01Bayesian Statistics - Module OverviewHow Bayesian thinking transforms ML - uncertainty quantification, priors as regularization, probabilistic programming, and principled model comparison. Module map and learning objectives.02Bayesian vs Frequentist StatisticsThe philosophical divide between Bayesian and frequentist probability - concrete examples, when each approach is better, and how the choice shapes ML system design.03Prior and Posterior DistributionsChoosing priors, conjugate distributions, posterior derivation via Bayes theorem, MAP estimation, and sensitivity analysis - the foundations of practical Bayesian ML.04Bayesian UpdatingSequential Bayesian updating, online learning, the Beta-Bernoulli stream, and the Kalman filter as Bayesian updating - how beliefs evolve as data arrives.05Markov Chain Monte CarloWhy MCMC is needed, Metropolis-Hastings algorithm, Gibbs sampling, convergence diagnostics (R-hat, trace plots), and practical Bayesian inference with PyMC.06Variational InferenceELBO derivation, mean-field variational inference, VI vs MCMC tradeoffs, the reparameterization trick, and variational autoencoders - scalable approximate Bayesian inference.07Gaussian ProcessesGP priors over functions, kernel functions (RBF, Matérn, periodic), GP regression with posterior mean and variance, hyperparameter optimization, and Bayesian optimization for ML hyperparameter tuning.08Hierarchical ModelsHierarchical Bayesian models, partial pooling, multilevel regression, and the connection to multi-task learning - sharing information across groups with sparse data.09Bayesian Model ComparisonBayes factors, marginal likelihood, BIC and AIC from a Bayesian perspective, Occam's razor via model evidence, and practical model selection in ML.
