Computational Statistics

  1. Bayesian Computational Methods
    1. Foundations of Bayesian Inference
      1. Bayes' Theorem
        1. Prior-to-Posterior Updating
          1. Likelihood Principle
          2. Prior, Likelihood, and Posterior Distributions
            1. Prior Specification
              1. Likelihood Construction
                1. Posterior Derivation
                2. Posterior Summarization
                  1. Posterior Mean and Variance
                    1. Credible Intervals
                      1. Marginal Distributions
                        1. Posterior Predictive Distributions
                        2. Conjugate Priors
                          1. Exponential Family Conjugacy
                            1. Common Conjugate Pairs
                              1. Computational Advantages
                            2. Challenges in Bayesian Computation
                              1. High-Dimensional Integrals
                                1. Curse of Dimensionality
                                  1. Analytical Intractability
                                  2. Intractable Posterior Distributions
                                    1. Non-Standard Forms
                                      1. Multimodal Posteriors
                                      2. Approximate Bayesian Computation
                                        1. ABC Algorithm
                                          1. Distance Metrics
                                            1. Tolerance Selection
                                          2. Markov Chain Monte Carlo (MCMC)
                                            1. Theory of Markov Chains
                                              1. State Space
                                                1. Transition Kernel
                                                  1. Stationarity and Ergodicity
                                                    1. Detailed Balance
                                                      1. Irreducibility and Aperiodicity
                                                      2. The Metropolis-Hastings Algorithm
                                                        1. Algorithm Structure
                                                          1. Proposal Distributions
                                                            1. Random Walk Proposals
                                                              1. Independence Sampler
                                                                1. Langevin Proposals
                                                                2. Acceptance-Rejection Step
                                                                  1. Tuning the Algorithm
                                                                    1. Acceptance Rate
                                                                      1. Adaptive Tuning
                                                                        1. Optimal Scaling
                                                                      2. The Gibbs Sampler
                                                                        1. Full Conditional Distributions
                                                                          1. Systematic vs Random Scan
                                                                            1. Blocked Gibbs Sampling
                                                                              1. Convergence Properties
                                                                              2. Advanced MCMC Algorithms
                                                                                1. Hamiltonian Monte Carlo (HMC)
                                                                                  1. Hamiltonian Dynamics
                                                                                    1. Leapfrog Integrator
                                                                                      1. Tuning Step Size and Trajectory Length
                                                                                      2. No-U-Turn Sampler (NUTS)
                                                                                        1. Automatic Path Length Selection
                                                                                          1. Dual Averaging
                                                                                          2. Reversible-Jump MCMC (RJMCMC)
                                                                                            1. Model Selection and Variable Dimension
                                                                                              1. Birth-Death Processes
                                                                                              2. Parallel Tempering
                                                                                                1. Temperature Ladders
                                                                                                  1. Replica Exchange
                                                                                              3. MCMC Diagnostics
                                                                                                1. Assessing Convergence
                                                                                                  1. Trace Plots
                                                                                                    1. Autocorrelation Plots
                                                                                                      1. Gelman-Rubin Diagnostic
                                                                                                        1. Effective Sample Size
                                                                                                          1. Geweke Diagnostic
                                                                                                          2. Burn-in Period
                                                                                                            1. Determining Burn-in Length
                                                                                                              1. Multiple Chain Strategies
                                                                                                              2. Assessing Model Fit
                                                                                                                1. Posterior Predictive Checks
                                                                                                                  1. Deviance Information Criterion (DIC)
                                                                                                                    1. Widely Applicable Information Criterion (WAIC)
                                                                                                                      1. Cross-Validation for Bayesian Models
                                                                                                                    2. Variational Inference
                                                                                                                      1. Mean Field Variational Bayes
                                                                                                                        1. Variational Families
                                                                                                                          1. ELBO Optimization
                                                                                                                          2. Automatic Differentiation Variational Inference
                                                                                                                            1. Gradient-Based Optimization
                                                                                                                              1. Reparameterization Tricks

                                                                                                                          Previous

                                                                                                                          4. Numerical Optimization in Statistics

                                                                                                                          Go to top

                                                                                                                          Next

                                                                                                                          6. High-Dimensional Data Analysis