Oberwolfach Seminar: Algebraic Statistics

Date:

May 11th - May 17th, 2008

Organizers:

Mathias Drton, Chicago

Bernd Sturmfels, Berkeley

Seth Sullivant, Cambridge

Programme:

How does an algebraic geometer studying secant varieties further the understanding of hypothesis tests in statistics? Why would a statistician working on factor analysis raise open problems about determinantal varieties? Connections of this type are at the heart of the new field of "algebraic statistics". In this field, mathematicians and statisticians come together to solve statistical inference problems using concepts from algebraic geometry as well as related computational and combinatorial techniques. The goal of this seminar is to introduce newcomers from the different camps to algebraic statistics. The introduction will be centered around the following three observations:

• many important statistical models correspond to algebraic or semi-algebraic sets of parameters;
• the geometry of these parameter spaces determines the behaviour of widely used statistical inference procedures;
• computational algebraic geometry can be used to study parameter spaces and other features of statistical models.

There will be three one-hour lectures per day, at 9:30, 11:00 and 16:00. The last lecture of each day will lead into a working session, where the participants will discuss the lectures, work on exercises, and compute examples.

The only exception is Wednesday afternoon, which will be free for the traditional Oberwolfach hike. On Monday evening, after dinner, there will be an "Open Mike Session" where participants are invited to give short presentations (5 to 10 minutes) about their own research interests. Further evening sessions will emerge spontaneously.

The five days are organized by themes. In what follows we list the subjects of the 14 lectures, as well as some suggested reading.

• MONDAY: Markov Bases
  9:30 Mathias: "Hypothesis Tests in Contingency Tables"
  11:00 Seth: "Markov bases and Polytopes of Graphical Models"
  16:00 Bernd: "Computing Markov bases of Log-linear Models"
• TUESDAY: Likelihood Inference
  9:30 Seth: "Likelihood Function in Discrete and Normal Models"
  11:00 Bernd: "Solving the Likelihood Equations"
  16:00 Mathias: "Likelihood Ratio Tests"
• WEDNESDAY: Bayesian Inference
  9:30 Mathias: "Asymptotics of Bayesian Integrals"
  11:00 Discussion: "What is Algebraic Statistics?"
• THURSDAY: Conditional Independence
  9:30 Seth: "Conditional Independence"
  11:00 Mathias: "Graphical Models"
  16:00 Bernd: "The Hammersley-Clifford Theorem"
• FRIDAY: Hidden Variables
  9:30 Seth: "Secant Varieties in Statistics"
  11:00 Mathias: "Factor Analysis"
  16:00 Bernd: "Marginal Likelihood Integrals"

Prerequisites:

* Cox, David; Little, John; O'Shea, Donal. Ideals, Varieties, and Algorithms. Undergraduate Texts in Mathematics. Springer.
* Drton, Mathias; Sullivant, Seth. Algebraic statistical models. Statistica Sinica 17 (2007), 1273--1297.
* Pachter, Lior; Sturmfels, Bernd. Algebraic statistics for computational biology, Cambridge Univ. Press, New York, 2005. (Chapter 1 and 3)
* Sturmfels, Bernd. Solving systems of polynomial equations. CBMS Regional Conference Series in Mathematics, 97. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2002. (Chapter 8)

Literature:

* Catanese, Fabrizio; Hosten, Serkan; Khetan, Amit; Sturmfels, Bernd. The maximum likelihood degree. Amer. J. Math. 128 (2006), no. 3, 671--697.
* Diaconis, Persi; Sturmfels, Bernd. Algebraic algorithms for sampling from conditional distributions. Ann. Statist. 26 (1998), no. 1, 363--397.
* Drton, Mathias; Sturmfels, Bernd; Sullivant, Seth. Algebraic factor analysis: tetrads, pentads and beyond. Probab. Theory Related Fields 138 (2007), no. 3-4, 463--493.
* Drton, Mathias. Likelihood ratio tests and singularities. Ann. Statist. (200x), to appear. arXiv:math.ST/0703360.
* Garcia, Luis David; Stillman, Michael; Sturmfels, Bernd. Algebraic geometry of Bayesian networks. J. Symbolic Comput. 39 (2005), no. 3-4, 331--355.
* Geiger, Dan; Meek, Christopher; Sturmfels, Bernd. On the toric algebra of graphical models. Ann. Statist. 34 (2006), no. 3, 1463--1492.
* Hosten, Serkan; Khetan, Amit; Sturmfels, Bernd. Solving the likelihood equations. Found. Comput. Math. 5 (2005), no. 4, 389--407.

Deadline for applications:

April 1st, 2008

The seminars take place at the Mathematisches Forschungsinstitut Oberwolfach. The number of participants is restricted to 24. The Institute covers accommodation and food. Travel expenses cannot be reimbursed. Applications including

• full name and address, including e-mail address
• present position, university
• name of supervisor of Ph.D. thesis
• a short summary of previous work and interest

should be sent as hard copy or by e-mail (.ps or .pdf file) to:

Prof. Dr. Gert-Martin Greuel
Universität Kaiserslautern
Fachbereich Mathematik
Erwin Schrödingerstr.
67663 Kaiserslautern, Germany
.