Bayesian Statistics

  1. Advanced Topics
    1. Bayesian Time Series Analysis
      1. Time Series Fundamentals
        1. Stationarity and Non-stationarity
          1. Autocorrelation Structure
            1. Trend and Seasonality
            2. State-Space Models
              1. State Equation
                1. Observation Equation
                  1. Kalman Filter
                    1. Particle Filters
                    2. Dynamic Linear Models
                      1. Local Level Models
                        1. Local Trend Models
                          1. Seasonal Models
                            1. Regression with Time-Varying Coefficients
                            2. Autoregressive Models
                              1. AR Models
                                1. VAR Models
                                  1. Bayesian Estimation
                                  2. Forecasting
                                    1. Point Forecasts
                                      1. Interval Forecasts
                                        1. Forecast Evaluation
                                      2. Bayesian Survival Analysis
                                        1. Survival Data Characteristics
                                          1. Time-to-Event Data
                                            1. Censoring Mechanisms
                                              1. Truncation
                                              2. Survival Functions
                                                1. Survival Function
                                                  1. Hazard Function
                                                    1. Cumulative Hazard
                                                    2. Parametric Survival Models
                                                      1. Exponential Model
                                                        1. Weibull Model
                                                          1. Log-Normal Model
                                                            1. Bayesian Inference
                                                            2. Semi-Parametric Models
                                                              1. Cox Proportional Hazards Model
                                                                1. Bayesian Cox Model
                                                                  1. Partial Likelihood
                                                                  2. Accelerated Failure Time Models
                                                                    1. Model Specification
                                                                      1. Bayesian Implementation
                                                                    2. Gaussian Processes
                                                                      1. Definition and Properties
                                                                        1. Stochastic Process Definition
                                                                          1. Mean and Covariance Functions
                                                                            1. Finite-Dimensional Distributions
                                                                            2. Covariance Functions
                                                                              1. Squared Exponential
                                                                                1. Matérn Family
                                                                                  1. Periodic Functions
                                                                                    1. Composite Kernels
                                                                                    2. Gaussian Process Regression
                                                                                      1. Predictive Distribution
                                                                                        1. Hyperparameter Inference
                                                                                          1. Model Selection
                                                                                          2. Classification with Gaussian Processes
                                                                                            1. Latent Function Approach
                                                                                              1. Approximate Inference Methods
                                                                                              2. Computational Considerations
                                                                                                1. Scalability Issues
                                                                                                  1. Sparse Approximations
                                                                                                    1. Inducing Points
                                                                                                  2. Bayesian Nonparametrics
                                                                                                    1. Motivation for Nonparametric Methods
                                                                                                      1. Infinite-Dimensional Parameter Spaces
                                                                                                        1. Model Flexibility
                                                                                                          1. Avoiding Parametric Assumptions
                                                                                                          2. Dirichlet Process
                                                                                                            1. Definition and Properties
                                                                                                              1. Stick-Breaking Construction
                                                                                                                1. Chinese Restaurant Process
                                                                                                                  1. Pólya Urn Model
                                                                                                                  2. Dirichlet Process Mixture Models
                                                                                                                    1. Clustering Applications
                                                                                                                      1. Infinite Mixture Models
                                                                                                                        1. Posterior Inference
                                                                                                                        2. Other Nonparametric Priors
                                                                                                                          1. Beta Process
                                                                                                                            1. Indian Buffet Process
                                                                                                                              1. Pitman-Yor Process
                                                                                                                              2. Applications
                                                                                                                                1. Density Estimation
                                                                                                                                  1. Clustering
                                                                                                                                    1. Topic Modeling
                                                                                                                                  2. Approximate Bayesian Computation
                                                                                                                                    1. Motivation and Use Cases
                                                                                                                                      1. Intractable Likelihoods
                                                                                                                                        1. Simulation-Based Models
                                                                                                                                          1. Complex Stochastic Models
                                                                                                                                          2. ABC Framework
                                                                                                                                            1. Summary Statistics
                                                                                                                                              1. Distance Measures
                                                                                                                                                1. Tolerance Levels
                                                                                                                                                2. ABC Algorithms
                                                                                                                                                  1. Rejection ABC
                                                                                                                                                    1. ABC with MCMC
                                                                                                                                                      1. Sequential Monte Carlo ABC
                                                                                                                                                        1. Adaptive ABC
                                                                                                                                                        2. Theoretical Properties
                                                                                                                                                          1. Consistency Results
                                                                                                                                                            1. Convergence Analysis
                                                                                                                                                            2. Limitations and Challenges
                                                                                                                                                              1. Curse of Dimensionality
                                                                                                                                                                1. Summary Statistic Selection
                                                                                                                                                                  1. Computational Efficiency
                                                                                                                                                                2. Variational Inference
                                                                                                                                                                  1. Motivation
                                                                                                                                                                    1. Scalability Issues with MCMC
                                                                                                                                                                      1. Optimization vs. Sampling
                                                                                                                                                                        1. Large-Scale Applications
                                                                                                                                                                        2. Variational Approximation
                                                                                                                                                                          1. KL Divergence Minimization
                                                                                                                                                                            1. Evidence Lower Bound
                                                                                                                                                                              1. Variational Family Selection
                                                                                                                                                                              2. Mean-Field Variational Bayes
                                                                                                                                                                                1. Factorization Assumptions
                                                                                                                                                                                  1. Coordinate Ascent Algorithm
                                                                                                                                                                                    1. Convergence Properties
                                                                                                                                                                                    2. Advanced Variational Methods
                                                                                                                                                                                      1. Structured Variational Families
                                                                                                                                                                                        1. Normalizing Flows
                                                                                                                                                                                          1. Variational Autoencoders
                                                                                                                                                                                          2. Advantages and Limitations
                                                                                                                                                                                            1. Computational Efficiency
                                                                                                                                                                                              1. Approximation Quality
                                                                                                                                                                                                1. Uncertainty Quantification Issues
                                                                                                                                                                                                2. Applications
                                                                                                                                                                                                  1. Large-Scale Models
                                                                                                                                                                                                    1. Online Learning
                                                                                                                                                                                                      1. Deep Learning Integration

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                                                                                                                                                                                                  8. Bayesian Regression Models

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                                                                                                                                                                                                  1. Foundations of Bayesian Inference