Bayesian Statistics

  1. Single-Parameter Models
    1. Introduction to Conjugacy
      1. Definition of a Conjugate Prior
        1. Mathematical Definition
          1. Exponential Family Connections
            1. Examples of Conjugate Pairs
            2. Computational Advantages
              1. Analytical Posterior Forms
                1. Simplification of Calculations
                  1. Closed-Form Solutions
                  2. Limitations of Conjugacy
                    1. Restrictive Prior Choices
                      1. Real-World Applicability
                    2. Common Conjugate Families
                      1. Beta-Binomial Model
                        1. Model Structure
                          1. Beta Prior Distribution
                            1. Binomial Likelihood
                              1. Posterior Beta Distribution
                                1. Modeling Proportions
                                  1. Example Applications
                                    1. Parameter Interpretation
                                    2. Gamma-Poisson Model
                                      1. Model Structure
                                        1. Gamma Prior Distribution
                                          1. Poisson Likelihood
                                            1. Posterior Gamma Distribution
                                              1. Modeling Count Data
                                                1. Rate Parameter Inference
                                                  1. Example Applications
                                                  2. Normal-Normal Model
                                                    1. Model Structure
                                                      1. Normal Prior Distribution
                                                        1. Normal Likelihood with Known Variance
                                                          1. Posterior Normal Distribution
                                                            1. Modeling Means
                                                              1. Precision and Variance Parameterization
                                                                1. Example Applications
                                                                2. Gamma-Exponential Model
                                                                  1. Model Structure
                                                                    1. Gamma Prior Distribution
                                                                      1. Exponential Likelihood
                                                                        1. Posterior Gamma Distribution
                                                                          1. Modeling Rates and Lifetimes
                                                                            1. Example Applications
                                                                            2. Inverse-Gamma-Normal Model
                                                                              1. Model Structure for Variance
                                                                                1. Inverse-Gamma Prior
                                                                                  1. Normal Likelihood with Unknown Variance
                                                                                    1. Posterior Inverse-Gamma Distribution
                                                                                  2. Summarizing the Posterior Distribution
                                                                                    1. Point Estimates
                                                                                      1. Posterior Mean
                                                                                        1. Calculation and Interpretation
                                                                                          1. Optimality Properties
                                                                                          2. Posterior Median
                                                                                            1. Calculation and Interpretation
                                                                                              1. Robustness Properties
                                                                                              2. Posterior Mode
                                                                                                1. Maximum a Posteriori Estimation
                                                                                                  1. Calculation and Interpretation
                                                                                                    1. Relationship to Maximum Likelihood
                                                                                                  2. Interval Estimates
                                                                                                    1. Credible Intervals
                                                                                                      1. Definition and Interpretation
                                                                                                        1. Probability Statements
                                                                                                        2. Highest Posterior Density Intervals
                                                                                                          1. Definition and Calculation
                                                                                                            1. Shortest Intervals
                                                                                                            2. Equal-Tailed Intervals
                                                                                                              1. Definition and Calculation
                                                                                                                1. Symmetric Intervals
                                                                                                                2. Comparison to Confidence Intervals
                                                                                                                  1. Differences from Frequentist Intervals
                                                                                                                    1. Interpretation Advantages
                                                                                                                      1. Practical Implications
                                                                                                                    2. Posterior Uncertainty Quantification
                                                                                                                      1. Posterior Variance
                                                                                                                        1. Posterior Standard Deviation
                                                                                                                          1. Coefficient of Variation
                                                                                                                        2. Posterior Predictive Distribution
                                                                                                                          1. Predicting New Observations
                                                                                                                            1. Definition and Purpose
                                                                                                                              1. Calculation Methods
                                                                                                                                1. Integration over Posterior
                                                                                                                                2. Averaging over Posterior Uncertainty
                                                                                                                                  1. Parameter Uncertainty Propagation
                                                                                                                                    1. Implications for Prediction
                                                                                                                                      1. Prediction Intervals
                                                                                                                                      2. Model Checking with Predictive Distribution
                                                                                                                                        1. Posterior Predictive Checks
                                                                                                                                          1. Discrepancy Measures

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                                                                                                                                      1. Foundations of Bayesian Inference

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                                                                                                                                      3. Multi-Parameter Models