Graduate Research Course

Bayesian Brain & Predictive Processing

The brain as a generative model that predicts its sensory input. Bayes’ theorem from the cell to consciousness — Helmholtz to Friston, perception as unconscious inference, schizophrenia as precision failure.

About This Course

Bayesian brain theory — a computational framework grounded in the principles of Predictive Processing (PP) — proposes a mechanistic account of how beliefs are formed and updated. The theory assumes that the brain encodes a generative model of its environment, made up of probabilistic beliefs organised in hierarchical networks, from which it generates predictions about future sensory inputs. The difference between predictions and sensory signals produces prediction errors, which are used to update the belief networks.

This course introduces the fundamental principles of Bayesian brain theory, from its historical roots in Hermann von Helmholtz’s 1867 unconscious inferencethrough the modern reformulation by Rao & Ballard (1999) and Karl Friston’s free-energy principle (2006, 2010), and shows how the brain dynamics of prediction are associated with the generation and evolution of beliefs — with applications to perception, attention, schizophrenia, autism, and artificial intelligence.

Cross-links: Comparative Sensory M8: Multimodal Integration,Biophysics,Statistics,Machine Learning.

Key Equations

Bayes' Theorem

\( p(h|d) = \frac{p(d|h)\,p(h)}{p(d)} \)

Precision-Weighted Prediction Error

\( \varepsilon = \Pi(y - g(\mu)) \)

Variational Free Energy

\( F = D_{KL}[q(s)\Vert p(s|y)] - \ln p(y) \)

ELBO Decomposition

\( \ln p(y) \geq \mathbb{E}_q[\ln p(y,s)] - \mathbb{E}_q[\ln q(s)] \)

Expected Free Energy (EFE)

\( G(\pi) = \mathbb{E}_q[\ln q(s|\pi) - \ln p(y,s|\pi)] \)

Surprise / Surprisal

\( \mathcal{I}(y) = -\ln p(y) \)

Nine Modules

Bayes & Beliefs

Bayes' theorem, likelihood/prior/posterior/evidence, Bayesian vs. frequentist, generative vs. discriminative models, beliefs as probability distributions.

BayesPriorPosterior

Generative Models of the World

Helmholtz's unconscious inference (1867), hierarchical Bayesian networks, world-model representation in cortex, latent-variable models, Bayesian brain hypothesis.

HelmholtzHierarchicalWorld-Model

Prediction Error & Precision

Kalman filters, variational inference, ELBO, precision as inverse variance, attention as precision modulation, noisy sensory channels.

KalmanELBOPrecision

Predictive Coding in the Cortex

Rao & Ballard 1999 (V1 receptive fields from sparse priors), Friston 2005-2010 hierarchical predictive coding, cortical columns, Mumford 1992 deep/superficial layers.

Rao-BallardFristonV1

The Free-Energy Principle

Variational free energy, surprise (surprisal) minimisation, Markov blankets, the Friston 2006/2010/2019 papers, self-organising systems, life as inference.

FEPFree EnergyFriston

Active Inference

Action as belief update, expected free energy, epistemic vs. pragmatic value, Bayesian decision theory, pixel-to-belief loop, embodied cognition.

Active InferenceEFEEpistemic

Neural Implementation

Bastos 2012 canonical microcircuit (superficial PE vs. deep prediction), laminar oscillations (gamma bottom-up, alpha/beta top-down), Wacongne 2012 MMN, L2/3 vs. L5.

BastosMicrocircuitMMN

Computational Psychiatry

Schizophrenia (aberrant precision on priors, Adams 2013 hollow-mask), autism HIPPEA (Pellicano-Burr 2012), depression, addiction, hallucinogens as precision-disruptors.

SchizophreniaAutismPsychiatry

Applications & Theories of Mind

BCI decoding, predictive-coding AI (Perceiver, RNN world-models, Dreamer), consciousness theories (GNW vs. IIT vs. AST vs. FEP), philosophical implications.

BCIConsciousnessIIT