Relevant textbooks:

1. "Lectures on Algebraic Statistics" Drton, Sturmfels, Sullivant

List of seminar topics and relevant sections of the book:

• Markov bases: hypothesis tests for contingency tables: sec.1.1
• Markov bases of hierarchical log-linear models: sec.1.2
• Graphical models/ conditional independence models: Parts of Chp 3

2. "Markov Bases in Algebraic Statistics" Aoki, Hara, Takemura (downloadable from Springerlink!)

(**Note from the book's preface: "This book is intended for statisticians with minimal backgrounds in algebra. "**)
List of seminar topics and relevant sections of the book:

• Exact Tests for Contingency Tables and Discrete Exponential Families: Chp 1
• Markov Chain Monte Carlo Methods over Discrete Sample Space: Chp 2
• Algebra background (with view toward Markov bases!): Chapters 3 and 4

Problems and suggested related papers (can be found on the arXiv):

1. (topic A) Generating Markov sub-bases efficiently
   • "Markov Chains, Quotient Ideals, and Connectivity with Positive Margins", Chen Dinwoodie and Yoshida ***Recommended reading in preparation for Rudy's visit in April
   • "Markov bases and subbases for bounded contingency tables", Fabio Rapallo, Ruriko Yoshida
   • "Connecting tables with zero-one entries by a subset of a Markov basis", Hisayuki Hara, Akimichi Takemura
   • "Running Markov chain without Markov basis", Hisayuki Hara, Satoshi Aoki, Akimichi Takemura
   • ..
2. (topic A) Using algebraic statistics (Markov bases; implicitization) for model selection or factorial design
   • "Algebraic Statistics in Model Selection", Luis David Garcia (2004)
   • "Bayesian model choice and information criteria in sparse generalized linear models", Rina Foygel, Mathias Drton
   • "Markov basis for design of experiments with three-level factors", Satoshi Aoki, Akimichi Takemura
   • "Design and analysis of fractional factorial experiments from the viewpoint of computational algebraic statistics", Satoshi Aoki, Akimichi Takemura
   • ..
3. (topic B) Understanding the geometry of graphical mixture models in biology (phylogenetcs)
• "Phylogenetic Algebraic Geometry", Nicholas Eriksson, Kristian Ranestad, Bernd Sturmfels, Seth Sullivant
  ( arXiv:math/0407033 , overview paper, points to open problems and several topics)
• "Geometry of the Kimura 3-parameter model", Marta Casanellas, Jesus Fernandez-Sanchez
• "On the ideals of equivariant tree models", Jan Draisma, Jochen Kuttler
• " Bounded-rank tensors are defined in bounded degree", Jan Draisma, Jochen Kuttler
• "The space of phylogenetic mixtures for equivariant models", Marta Casanellas, Jesus Fernandez-Sanchez, Anna Kedzierska
• "Identifiability of Large Phylogenetic Mixture Models", John A. Rhodes, Seth Sullivant
• "When Do Phylogenetic Mixture Models Mimic Other Phylogenetic Models?", Elizabeth S. Allman, John A. Rhodes, Seth Sullivant
4. (topic B) Parameter identifiability problems
   • "Binary hidden Markov models and varieties", Andrew J. Critch
   • "Identifying Causal Effects with Computer Algebra", Luis David García-Puente, Sarah Spielvogel, Seth Sullivant (2010)
   • "Parameter identifiability in a class of random graph mixture models", Elizabeth S. Allman, Catherine Matias, John A. Rhodes
   • "Identifiability of parameters in latent structure models with many observed variables", Elizabeth S. Allman, Catherine Matias, John A. Rhodes
   • "Identifying evolutionary trees and substitution parameters for the general Markov model with invariable sites", Elizabeth S. Allman, John A. Rhodes
   • ..