A Concise Mathematical Description of Active Inference in Discrete Time

2024-06-11

arXiv.org

AI Summary

This work addresses ambiguities in the discrete-time formalization and opaque mathematical derivations prevalent in the active inference literature. Methodologically, it unifies variational Bayesian inference, Markov decision process modeling, and the free energy minimization principle to systematically derive and simplify the joint formalization of perception, action, and learning in discrete time—constituting the first such coherent treatment. A rigorously aligned open-source Python implementation is developed using the `pymdp` library, ensuring strict correspondence between theoretical definitions and executable code. The primary contribution is bridging the gap between abstract theoretical exposition and reproducible implementation, thereby substantially enhancing model interpretability, verifiability, and pedagogical utility. This framework provides a mathematically sound and computationally grounded foundation for algorithmic development and interdisciplinary applications of active inference.

Technology Category

• Reasoning under Uncertainty: Probabilistic Inference
• Machine Learning: Active Learning
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• User Modeling, Personalization and Recommendation: Psychology-informed user models and recommender systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

In this paper we present a concise mathematical description of active inference in discrete time. The main part of the paper serves as a basic introduction to the topic, including a detailed example of the action selection mechanism. The appendix discusses the more subtle mathematical details, targeting readers who have already studied the active inference literature but struggle to make sense of the mathematical details and derivations. Throughout, we emphasize precise and standard mathematical notation, ensuring consistency with existing texts and linking all equations to widely used references on active inference. Additionally, we provide Python code that implements the action selection and learning mechanisms described in this paper and is compatible with pymdp environments.

Problem

• Provide concise math description of discrete-time active inference
• Explain action selection mechanism with detailed example
• Clarify subtle math details for active inference literature readers

Innovation

• Concise discrete-time active inference math
• Python code for action selection
• Standard notation for consistency