This chapter introduces the Bayesian reflex -- an analogy with the autonomic nervous system -- as a unifying framework for online learning in AI. Bayesian online algorithms automatically maintain equilibrium in dynamic environments via three mechanisms: belief maintenance through probabilistic representations, sequential updating via Bayes' theorem, and uncertainty-driven action balancing exploration and exploitation. We survey online Bayesian methods, highlighting two computational principles: the look-up table principle for sequential inference in function space, and the ellipsoidal decomposition framework for nearly exact i.i.d. sampling from arbitrary posteriors. These principles are generalized across dynamic emulation, nonparametric state-space models, circular time series, inverse regression for climate model evaluation, and deep architectures via Recursive Gaussian Processes. Decision-making is explored via Thompson sampling and restless bandits. We extend the framework to assess infinite series convergence (applied to climate dynamics and the Riemann Hypothesis), model prime number distributions leading to the discovery of 184 strong Mersenne prime candidates, detect stationarity, and characterize point processes. The Bayesian reflex provides a foundational infrastructure for adaptive AI that continuously learns in a complex world.

翻译：本章介绍贝叶斯反射——类比自主神经系统——作为人工智能在线学习的统一框架。贝叶斯在线算法通过三种机制自动维持动态环境中的平衡：基于概率表示维护信念、通过贝叶斯定理进行序贯更新，以及由不确定性驱动的探索与利用平衡行为。我们综述了在线贝叶斯方法，重点阐述了两种计算原理：函数空间中用于序贯推理的查找表原理，以及近乎精确地从任意后验分布中进行独立同分布采样的椭球分解框架。这些原理被推广应用于动态仿真、非参数状态空间模型、循环时间序列、气候模型评估中的逆回归，以及通过递归高斯过程实现的深度架构。决策方面通过汤普森采样和动态赌博机问题进行探讨。我们将该框架扩展至评估无穷级数收敛性（应用于气候动力学和黎曼猜想）、建模素数分布（由此发现184个强梅森素数候选数）、检测平稳性以及刻画点过程。贝叶斯反射为持续学习复杂世界中的自适应人工智能提供了基础性基础设施。