Grégoire Sergeant-Perthuis

Propriété d'intersection, décomposition d'interaction, optimisation régionalisée et applications.: Replacer des constructions probabilistes et statistiques dans le cadre de la théorie des catégories est un moyen de les unifier et de les étendre tout en les rendant plus modulables pour des applications concrètes; c'est aussi une manière d'identifier ce qu'elles ont de topologique. La première partie de cette thèse est consacrée à l'étude du problème des marginales et à la description des mesures de Gibbs de systèmes statistiques. Ces problèmes peuvent être reformulés et étendus dans le cadre de la théorie des catégories, respectivement dans la catégorie des espaces mesurables et dans celles des noyaux de probabilité comme initiés par Giry dans "A categorical approach to probability theory". Nous n'avons pas connaissance de travaux antérieurs qui proposent une présentation catégorique des mesures de Gibbs; ce point de vue permet d'exhiber de grandes similarités entre le problème des marginales et la caractérisation des mesures de Gibbs, les deux ayant la même interprétation topologique. Une contribution importante au problème des marginales est le résultat de Kellerer, dans "Masstheoretische Marginalprobleme", pour le problème des marginales linéarisé qui repose sur une construction algébrique que Lauritzen nomme la décomposition en sous-espaces d'interaction dans son livre de référence "Graphical Models". Dans ce travail, nous étendons cette construction, extension que nous appellerons décomposition d'interaction, dans plusieurs directions ce qui permet d'obtenir de nouveaux résultats pour le problème des marginales et la caractérisation des mesures de Gibbs dans les contextes énoncés précédemment. Les décompositions d'interaction n'existent pas toujours, c'est pourquoi nous introduisons plusieurs conditions intrinsèques qui caractérisent leur existence: ce sont les propriétés d'intersection, nommées ainsi en référence à la propriété d'intersection pour les graphoïdes. Dans une autre direction on se propose, dans le même esprit que Yedidia, Freeman, Weiss dans "Constructing free-energy approximations and generalized belief propagation algorithms", de donner une procédure pour reconstruire des problèmes d'optimisation globaux à partir de problèmes locaux; nous appellerons cette procédure l'optimisation régionalisée en référence à leur généralisation de l'énergie libre de Bethe, les approximations de l'énergie libre basées sur des régions. Une application de l'étude de cette procédure est la formulation d'une Analyse en Composantes Principales adaptées à des données filtrées telles que les séries temporelles; une autre application est l'extension des approximations de l'énergie libre basées sur des régions de Yedidia, Freeman, Weiss aux collections de mesures de probabilités dont les règles de compatibilité sont données par un diagramme dans la catégorie des noyaux de probabilités. Nous pensons qu'une telle procédure trouvera de nombreuses applications concrètes notamment pour la modélisation des mouvements du ribosomes pendant la traduction de l'ARN messager mais aussi pour l'expression des émotions. Des propriétés de la décomposition d'interaction permettent de donner un algorithme très simple pour trouver les points critiques de la fonction de coût sous-jacente à l'optimisation régionalisée.

According to the Projective Consciousness Model (PCM), in human spatial awareness, 3-dimensional projective geometry structures information integration and action planning through perspective taking within an internal representation space. The way different perspectives are related to and transform a world model defines a specific perception and imagination scheme. In mathematics, such a collection of transformations corresponds to a ‘group’, whose ‘actions’ characterize the geometry of a space. Imbuing world models with a group structure may capture different agents’ spatial awareness and affordance schemes. We used group action as a special class of policies for perspective-dependent control. We explored how such a geometric structure impacts agents’ behaviors, comparing how the Euclidean versus projective groups act on epistemic value in active inference, drive curiosity, and exploration. We formally demonstrate and simulate how the groups induce distinct behaviors in a simple search task. The projective group’s nonlinear magnification of information transformed epistemic value according to the choice of frame, generating behaviors of approach toward objects with uncertain locations due to limited sampling. The Euclidean group had no effect on epistemic value: no action was better than the initial idle state. In structuring a priori an agent’s internal representation, we show how geometry can play a key role in information integration and action planning. Our results add further support to the PCM.

This article presents a concise mathematical formulation of Integrated Information Theory (IIT), aimed at making the theory more accessible. IIT is one of the most influential theories of consciousness, and from a computational point of view, it can be difficult and time-consuming to find a clear presentation of the technical details of IIT. Our presentation builds upon the work by Kleiner and Tull that presents IIT in a clear and unified mathematical framework. We propose in this paper a synthesis that highlights the core formalisms of IIT while setting aside the more philosophical aspects, such as the interpretation of IIT's axioms and postulates. The focus is squarely on the mathematical structure of IIT, utilizing basic but central concepts from probability theory, such as Markov kernels and conditional independence, to articulate how IIT formalizes consciousness. The article discusses the `cutting' of interactions within a system to isolate and evaluate its integrated information, a procedure central to IIT's procedure for quantifying consciousness. By distilling IIT to its mathematical essence, the article aims to foster broader understanding and stimulate further discussion about the theory's potential as a model for consciousness, inviting future explorations into its relationships with other theories and its implications for understanding conscious processes.

This document is a summary of various results related to a compositional/categorical approach to rigorous Statistical Mechanics. In particular, we present a categorical formulation of specifications that we call A-specifications and their Gibbs measures. We give a characterization of extreme Gibbs measures of A-specifications and an entropy functional for A-specifications, along with an associated variational principle and message passing algorithm to 'solve' the variational principle.

In this document, we aim to gather various results related to a compositional/categorical approach to rigorous Statistical Mechanics. Rigorous Statistical Mechanics is centered on the mathematical study of statistical systems. Central concepts in this field have a natural expression in terms of diagrams in a category that couples measurable maps and Markov kernels. We showed that statistical systems are particular representations of partially ordered sets (posets), that we call A-specifications, and expressed their phases, i.e., Gibbs measures, as invariants of these representations. It opens the way to the use of homological algebra to compute phases of statistical systems. Two central results of rigorous Statistical Mechanics are, firstly, the characterization of extreme Gibbs measures as it relates to the zero-one law for extreme Gibbs measures, and, secondly, their variational principle which states that for translation invariant Hamiltonians, Gibbs measures are the minima of the Gibbs free energy. We showed how the characterization of extreme Gibbs measures extends to A-specifications; we proposed an Entropy functional for A-specifications and gave a message-passing algorithm, that generalized the belief propagation algorithm of graphical models, to find critical points of the associated variational free energy.

In recent work we proposed to follow a categorical approach to Statistical Mechanics. In this paper we continue in this direction and give a categorical formulation of the decomposition of Gibbs measures by extending the characterization of extreme Gibbs measures to the categorical setting.

In human spatial awareness, 3-D projective geometry structures information integration and action planning through perspective taking within an internal representation space. The way different perspectives are related and transform a world model defines a specific perception and imagination scheme. In mathematics, such collection of transformations corresponds to a 'group', whose 'actions' characterize the geometry of a space. Imbuing world models with a group structure may capture different agents' spatial awareness and affordance schemes. We used group action as a special class of policies for perspective-dependent control. We explored how such geometric structure impacts agents' behavior, comparing how the Euclidean versus projective groups act on epistemic value in active inference, drive curiosity, and exploration behaviors. We formally demonstrate and simulate how the groups induce distinct behaviors in a simple search task. The projective group's nonlinear magnification of information transformed epistemic value according to the choice of frame, generating behaviors of approach toward an object of interest. The projective group structure within the agent's world model contains the Projective Consciousness Model, which is know to capture key features of consciousness. On the other hand, the Euclidean group had no effect on epistemic value : no action was better than the initial idle state. In structuring a priori an agent's internal representation, we show how geometry can play a key role in information integration and action planning.

Consciousness has been described as acting as a global workspace that integrates perception, imagination, emotion and action programming for adaptive decision making. The mechanisms of this workspace and their relationships to the phenomenology of consciousness need to be further specified. Much research in this area has focused on the neural correlates of consciousness, but, arguably, computational modeling can better be used toward this aim. According to the Projective Consciousness Model (PCM), consciousness is structured as a viewpoint-organized, internal space, relying on 3D projective geometry and governed by the action of the Projective Group as part of a process of active inference. The geometry induces a group-structured subjective perspective on an encoded world model, enabling adaptive perspective taking in agents. Here, we review and discuss the PCM. We emphasize the role of projective mechanisms in perception and the appraisal of affective and epistemic values as tied to the motivation of action, under an optimization process of Free Energy minimization, or more generally stochastic optimal control. We discuss how these mechanisms enable us to model and simulate group-structured drives in the context of social cognition and to understand the mechanisms underpinning empathy, emotion expression and regulation, and approach–avoidance behaviors. We review previous results, drawing on applications in robotics and virtual humans. We briefly discuss future axes of research relating to applications of the model to simulation- and model-based behavioral science, geometrically structured artificial neural networks, the relevance of the approach for explainable AI and human–machine interactions, and the study of the neural correlates of consciousness.

Relating explicit psychological mechanisms and observable behaviours is a central aim of psychological and behavioural science. One of the challenges is to understand and model the role of consciousness and, in particular, its subjective perspective as an internal level of representation (including for social cognition) in the governance of behaviour. Toward this aim, we implemented the principles of the Projective Consciousness Model (PCM) into artificial agents embodied as virtual humans, extending a previous implementation of the model. Our goal was to offer a proof-of-concept, based purely on simulations, as a basis for a future methodological framework. Its overarching aim is to be able to assess hidden psychological parameters in human participants, based on a model relevant to consciousness research, in the context of experiments in virtual reality. As an illustration of the approach, we focused on simulating the role of Theory of Mind (ToM) in the choice of strategic behaviours of approach and avoidance to optimise the satisfaction of agents’ preferences. We designed a main experiment in a virtual environment that could be used with real humans, allowing us to classify behaviours as a function of order of ToM, up to the second order. We show that agents using the PCM demonstrated expected behaviours with consistent parameters of ToM in this experiment. We also show that the agents could be used to estimate correctly each other’s order of ToM. Furthermore, in a supplementary experiment, we demonstrated how the agents could simultaneously estimate order of ToM and preferences attributed to others to optimize behavioural outcomes. Future studies will empirically assess and fine tune the framework with real humans in virtual reality experiments.