k e H a y a s h i | | 2 0 2 5 . 0 3 . 1 5 P a g e . 0 A New Mathematical Theory of Communication CPC Spring Camp 2025 Yusuke Hayashi (AI Alignment Network)
(humans and AI) cannot be presumed to engage solely in cooperative behaviors. • CPC framework describes how collective can emerge new symbols, social agreements, and scientific insights through language-mediated communication between agents. • The framework also identifies the specific conditions that enable consensus formation between humans and AI. Why Study Collective Predictive Coding (CPC)? : Research Motivation Human AI Agent Symbol Emergence
framework that emerged in 2024, developed primarily in Japan • Extends beyond individual Predictive Coding (PC) to social dynamics • Explains how shared symbols emerge through collective interaction • Potential to revolutionize our understanding of language, culture, and science From Japanese Research to Global Scientific Discourse Collective Predictive Coding (CPC): A Novel Theory Transcending Borders I am listed among the authors of several academic publications.
Research Currently Limited by Language Barriers • Despite being a new theoretical framework that emerged in 2024, CPC proposed by Professor Tadahiro Taniguchi has been featured in major Japanese media outlets, including the scientific magazine "Nikkei Science" and the Nikkei newspaper. • My name is also mentioned, very briefly, short, in these articles. • On the other hand, foreign media have not yet paid attention to CPC up to the present time.
breaking international barriers Recent Breakthrough: International Recognition Begins • February 11th, 2025: Professor Tadahiro Taniguchi, delivered a presentation on CPC to Professor Karl Friston, the originator of the Free Energy Principle. Professor Friston offered enthusiastic praise to Professor Taniguchi. This marked the first major recognition from the Western theoretical neuroscience community.
breaking international barriers Recent Breakthrough: International Recognition Begins • March 15th, 2025: Collective Predictive Coding Hypothesis: Symbol Emergence as Decentralized Bayesian Inference (i.e., the paper on "the CPC hypothesis"), has been selected as one of the best papers (outstanding paper) of 2024 in Frontiers in Robotics and AI.
breaking international barriers Recent Breakthrough: International Recognition Begins • March 15th, 2025: Collective Predictive Coding Hypothesis: Symbol Emergence as Decentralized Bayesian Inference (i.e., the paper on "the CPC hypothesis"), has been selected as one of the best papers (outstanding paper) of 2024 in Frontiers in Robotics and AI. We have finally completed the explanation of the current situation regarding CPC. Let's return to the scientific discussion.
A Mathematical Theory of Communication The Fundamental Problem of Communication is that of Reproducing A Message Selected… The sender and receiver in communication share the same codebook from the beginning. Therefore, Shannon's communication theory does not need to consider the mechanism of symbol emergence or semantic aspects of communication. Sender Receiver
Rate-Distortion Theory within Deep Generative Models Code Book Code Book VAE latent spaces contain semantic structures partially interpretable by humans. However, this interpretability is limited—most meanings encoded in these vectors remain accessible only to the VAE system itself. This phenomenon resembles Wittgenstein's concept of a "private language”.
Coding? • Models how agents predict observations from latent states • Inference and Generative model: • Key Idea: Agents minimize prediction errors independently • PC Free Energy Formula: • Components • (A) Individual Regularization: Keeps latent states close to priors • (B) Prediction Accuracy: Reduces surprise in observations Predictive Coding: The Foundation Predictive Coding (PC) Overview World model
emergence occurs through collective predictive coding • These symbols (external representations) coordinate the community's collective understanding • Individual mental models (latent states ) update based on shared knowledge • Extended inference and generative model with : • Probabilistic graphical models representing the CPC hypothesis. The CPC Hypothesis: How Symbols Emerge The Collective Predictive Coding (CPC) Hypothesis
emergence occurs through collective predictive coding • These symbols (external representations) coordinate the community's collective understanding • Individual mental models (latent states ) update based on shared knowledge • Extended inference and generative model with : • Probabilistic graphical models representing the CPC hypothesis. The CPC Hypothesis: How Symbols Emerge The Collective Predictive Coding (CPC) Hypothesis
models individual cognition without coordination mechanism • CPC Extends PC by adding: • Shared external representation ( ) (papers, symbols, equations, models) • Communication channels between agents • Collective regularization of individual beliefs • Core Innovation: Formalizing how collective knowledge shapes individual understanding From PC to CPC: The Key Extension Introducing Collective Predictive Coding (CPC)
Formula: • Components • (A) Individual Regularization: Keeps latent states close to priors • (B) Prediction Accuracy: Reduces surprise in observations • CPC Free Energy Formula: • Components • (C) Collective Regularization: Aligns ( ) with all agents • (D) Individual Regularization (similar to PC, but w.r.t. ( )) • (E) Prediction Accuracy (similar to PC) Free Energy Components in CPC CPC Free Energy
Formula: Extending Rate-Distortion theory through Collective Regularization • Probabilistic graphical models representing the CPC hypothesis. The CPC Hypothesis: Formalizing how collective knowledge shapes individual understanding A New Mathematical Theory of Communication ?
how collective knowledge shapes individual understanding • CPC Free Energy Formula: Extending Rate-Distortion theory through Collective Regularization • Probabilistic graphical models representing the CPC hypothesis. A New Mathematical Theory of Communication ? When viewing a collective as a single meta- agent, an interesting relationship emerges: what constitutes internal representations for the meta-agent becomes external representations for individual agents. CPC theory successfully describes symbol emergence systems and meaning generation mechanisms, truly transcending Shannon's information theory by addressing the fundamental processes of semantic creation.
Dynamics: Posterior can be computed via Langevin equation. • Fokker-Planck equation: • Stationary distribution: • Fokker-Planck equation validates using Langevin for Bayesian inference. Linking Bayesian Inference to Stochastic Dynamics From Bayesian Updates to Dynamics The Fokker-Planck Perspective
• CPC Dynamics: • Difference: adds a control force. Drives agents to align with shared representation . • PC Generative model • PC Dynamics • CPC Generative model • CPC Dynamics How Dynamics Differ: PC vs. CPC Comparing PC and CPC Langevin Dynamics
• Drives agents to align with shared representation ( ) • Acts as a soft constraint for coordination • Implication: Enables collective behavior beyond individual optimization • Shared external representation as coordination channel • Soft constraints through collectively regularized free energy • Bidirectional influence between human and AI through ( ) • Double-Edged Sword: • Positive: ( ) can embed ethical norms • Negative: Risk of manipulation if ( ) is corrupted • Need: Security, transparency, and governance of shared representations • Empirical validation in multi-agent systems • Human-AI co-learning experiments • Modeling evolving ( ) (e.g., symbol emergence) • Robust approaches to communication constraints The Emergence of Control Force in CPC Mechanism for Human-AI Coordination Ethical and Safety Implications Open Challenges and Future Research A 3D scatter plot displaying colorful particle trajectories resembling the Langevin dynamics of entities such as humans or AI agents. These particles converge to a specific equilibrium state, akin to glass beads settling at the bottom of a bottle, with the control force influencing which equilibrium state they ultimately reach.
with a collective force for human-AI synergy • Theoretical grounding via Bayesian-Langevin-Fokker-Planck links • The control force mechanism enables collective alignment • Framework offers promising path for human-AI symbiosis • PSS workshop reviewers, Prof. Tadahiro Taniguchi and My Grandmother Yoshiko Hayashi. • : [email protected] • : @hayashiyus Key Insights Acknowledgments
(dog) ① Perceptual inference External world (the dog owner) What is Predictive Coding / Active Inference? Collective Predictive Coding as Active Inference
(dog) ① Perceptual inference External world (the dog owner) Sensory inputs/ Rewards What is Predictive Coding / Active Inference? Collective Predictive Coding as Active Inference
Rewards FEP agent (dog) ① Perceptual inference External world (the dog owner) What is Predictive Coding / Active Inference? Collective Predictive Coding as Active Inference
Rewards FEP agent (dog) ① Perceptual inference Internal representations External world (the dog owner) What is Predictive Coding / Active Inference? Collective Predictive Coding as Active Inference
Rewards Perceptual inference FEP agent (dog) ① Perceptual inference Internal representations External world (the dog owner) Parameters What is Predictive Coding / Active Inference? Collective Predictive Coding as Active Inference
Sensory inputs/ Rewards Perceptual inference FEP agent (dog) Action policies ② Active inference External world (the dog owner) Parameters What is Predictive Coding / Active Inference? Collective Predictive Coding as Active Inference
Sensory inputs/ Rewards Action policies Hidden states Active inference FEP agent (dog) ② Active inference External world (the dog owner) Parameters Parameters What is Predictive Coding / Active Inference? Collective Predictive Coding as Active Inference
Sensory inputs/ Rewards Action policies Hidden states Parameters Parameters Active inference External world (the dog owner) FEP agent (dog) ③ Learning / Parameter optimization Parameters Learning What is Predictive Coding / Active Inference? Collective Predictive Coding as Active Inference
infere nce The hidden states of the external world are inferred from sensory inputs based on a generative model. In this process, the internal representation that minimizes free energy is selected. ② Acti ve i nferenc e Through actions, a FEP agent can intervene in the external environment to obtain the desired sensory input. This is known as active inference, where actions are chosen to minimize free energy. ③ Learning / Parameter optimization Alongside perception and action, the generative model itself is also updated. This update is similarly determined to minimize free energy. Free energy/Objective function Expected free energy Calculate the expected value for sequential data. The minimization of free energy and expected free energy drives perception, action, and learning. What is Predictive Coding / Active Inference? Collective Predictive Coding as Active Inference
free energy/Objective function Note that the collective regularization term cannot be expressed as a sum of terms for individual agents. The global/collective representation acts as an interactive force or bond that unites the entire collective through interactions among individual agents. What is Predictive Coding / Active Inference? Collective Predictive Coding as Active Inference
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