Class: | Wed 11:40am-12:55pm | Languages 312 |

Prof: | Robert L. Wolpert | (wolpert@stat.duke.edu) |

Office: | Old Chem 211c, 684-3275 | |

OH: | Mon 4:15-5:00pm |

Tentative Schedule ACES

Methods: Introduction (ps, pdf); Brownian Motion (ps, pdf); Isotropic covariance functions (ps, pdf);

Gaussian models with specified covariance functions (ps, pdf); Markov Chains (ps, pdf); Simulating Lévy RF's (ps, pdf)

Applications: Bioabundance (ps, pdf); Epidemiology (talk ps, pdf, paper ).

In parametric statistical analysis each uncertain feature of a statistical
model (such as a regression function or probability density function) is
taken to belong to a finite-dimensional family indexed by a parameter qÎQÍ**R**^{d}. In the Bayesian
approach, inference is based on prior and posterior distributions on this
finite-dimensional space.

For nonparametric analysis the uncertain quantities (regression functions,
probability density functions, etc.) might be functions or measures, and so a
Bayesian analysis will require that we place probability distributions on
families of functions or measures- thus both prior and posterior
distributions will be *stochastic processes* or *random fields*.

In this course we will develop elements of the theory of stochastic processes and random fields, tailored to applications in nonparametric Bayesian statistics. These will include independent increment stochastic processes and random fields and their stochastic integrals. We will apply these ideas to the problems of nonparametric Bayesian statistical analysis. We will consider Brownian motion and Lévy processes including the gamma and stable processes (and hence their relatives such as the Dirichlet process), and will study the use of them and their stochastic integrals and moving averages as prior distributions in regression, estimation, and prediction problems.

Example applications will include spatial epidemiology, spatial biodiversity, and the statistical solution to certain inverse problems.

There is no textbook for the course. Recommended optional references include Introduction to Stochastic Processes by Gregory F Lawler or Introduction to Stochastic Processes by Erhan Çinlar for an introduction to Markov chains, discrete-time martingales, and Brownian motion; Diffusions, Markov processes, and martingales by LCG Rogers and David Williams, for Lévy processes and continuous-time martingales.

Students are expected to be (or become) comfortable with probability theory at the level of STA214 or STA205 and with computer programming in R, S-Plus, MatLab, or C. The course will include lectures on some of the topics below, or related ones, selected to meet students' interests.

**Introduction to stochastic processes**- Function spaces
- Martingales, Markov Processes
- Properties

**Brownian motion and Gaussian Processes**- Properties
- Construction (incl. Karhunen-Loève)
- Intro to Stochastic Integration and Diffusions
- Spectral representations

**Distribution-Valued Random Variables**- Hilbert Spaces
- Sobolev Spaces of Tempered Distributions
- The Free Euclidean Field and the O-U Process

**Independent-Increment Processes**- Lévy-Khinchine Characterization
- Construction via Inverse Lévy Measure algorithm
- Moving averages, Stochastic Integrals
- Gamma, a-Stable, and Related Processes and Random Fields
- Compensation

**Dirichlet & Related Processes**- Partition Refinement Construction
- Stick-breaking Algorithms
- Inconsistency Results
- Relation to Gamma Processes and Random Fields
- Polya Trees

**Nonparametric Statistics**- Inference for Point Processes I: Bioabundance and Biodiversity
- Inference for Point Processes II: Spatial Epidemiology
- Statistical approaches to Inverse Problems
- Nonparametric Survival Analysis

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