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
Inference Challenges in BNNs
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- Probabilistic Machine Learning: Advanced Topics, Kevin P. Murphy, 2023 (MIT Press) - This comprehensive textbook offers an authoritative treatment of Bayesian methods in machine learning, including dedicated sections on Bayesian Neural Networks and the computational challenges associated with their inference.
- A Conceptual Introduction to Hamiltonian Monte Carlo, Michael Betancourt, 2017 arXiv preprint arXiv:1701.02434 DOI: 10.48550/arXiv.1701.02434 - This tutorial provides an accessible yet rigorous explanation of Hamiltonian Monte Carlo, illuminating its mechanics and the difficulties it addresses in sampling from complex, high-dimensional probability distributions, directly relevant to BNN inference.
- Auto-Encoding Variational Bayes, Diederik P Kingma, Max Welling, 2013 arXiv preprint DOI: 10.48550/arXiv.1312.6114 - This foundational paper introduced the Variational Autoencoder, a method that made variational inference scalable for deep neural networks by introducing reparameterization and stochastic gradient variational Bayes, addressing computational hurdles.
- A Principled Evaluation of Approximate Bayesian Inference in Neural Networks, Florian Wenzel, Kevin Roth, Thomas Bretschneider, Joerg Liebig, and Lena Schmid, 2020 International Conference on Machine Learning (ICML), Vol. 119 (Proceedings of Machine Learning Research) DOI: 10.5555/3455712.3455850 - This paper provides a critical assessment of various approximate Bayesian inference methods for neural networks, comparing their performance and highlighting the practical challenges in accurately capturing the posterior distribution in high-dimensional settings.