Prof: | Robert L. Wolpert |
wolpert@stat.duke.edu | OH: Mon 2:00-3:00pm, 211c Old Chem | |||

TA: | Xi Chen |
xc54@duke.edu | OH: Wed 1:30-2:30pm, 211a Old Chem | |||

Class: | Tue/Thu 11:45a-1:00p, Old Chem 123 | |||||

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

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 14 | Intro: Probability distributions & Function spaces | ||||

Jan 19-21 | Brownian Motion construction | Einstein 1905 in English or original German | |||

Jan 26-28 | No class (Volcano workshop at Kilauea) | ||||

Feb 02-04 | Stochastic Integration & Diffusions | ||||

Feb 09-11 | Dirichlet Sobolev spaces & Brownian Bridge | hw1 | Feb 25 | ||

II. Gaussian Processes and Computer Emulation | |||||

Feb 16-18 | Isotropic Covariance Functions & Gaussian Procs | ||||

Feb 23-25 | Sacks/Welch/Mitchell & Kennedy/O'Hagan & c. | ||||

Mar 01-03 | Guest Lecture by Mengyang Gu, plus ARMA | hw2 | Mar 22 | ||

III. ID Dist'ns & Lévy Processes | |||||

Mar 08-10 | ID & Lévy | ||||

--- Spring Recess (Mar 12-20) --- | |||||

Mar 22-24 | LARK Models and Lévy Random Fields | ||||

Mar 29-31 | Spatial Extremes & Smith Models | ||||

Apr 05-07 | No class (SIAM/ASA Conference in Lausanne) | ||||

IV. Exotic Time Series | |||||

Apr 12-14 | Non-Gauss. ID AR(1)-like Procs: Ga, Po, NB | hw3 | Apr 19 | ||

Birth & Death, NB Inference | |||||

Apr 19 | Non-Gauss. Non-Stat. Irreg. Observed Procs |

- 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.