Bayesian Statistics

  1. Bayesian Computation
    1. The Challenge of High-Dimensional Integrals
      1. Curse of Dimensionality
        1. Intractability in Complex Models
          1. Analytical vs. Numerical Solutions
            1. Approximation Necessity
            2. Deterministic Approximation Methods
              1. Grid Approximation
                1. Procedure and Implementation
                  1. Limitations and Scalability
                    1. Computational Complexity
                    2. Quadrature Methods
                      1. Gaussian Quadrature
                        1. Adaptive Quadrature
                          1. Applicability and Limitations
                            1. Sparse Grid Methods
                            2. Laplace Approximation
                              1. Normal Approximation to Posterior
                                1. Mode Finding
                                  1. Accuracy and Limitations
                                2. Simulation-Based Methods
                                  1. Monte Carlo Integration
                                    1. Basic Principles
                                      1. Random Sampling
                                        1. Estimating Posterior Quantities
                                          1. Law of Large Numbers
                                            1. Central Limit Theorem Applications
                                            2. Direct Sampling Methods
                                              1. Inverse Transform Sampling
                                                1. Rejection Sampling
                                                  1. Acceptance-Rejection Methods
                                                  2. Importance Sampling
                                                    1. Importance Weights
                                                      1. Weighting Schemes
                                                        1. Variance Reduction
                                                          1. Effective Sample Size
                                                            1. Self-Normalized Importance Sampling
                                                          2. Markov Chain Monte Carlo
                                                            1. The Logic of MCMC
                                                              1. Markov Chains in Bayesian Inference
                                                                1. Sampling from Complex Posteriors
                                                                  1. Ergodic Theory Foundations
                                                                  2. Properties of Markov Chains
                                                                    1. State Space and Transition Kernels
                                                                      1. Irreducibility
                                                                        1. Definition and Importance
                                                                          1. Communication Classes
                                                                          2. Aperiodicity
                                                                            1. Definition and Importance
                                                                              1. Period of States
                                                                              2. Stationarity
                                                                                1. Stationary Distribution
                                                                                  1. Invariant Distribution
                                                                                  2. Ergodicity
                                                                                    1. Ergodic Theorem
                                                                                      1. Convergence to Stationarity
                                                                                    2. MCMC Diagnostics
                                                                                      1. Assessing Convergence
                                                                                        1. Visual Diagnostics
                                                                                          1. Formal Tests
                                                                                          2. Trace Plots
                                                                                            1. Interpretation
                                                                                              1. Mixing Assessment
                                                                                              2. Autocorrelation Analysis
                                                                                                1. Autocorrelation Function
                                                                                                  1. Integrated Autocorrelation Time
                                                                                                  2. Gelman-Rubin Diagnostic
                                                                                                    1. Multiple Chain Comparison
                                                                                                      1. Potential Scale Reduction Factor
                                                                                                        1. Interpretation and Thresholds
                                                                                                        2. Effective Sample Size
                                                                                                          1. Calculation Methods
                                                                                                            1. Relevance to Inference
                                                                                                              1. Monte Carlo Standard Error
                                                                                                            2. MCMC Implementation Issues
                                                                                                              1. Burn-in Period
                                                                                                                1. Purpose and Determination
                                                                                                                  1. Discarding Initial Samples
                                                                                                                  2. Thinning
                                                                                                                    1. Reducing Autocorrelation
                                                                                                                      1. Storage Considerations
                                                                                                                        1. Trade-offs
                                                                                                                        2. Chain Length Determination
                                                                                                                          1. Convergence vs. Efficiency
                                                                                                                            1. Practical Guidelines

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                                                                                                                      3. Multi-Parameter Models

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                                                                                                                      5. MCMC Algorithms