Bayesian Statistics

  1. MCMC Algorithms
    1. The Metropolis-Hastings Algorithm
      1. Algorithm Development and History
        1. Algorithm Steps
          1. Proposal Generation
            1. Acceptance Decision
              1. State Update
              2. Proposal Distributions
                1. Random Walk Proposals
                  1. Independence Proposals
                    1. Tuning Parameters
                      1. Adaptive Proposals
                      2. Acceptance Probability
                        1. Calculation and Interpretation
                          1. Detailed Balance
                            1. Reversibility
                            2. Special Cases
                              1. Metropolis Algorithm
                                1. Independence Sampler
                                2. Tuning and Optimization
                                  1. Acceptance Rate Optimization
                                    1. Proposal Variance Tuning
                                      1. Adaptive Algorithms
                                    2. The Gibbs Sampler
                                      1. Algorithm Development
                                        1. Algorithm Steps
                                          1. Systematic Scan
                                            1. Random Scan
                                            2. Full Conditional Distributions
                                              1. Derivation Methods
                                                1. Sampling from Full Conditionals
                                                  1. Conjugacy Advantages
                                                  2. Blocked Gibbs Sampling
                                                    1. Motivation and Benefits
                                                      1. Block Selection Strategies
                                                        1. Implementation Considerations
                                                        2. Convergence Properties
                                                          1. Geometric Ergodicity
                                                            1. Mixing Properties
                                                            2. Limitations and Challenges
                                                              1. High Correlation Issues
                                                                1. Slow Mixing
                                                              2. Advanced MCMC Methods
                                                                1. Hamiltonian Monte Carlo
                                                                  1. Physical Motivation
                                                                    1. Hamiltonian Dynamics
                                                                      1. Leapfrog Integration
                                                                        1. Tuning Parameters
                                                                          1. Mass Matrix Selection
                                                                          2. No-U-Turn Sampler
                                                                            1. Adaptive Path Lengths
                                                                              1. Tree Building Algorithm
                                                                                1. Advantages over Standard HMC
                                                                                  1. Automatic Tuning
                                                                                  2. Slice Sampling
                                                                                    1. Algorithm Principle
                                                                                      1. Univariate Slice Sampling
                                                                                        1. Multivariate Extensions
                                                                                        2. Reversible Jump MCMC
                                                                                          1. Trans-dimensional Problems
                                                                                            1. Model Selection Applications
                                                                                            2. Parallel Tempering
                                                                                              1. Temperature Ladders
                                                                                                1. Replica Exchange
                                                                                                  1. Multimodal Distributions

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                                                                                              4. Bayesian Computation

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                                                                                              6. Hierarchical Models