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Prerequisites
Basic statistical theory, linear models, and Bayesian modeling at the level of STA214, STA215 and STA244. Some experience in a lower level programming language such as C or FORTRAN and a familiarity with UNIX workstations is assumed; experience with a high level language such as S-Plus or Matlab is useful, but not required.

Course Description (from the Duke Statistics course listings)
Techniques for maximization and integration suitable for computations in statistical analyses and particularly Bayesian inference. Quadrature methods; simulated annealing; Metropolis techniques; methods of Monte-Carlo and Gibbs sampling; error analysis, variance reduction, and comparisons with standard derivative-based optimization methods and quadrature-based integration methods.

An introduction to advanced statistical modeling and modern numerical methods useful in implementing statistical procedures for data analysis, model exploration, inference, and prediction. Methods are applied to substantial problems in discrete multivariate analysis, time series, econometrics, non-linear regression models, density estimation, applications with censored and missing data, hierarchical models, mixture modeling, and non-linear regressions.

Syllabus

1. Locating Modes -- Selected Optimization Techniques
Standard derivative and non-derivative based methods; the EM algorithm and simulated annealing.

2. Likelihood Approximations
Normal approximations; the delta method; Laplace approximations.

3. Numerical Integration 1: Quadrature
Quadrature techniques applied to marginalizing densities and estimating (their) moments.

4. Numerical Integration 2: Simulation-Based Methods
Non-iterative simulation-based techniques in numerical integration applied to marginalizing densities and estimating (their) moments. Monte Carlo integration; importance sampling.

5. Numerical Integration 3: Markov Chain Monte Carlo
Markov Chain Monte Carlo methods for inference. Metropolis-Hastings methods including Gibbs sampling; convergence and applications.

---Digressions---

A. Simulation and Monte Carlo Methods
Pseudo-random numbers, simulation techniques and Monte Carlo methods.

B. Assorted Numerical Methods
Computer representation and manipulation of numeric data and ramifications for statistical algorithms; matrix and linear computations.

Grading
Course grades will be based on weekly/biweekly assignments and a project. Assignments will involve programming in S-Plus with occasional exercises involving simple C (or Fortran--your choice) programming. The project will involve a substantive analysis of scientific interest that employs inferential methods described in class. Each student will be expected to submit a one page project proposal, due in class 10/14. A written project report will be due (tentatively) Monday of finals week (12/14).