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.

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