Prof: | Robert L. Wolpert |
wolpert@stat.duke.edu | OH: Mon 4:30-5:30pm, 211c Old Chem | |||

TA: | Yun Yang |
yy84@stat.duke.edu | OH: Mon 7:00-9:00pm, 211a Old Chem | |||

Class: | Tue/Thu 11:45a-1:00p, Soc Sci 124 | |||||

Text: | Rasmussen & Williams, | Gaussian Processes for Machine Learning | Free on-line version | |||

Opt'l: | Santner, Williams & Notz, | The Design and Analysis of Computer Experiments | ||||

Reed & Simon, | Functional Analysis, Volume I | |||||

Meyn & Tweedie, | Markov Chains and Stochastic Stability | |||||

Michael Stein, | Interpolation of Spatial Data |

Week | Topic | Homework | |
---|---|---|---|

I. Brownian Bridge and Related Topics |
Problems | Due | |

Jan 09 | Intro: Probability distributions & Function spaces | ||

Jan 14-16 | Motivation: Donsker & Kolmogorov-Smirnov | ||

Jan 21-23 | Dirichlet Sobolev spaces & Brownian Bridge | hw1 | Feb 18 |

Jan 28-30 | Free Sobolev & O-U, Karhunen-Loève | ||

Feb 04-06 | No class (SAMSI Workshop: Topological Analysis) | ||

Feb 11 | Reflection and Maxima of the Brownian Bridge | ||

Feb 13 | No class--- Snow Day. Homework postponed. | ||

II. Gaussian Processes and Computer Emulation | |||

Feb 18-20 | Isotropic Covariance Functions & Gaussian Procs | ||

Feb 25-27 | Sacks/Welch/Mitchell & Kennedy/O'Hagan & c. | ||

Mar 04-06 | Prediction & Inference for GPs (PCA notes) | ||

--- Spring Break (Mar 09-16) --- | |||

Mar 18 | Multiple Regression, GPs, and PCA | ||

Mar 20 | Case Study: Heavy Ion Collisions (Chris C-S) | ||

III. Exotic Time Series | |||

Mar 25-27 | Examples: Gaussian AR(p) & Linear B/D PPs | hw2 | Apr 10 |

Apr 01-03 | No class (SIAM/ASA Conference in Savannah) | ||

Apr 08-10 | Six different AR(1)-like Gamma Processes | notes1 | notes2 |

Apr 15 | Integer-valued AR(1)-like Processes or α-Stables |
hw3 | Apr 24 |

- Basics of Statical SPs: Donsker's Theorem and the
Brownian Bridge; Karhunen-Loève; Likelihood Ratios in
sequential procedures as martingales and random walks;
some function space background (L
_{2}Sobolev theory, Hilbert & Banach spaces), probability distributions on function spaces. - Inference and prediction for Gaussian Processes and Random Fields, and in particular on building GP emulators for modeling complex computer output;
- Exotic time series with Infinitely Divisible distributions including integer-valued autocorrelated AR(p)-like processes with Poisson or Negative Binomial marginals, useful for modeling autocorrelated count data, and positive-valued processes with Gamma or α-Stable marginals, useful for modeling correlated positive quantities or for use at intermediate stages of hierarchical models (e.g., for building more flexible Bayesian semi- or non-parametric models than is possible with Dirichlet processes).

Topics in the theory and modelling aspects of Stochastic Processes central to Bayesian statistical analysis. Topics vary from year to year but may include some of:

- Convergence theory of Markov chains in general state spaces;
- Convergence rates and mixing times for Markov chains;
- Stochastic approximation and adaptive estimators;
- Inference and prediction for Gaussian Processes and Random Fields;
- Modeling and Inference with Continuous-time Jump Processes;
- Theory, Application, and inference for Lévy Random Flights;
- Inference for Diffusions;
- Exotic time series with Infinitely Divisible distributions;
- Functional Data Analysis.