Markov Chain Monte Carlo: Theory and Application
[ Syllabus ]

• Introduction
  1. The problem: Bayesian inference and calculating expectations.
  2. Markov Chain Monte Carlo (MCMC).
  3. Implementation.
  4. Application.
• Markov Chain concepts
  1. Introduction.
  2. Rates of convergence.
  3. Estimation.
  4. The Gibbs sampler and Metropolis-Hastings algorithm.
• Full conditional distributions
  1. Deriving full conditionals.
  2. Sampling from full conditionals.
  3. Application.
• Strategies for improving MCMC
  1. Reparametrization.
  2. Random and adaptive direction sampling.
  3. Modifying stationary distribution.
• Implementing MCMC
  1. Determining the number of iterations.
  2. Software and implementation.
  3. Output analysis.
  4. Application.
• Generalized linear mixed models
  1. Frequentist ans Bayesian GLMs.
  2. Specification of random-effect distributions.
  3. Hyperpriors and estimation of hyperparameters.
  4. Examples.
• Hierarchical longitudinal modeling
  1. Clinical bacground.
  2. Model detail and MCMC implementation.
  3. Results.
• Bayesian mapping of disease
  1. Maximum likelihood estimation of relative risks.
  2. Hierarical Bayesian model of relative risks.
  3. Empirical Bayes estimation of relative risks.
  4. Fully Bayesian estimation of relative risks.
• Mixtures of distributions
  1. Introduction.
  2. The missing data structure.
  3. Gibbs sampling implementation.
  4. Convergence of algorithm.
  5. Testing for mixtures.
  6. Infinite mixtures and other extensions.

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