Basic Stucture

The Basis for this course is to develop an understanding of computational methods for statistics. This will require a fundamental understanding of models and statistical methodology.

Statistics 5314 will be a comprehensive course in simulation based sampling methodology. Modern statistics usually encompasses arbitrarily complex models, where the computer is a necessary component in analyzing data. In hierarchical approaches to modeling data, closed form probability density functions are rarely known, so that sampling based approaches become necessary for conducting statistical analyses. This course will not only cover these sampling methods but will also expose you to classes of valuable models where sampling based methods are necessary devices.

Probability does not exist. -De Finetti

Topics to be discussed in this course follow as:

• Monte Carlo methods,
• Associated Monte Carlo bounds,
• Inverse CDF sampling,
• Accept/Reject sampling,
• Importance sampling,
• and Markov Chain Monte Carlo.

A very large portion of the course will emphasize the MCMC class of methods (Gibbs and Metropolis-Hastings type methods). While each of these methods can be studied (with limited gains) through toy examples, we will be a little more adventurous in our route. Methods will be largely motivated through examples using a time series and dynamic modeling framework (Auto Regressive (AR), ARMA (AR+Moving Average), Hidden Markov (HM), and Stochastic Volatility models). While previous exposure to these models will be useful, it is unnecessary and will not be assumed. However, basic understandings of statistical inference from a Classical and Bayesian perspective will be assumed.

If time permits, we will also discuss:

• Model selection techniques,
• Coupling From The Past,
• Las Vegas methods (and associated bounds),
• The Reversible Jump Algorithm,
• Parallel Tempering.
• Semi-Parametric methods: Gaussian and Dirichlet processes.