Statistical Inference

  1. Introduction to Bayesian Inference
    1. Bayesian vs Frequentist Philosophy
      1. Different Interpretations of Probability
        1. Subjective vs Objective Probability
          1. Parameter as Random Variable
            1. Role of Prior Information
            2. Bayes' Theorem
              1. Mathematical Statement
                1. Components and Interpretation
                  1. Updating Beliefs with Data
                    1. Continuous Parameter Version
                    2. Components of Bayesian Analysis
                      1. Prior Distribution
                        1. Expressing Prior Beliefs
                          1. Types of Priors
                            1. Informative Priors
                              1. Non-informative Priors
                                1. Conjugate Priors
                                  1. Jeffreys Priors
                                2. Likelihood Function
                                  1. Role in Bayesian Analysis
                                    1. Same as in Frequentist Analysis
                                    2. Posterior Distribution
                                      1. Combining Prior and Likelihood
                                        1. Normalization Constant
                                          1. Posterior as Compromise
                                          2. Marginal Likelihood (Evidence)
                                            1. Role in Normalization
                                              1. Model Comparison
                                            2. Bayesian Estimation
                                              1. Posterior Point Estimates
                                                1. Posterior Mean
                                                  1. Posterior Median
                                                    1. Posterior Mode (MAP)
                                                      1. Loss Functions and Optimal Estimators
                                                      2. Credible Intervals
                                                        1. Definition and Interpretation
                                                          1. Equal-Tailed Intervals
                                                            1. Highest Posterior Density Intervals
                                                              1. Comparison with Confidence Intervals
                                                            2. Bayesian Hypothesis Testing
                                                              1. Bayes Factors
                                                                1. Definition and Calculation
                                                                  1. Interpretation and Evidence Scale
                                                                    1. Comparison with P-values
                                                                    2. Posterior Probabilities of Hypotheses
                                                                      1. Decision Theory Approach
                                                                      2. Computational Methods
                                                                        1. Analytical Solutions
                                                                          1. Numerical Integration
                                                                            1. Markov Chain Monte Carlo
                                                                              1. Gibbs Sampling
                                                                                1. Metropolis-Hastings Algorithm

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                                                                              10. Simple Linear Regression Inference

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                                                                              12. Non-parametric Methods