Advanced Bayesian Machine Learning
Chapter 1: Revisiting Bayesian Foundations
Probabilistic Modeling Principles
Bayes' Theorem in High Dimensions
Subjectivity and Objectivity in Prior Selection
Information Theory Connection: Entropy and KL Divergence
Computational Challenges in Bayesian Inference
Advanced Model Checking and Criticism
Chapter 2: Markov Chain Monte Carlo Methods
Principles of Monte Carlo Integration
Markov Chain Theory for MCMC
Metropolis-Hastings Algorithm Variants
Gibbs Sampling for Conditional Structures
Hamiltonian Monte Carlo (HMC) Mechanics
The No-U-Turn Sampler (NUTS)
Diagnosing MCMC Convergence and Performance
Hands-on Practical: Implementing Advanced MCMC
Chapter 3: Variational Inference Techniques
Optimization as Inference: The VI Perspective
Deriving the Evidence Lower Bound (ELBO)
Mean-Field Approximation Details
Coordinate Ascent Variational Inference (CAVI)
Stochastic Variational Inference (SVI) for Large Data
Black Box Variational Inference (BBVI)
Advanced Variational Families
Comparing MCMC and VI Strengths
Hands-on Practical: Scalable Variational Inference
Chapter 4: Gaussian Processes
Bayesian Non-parametric Modeling Introduction
Defining Gaussian Processes: Priors Over Functions
Covariance Functions (Kernels): Properties and Selection
Gaussian Process Regression Formulation
Hyperparameter Marginal Likelihood Optimization
Gaussian Process Classification Techniques
Approximations for Scalable Gaussian Processes
Hands-on Practical: Gaussian Process Modeling
Chapter 5: Advanced Topics in Probabilistic Graphical Models
Bayesian Networks: Structure Learning
Parameter Learning in Bayesian Networks
Advanced Inference in PGMs: Junction Tree Algorithm
Approximate Inference in Large PGMs
Latent Dirichlet Allocation (LDA): Bayesian Formulation
Inference for LDA: Collapsed Gibbs Sampling
Inference for LDA: Variational Bayes
Hands-on Practical: Topic Modeling with LDA
Chapter 6: Bayesian Deep Learning
Motivation for Bayesian Deep Learning
Bayesian Neural Networks (BNNs): Priors over Weights
Inference Challenges in BNNs
MCMC Methods for BNNs (e.g., Stochastic Gradient HMC)
Variational Inference for BNNs (e.g., Bayes by Backprop)
Uncertainty Estimation in BNNs
Variational Autoencoders (VAEs) as Probabilistic Models
Dropout as Approximate Bayesian Inference
Practical Training and Evaluation of BNNs
Hands-on Practical: Building a Bayesian Neural Network
Gibbs Sampling for Conditional Structures
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- Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images, Stuart Geman and Donald Geman, 1984 IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. PAMI-6 (IEEE) DOI: 10.1109/TPAMI.1984.4767596 - The seminal paper that introduced Gibbs sampling, laying the theoretical foundation for its use in various statistical and image processing applications.
- Bayesian Data Analysis, Andrew Gelman, John B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B. Rubin, 2013 (Chapman and Hall/CRC) - A comprehensive textbook on Bayesian inference, with detailed chapters (e.g., Chapter 11) on MCMC methods including Gibbs sampling, its algorithm, practical considerations, and extensions like blocked Gibbs sampling. 3rd edition.
- Monte Carlo Statistical Methods, Christian P. Robert and George Casella, 2004 (Springer) DOI: 10.1007/978-1-4757-4145-2 - A rigorous and extensive reference for Monte Carlo methods, providing dedicated chapters on the theoretical underpinnings and practical implementation of Gibbs sampling, including its convergence properties and various schemes. 2nd edition.
- Pattern Recognition and Machine Learning, Christopher M. Bishop, 2006 (Springer) DOI: 10.1007/978-0-387-44716-2 - Chapter 11 offers an accessible introduction to MCMC methods, with a clear explanation of Gibbs sampling, its relationship to Metropolis-Hastings, and practical aspects for its application in machine learning contexts.