Bayesian Statistics

  1. Hierarchical Models
    1. The Concept of Pooling
      1. No Pooling
        1. Separate Models for Each Group
          1. Independent Analysis
            1. Limitations and Inefficiencies
            2. Complete Pooling
              1. Single Model for All Data
                1. Ignoring Group Structure
                  1. When Appropriate
                  2. Partial Pooling
                    1. Hierarchical Modeling Approach
                      1. Borrowing Strength
                        1. Shrinkage Effects
                      2. Structure of Hierarchical Models
                        1. Multi-Level Architecture
                          1. Data Level
                            1. Observed Data and Likelihood
                              1. Within-Group Variation
                              2. Parameter Level
                                1. Group-Level Parameters
                                  1. Between-Group Variation
                                  2. Hyperparameter Level
                                    1. Priors on Group-Level Parameters
                                      1. Population-Level Parameters
                                      2. Hyperprior Level
                                        1. Priors on Hyperparameters
                                          1. Model Specification Completion
                                        2. Mathematical Formulation
                                          1. Conditional Independence Structure
                                            1. Exchangeability
                                              1. De Finetti's Theorem
                                                1. Hierarchical Likelihood
                                                2. Advantages of Hierarchical Modeling
                                                  1. Borrowing Strength Across Groups
                                                    1. Improved Estimates for Small Groups
                                                      1. Shrinkage Toward Population Mean
                                                      2. Modeling Complex Data Structures
                                                        1. Nested Structures
                                                          1. Crossed Random Effects
                                                            1. Handling Unbalanced Data
                                                            2. Uncertainty Propagation
                                                              1. Multiple Levels of Uncertainty
                                                                1. Proper Uncertainty Quantification
                                                              2. Common Hierarchical Models
                                                                1. Normal Hierarchical Model
                                                                  1. Model Specification
                                                                    1. Conjugate Analysis
                                                                      1. Shrinkage Properties
                                                                      2. Binomial Hierarchical Model
                                                                        1. Beta-Binomial Structure
                                                                          1. Overdispersion Modeling
                                                                          2. Poisson Hierarchical Model
                                                                            1. Gamma-Poisson Structure
                                                                              1. Count Data Applications
                                                                            2. Example Applications
                                                                              1. The Eight Schools Model
                                                                                1. Model Specification
                                                                                  1. Educational Testing Context
                                                                                    1. Interpretation of Results
                                                                                      1. Lessons Learned
                                                                                      2. Meta-Analysis
                                                                                        1. Combining Study Results
                                                                                          1. Between-Study Heterogeneity
                                                                                          2. Longitudinal Data Analysis
                                                                                            1. Repeated Measures
                                                                                              1. Individual Growth Curves

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                                                                                          5. MCMC Algorithms

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                                                                                          7. Model Checking and Selection