Statistics

1. Bayesian Statistics
   1. Bayesian Inference
      1. Basics of Bayesian Inference
         1. Bayes' Theorem
            1. Formula and Explanation
         2. Updating Beliefs
            1. Prior, Likelihood, Posterior
         3. Probability Model Construction
      2. Comparison to Frequentist Inference
         1. Differences in Philosophy
         2. Handling of Prior Information
         3. Interpretation of Parameters
      3. Real-world Examples
         1. Medical Diagnosis
         2. Decision Making under Uncertainty
   2. Prior and Posterior Distributions
      1. Priors
         1. Types of Priors
            1. Informative Priors
            2. Non-informative Priors
            3. Conjugate Priors
         2. Elicitation of Priors
            1. Expert Elicitation
            2. Empirical Methods
         3. Impact on Posterior Distribution
      2. Posterior Distributions
         1. Calculation using Bayes' Theorem
         2. Types of Posteriors
            1. Analytical Solutions
            2. Numerical Approximations
   3. Bayesian Estimation Techniques
      1. Point Estimation
         1. Maximum A Posteriori (MAP)
         2. Posterior Mode
      2. Interval Estimation
         1. Credible Intervals
         2. Highest Posterior Density (HPD) Intervals
      3. Bayesian Hypothesis Testing
         1. Bayes Factor
         2. Decision-theoretic Approaches
   4. Markov Chain Monte Carlo (MCMC)
      1. Introduction
         1. Purpose of MCMC
         2. Importance in Bayesian Computation
      2. Algorithms
         1. Gibbs Sampling
         2. Metropolis-Hastings Algorithm
         3. Hamiltonian Monte Carlo
      3. Convergence Diagnostics
         1. Trace Plots
         2. Autocorrelation Analysis
         3. Geweke Diagnostic
      4. Practical Considerations
         1. Choice of Priors
         2. Computational Challenges
   5. Hierarchical Bayesian Models
      1. Definition and Utility
         1. Multilevel Models
         2. Nesting and Non-nesting Structures
      2. Construction of Hierarchical Model
         1. Specifying Hyperparameters
         2. Partially Pooled Data
      3. Applications
         1. Longitudinal Data Analysis
         2. Spatial Data Models
   6. Bayesian Model Averaging
      1. Importance and Concept
      2. Avoiding Model Selection Bias
      3. Computational Techniques
         1. Reversible Jump MCMC
         2. Bridge Sampling
   7. Applications of Bayesian Methods
      1. Healthcare
         1. Bayesian Clinical Trials
         2. Adaptive Designs
      2. Finance
         1. Risk Assessment
         2. Portfolio Optimization
      3. Machine Learning
         1. Bayesian Neural Networks
         2. Gaussian Processes
      4. Environmental Science
         1. Climate Modeling
         2. Bayesian Spatiotemporal Models
   8. Criticism and Challenges
      1. Subjectivity of Priors
         1. Debate on Objectivity
      2. Computational Intensity
         1. High-dimensional Problems
         2. Advancements in Computing Power
      3. Interpretational Complexity
         1. Communicating Results to Non-statisticians
   9. Advanced Topics
      1. Bayesian Nonparametrics
         1. Dirichlet Process
         2. Gaussian Processes
      2. Bayesian Networks
         1. Structure Learning
         2. Node Relationships
      3. Sequential Bayesian Inference
         1. Particle Filters
         2. Real-time Data Processing