statistical physics, scoring how far a stochastic process is from being time-reversible. In this paper, we compute the entropy production of diffusion processes at non-equilibrium steady-state under the condition that the time-reversal of the diffusion remains a diffusion. We start by characterising the entropy production of both discrete and continuous-time Markov processes. We investigate the time-reversal of time-homogeneous stationary diffusions and recall the most general conditions for the reversibility of the diffusion property, which includes hypoelliptic and degenerate diffusions, and locally Lipschitz vector fields. We decompose the drift into its time-reversible and irreversible parts, or equivalently, the generator into symmetric and antisymmetric operators. We show the equivalence with a decomposition of the backward Kolmogorov equation considered in hypocoercivity theory, and a decomposition of the Fokker-Planck equation in GENERIC form. The main result shows that when the time-irreversible part of the drift is in the range of the volatility matrix (almost everywhere) the forward and time-reversed path space measures of the process are mutually equivalent, and evaluates the entropy production. When this does not hold, the measures are mutually singular and the entropy production is infinite. We verify these results using exact numerical simulations of linear diffusions. We illustrate the discrepancy between the entropy production of non-linear diffusions and their numerical simulations in several examples and illustrate how the entropy production can be used for accurate numerical simulation. Finally, we discuss the relationship between time-irreversibility and sampling efficiency, and how we can modify the definition of entropy production to score how far a process is from being generalised reversible.

cond-mat.stat-mech↗