## **I. Introduction** Bayes’ Rule, or **Bayes’ Theorem**, is a foundational principle in probability theory that provides a formal method for updating beliefs in light of new evidence. Originally formulated by Reverend Thomas Bayes in the eighteenth century, it has since become integral to modern statistics, machine learning, and cognitive neuroscience. --- ## **II. Formal Definition of Bayes’ Rule** Bayes’ Rule expresses the conditional probability of a hypothesis H given observed evidence E. It is written as: P(H|E) = \frac{P(E|H) \times P(H)}{P(E)} Where: - P(H|E): the **posterior probability** — the probability that hypothesis H is true given the evidence E; - P(E|H): the **likelihood** — the probability of observing evidence E if H were true; - P(H): the **prior probability** — the initial [[Belief]] in H before seeing any evidence; - P(E): the **marginal probability** — the overall probability of observing the evidence under all possible hypotheses. --- ## **III. Conceptual Interpretation** Bayes’ Rule formalises a process of **belief revision**. It allows a system—human or artificial—to: 1. Begin with an expectation or _prior_ about the world. 2. Encounter sensory data or _evidence_. 3. Integrate this evidence to produce an updated, _posterior_ understanding. This recursive process underpins rational inference and adaptive behaviour in uncertain environments. --- ## **IV. Application to the “Bayesian Brain” Hypothesis** The **Bayesian brain** framework in cognitive [[Neuroscience]] extends Bayes’ Rule to describe how the brain manages [[perception]], action, and cognition. According to this model: - The brain continually generates **priors**—expectations about sensory input—based on past experience. - Incoming sensory data are treated as **evidence**. - The brain computes a **posterior prediction** that reconciles expectations with the new input, thereby minimising **prediction error**. In this sense, the brain functions as a _[[hierarchical]] Bayesian inference machine_, constantly revising its internal model of the world in response to probabilistic evidence. --- ## **V. Theoretical Significance** The integration of Bayes’ Rule into neuroscience provides a unifying framework for understanding perception and cognition not as passive reception of stimuli, but as **active hypothesis testing**. Perception becomes the brain’s best probabilistic guess about the causes of its sensory input, grounded in Bayesian computation. --- **In summary:** > Bayes’ Rule offers the mathematical foundation for the Bayesian brain theory, which conceives the mind as a probabilistic inference system—one that continuously updates its beliefs about the world to maintain coherence between internal models and external sensory data. --- In essence, Bayes’ Rule provides the mathematical foundation for the [[Predictive Processing]] framework, portraying the brain as a dynamic inference system that continuously updates its internal models of the world in light of new sensory evidence. `Concepts:` `Knowledge Base:`