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
Metropolis-Hastings Algorithm Variants
Was this section helpful?
- 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) DOI: 10.1201/b16018 - Definitive textbook covering theoretical and practical aspects of Bayesian modeling, with extensive treatment of MCMC algorithms, including Metropolis-Hastings and its advanced variants.
- Optimal Scaling of Discrete Approximations to Langevin Diffusions, Gareth O. Roberts, Jeffrey S. Rosenthal, 1998 Journal of the Royal Statistical Society: Series B (Statistical Methodology), Vol. 60 (John Wiley & Sons) DOI: 10.1111/1467-9868.00123 - A seminal paper that introduces the Metropolis-Adjusted Langevin Algorithm (MALA) and rigorously analyzes its optimal acceptance rates and scaling behavior in high dimensions.
- Monte Carlo Statistical Methods, Christian P. Robert and George Casella, 2004 (Springer) DOI: 10.1007/978-1-4757-4145-1 - This textbook provides a comprehensive theoretical foundation for Monte Carlo methods, with dedicated chapters on Markov Chain Monte Carlo algorithms, including detailed explanations of Metropolis-Hastings and its variations.
- Examples of adaptive MCMC, Gareth O. Roberts, Jeffrey S. Rosenthal, 2009 Journal of Computational and Graphical Statistics, Vol. 18 (Taylor & Francis) DOI: 10.1198/jcgs.2009.06134 - Discusses the design and properties of adaptive Metropolis-Hastings algorithms, which are crucial for automatically tuning proposal distributions for improved efficiency.