Algebraic Statistics 

Algebra has seen many applications in statistics, but it is only rather recently that computational algebraic geometry and related techniques in algebra and combinatorics have been used to study statistical models and inference problems. The main idea is that polynomials and ratios of polynomials appear in statistics and probability under various guises, in model representations as well as in inferential procedures. This links to some basic concepts in algebraic geometry.

This use of computational algebraic geometry was initiated in work on exact tests for models for contingency tables Diaconis and Sturmfels [1998] (see below). Another line of work was initiated by the group of researchers working at the University of Warwick: Wilfrid Kendall, Giovanni Pistone, Eva Riccomagno, Raffaella Settimi, Jim Q. Smith and Henry Wynn. Their research mainly focused on experimental design (see e.g. Pistone and Wynn [1996]) and other applications of computer algebra in statistics (see Kendall [1993], Settimi and Smith [1998]). This finally led to the monograph Pistone et al. [2001] where the name “algebraic statistics” was used for the first time. From this point the field largely expanded…


WOGAS3 Workshop on Geometric and Algebraic Statistics 3: 5-7 April 2011 at Warwick. Read more. 

  • WOGAS2 Workshop on Geometric and Algebraic Statistics 2: 6-7 Apr 2010 at Warwick. Read more.
  • WOGAS Workshop on Geometric and Algebraic Statistics: 13 Jul 2009 at Open University. Read more.

Books on algebraic statistics:

  1. Paolo Gibilisco, Eva Riccomagno, Maria-Piera Rogantin, Henry P. Wynn eds., Algebraic and Geometric Methods in Statistics, Cambridge University Press, 2009.
  2. Sumio Watanabe. Algebraic Geometry and Statistical Learning Theory, Cambridge University Press, 2009.
  3. Mathias Drton, Bernd Sturmfels, Seth Sullivant. Lectures on Algebraic Statistics, Oberwolfach Seminars vol. 39, Birkhäuser, 2009.
  4. Statistica Sinica, vol. 17, number 4, October 2007. Special issue on algebraic statistics.
  5. Bernd Sturmfels, Lior Patcher, eds.. Algebraic Statistics for Computational Biology. Cambridge University Press, 2005.
  6. Giovanni Pistone, Eva Riccomagno, Henry P. Wynn. Algebraic Statistics: Computational Commutative Algebra in Statistics. Chapman&Hall, 2001.


  1. Persi Diaconis, Bernd Sturmfels. Algebraic algorithms for sampling from conditional distributions, Annals of Statistics, 1998.
  2. Giovanni Pistone and Henry P. Wynn. Generalised confounding with Grobner bases. Biometrika, 83(3):653, 1996.
  3. Wilfrid S. Kendall. Computer algebra in probability and statistics. Statistica Neerlandica, 47:9–9, 1993.
  4. Raffaella Settimi and Jim Q. Smith. Geometry, moments and conditional independence trees with hidden variables. Ann. Statist., 28(4):1179–1205, 2000.

Online tutorials:

  1. AS Shourt Course – Seth Sullivant

Interesting links :

  1. 2008-09 Program on Algebraic Methods in Systems Biology and Statistics
  2. Algebra for Biology and Statistics
  3. Seth Sullivant – the web page of Seth Sullivant is a good reference on what’s happening in algebraic statistics
  4. Algebraic Statistics Short Course – These lectures will provide an introduction to algebraic statistics, emphasizing both the interesting algebraic questions that arise and the statistical consequences of the algebraic analysis.
  5. IMA Thematic Year on Applications of Algebraic Geometry Applications in Biology, Dynamics, and Statistics – number of tutorials and more advanced talks on the topic.

People in Algebraic/Geometric Statistics (to be updated):

  1. Frank Critchley
  2. Maria Angelica Cueto
  3. Mathias Drton
  4. Thomas Kahle
  5. Shaowei Lin
  6. Paul Marriot
  7. Hugo Maruri-Aguilar
  8. Giovanni Pistone
  9. Johannes Rauh
  10. Eva Riccomagno
  11. Eduardo Sáenz de Cabezón Irigaray
  12. Anne J. Shiu
  13. Bernd Sturmfels
  14. Seth Sullivant
  15. Caroline Uhler
  16. Henry P. Wynn
  17. Ruriko Yoshida



forthcoming activities:
past activities (since October 2007):
  • Talk@Institut Mittag-Leffler Seminar (3. February 2011) at Institut Mittag-Leffler, Stockholm, Sweden. Probabilistically motivated coordinates in the analysis of phylogenetic tree models. (pdf)
  • Two lectures@Seminar on algebraic models of phylogenetic trees (4. and 11. January 2011) in the Department of Mathematics, Warsaw University. Algebraic statistics of phylogenetic tree models.
  • Talk@Statistics Seminar, at Queen Mary, London. (18. November 2010).
  • Talk@YRM. (9. November 2010) at Warwick University. Cumulants and beyond.
  • Orthogonal Polynomials, applications in Statistics and Stochastic Processes. (12.- 15. July 2010) at Warwick University.
  • Talk@SIMAI 2010. (21.- 25. June 2010) at Università di Cagliari, Sardinia, Italy. Posets, Möbius functions and tree-cumulants.
  • Talk@Higher Dimensional Birational Geometry. (16.- 19. June 2010) Warwick. Real Log Canonical Thresholds and Asymptotic Approximation of Laplace Integrals.
  • Poster@9th Valencia International Meeting on Bayesian Statistics 2010. (3.- 9. June 2010) Benidorm, Alicante, Spain. Asymptotic model selection and identifiability of directed tree models with hidden variables. (pdf).
  • Poster@Model Uncertainty at Warwick. (30. May – 1. June 2010) Warwick University.
  • Talk@Research Students’ Conference 2010 (12.-15. April 2010) at University of Warwick. Posets, Möbius functions and tree-cumulants.
  • Invited talk@WOGAS2 Workshop on Geometric and Algebraic Statistics 2. (6.-7. April 2010) at University of Warwick. Posets, Möbius functions and tree-cumulants.
  • Visitor at MSRI, Berkeley, CA (2. – 18. October 2009).
  • Workshop on Geometric and Algebraic Statistics. (13. July 2009) at The Open University, Milton Keynes, UK.
  • Talk@YRM. (25. May 2009) at Warwick. Asymptotic Bayesian model selection: what may go wrong? (plots)(pdf).
  • Poster@The Workshop on Graphical Models and Genetic Applications. (15. – 17. April 2009) at Warwick.
  • Talk@RSC 2009. (23. – 26. March 2009) in Lancaster. Asymptotic model selection under non standard conditions. (pdf).
  • Workshop on Markov-Chain Monte-Carlo. (16. – 20. March 2009) at Warwick.
  • Talk@The Workshop on Algebraic Statistics. (14. – 18. December 2008) at MSRI, Berkeley, CA, USA. Semi-algebraic description of conditional independence tree models for binary data (pdf, video).
  • Talk@Seminarium MCMC. (13. October 2008) at MIM, Warsaw University, Warsaw.
  • Poster@The Opening Workshop for 2008-09 Program on Algebraic Methods in Systems Biology and Statistics. (14. – 17. September 2008) at SAMSI, NC, USA. I was a short time visitor at SAMSI until 20. September 2008.
  • Poster@Workshop on Applied Mathematics/Statistics Interface, Warwick (11. June 2008). Semi-algebraic aspects of binary general Markov models on trees.
  • Talk@WAG Workshop on Commutative Algebra, Complexes and Computer Algebra, Warwick (2. June 2008). Statistical models are algebraic varieties (pdf).
  • Talk@YRM Warwick (27. May 2008). Geometric aspects of statistical models for discrete data.
past courses:
  • Algebraic Geometry School in Łukęcin, Poland. (6. – 12. September 2009) – Algebraic torus actions (K Altmann, J Hausen).
  • APTS@Glasgow (1. – 5. September 2008) – two modules: Spatial and Longitudinal Data Analysis (P J Diggle), Nonparametric Smoothing (A Delaigle).
  • APTS@Bristol (7. – 11. July 2008) – two modules: Applied Stochastic Processes (W S Kendall), Computationally-Intensive Statistical Methods (B D Ripley).
  • Oberwolfach Seminar (11. May – 17. May 2008), Algebraic Statistics (M Drton, B Sturmfels, S Sullivant
  • APTS@Oxford (April 2008) – two modules: Statistical Modelling (A C Davison and J J Forster), Statistical Asymptotics (G A Young).
  • APTS@Warwick (December 2007) – two modules: Statistical Inference (D R Cox and D Firth), Statistical Computing (S N Wood).
  • EABCN Training School (10. Sep – 14. Sep 2007) – Forecast Evaluation and Related Topics in Applied Time Series (K West), hosted by Banco de Portugal, Lisbon.
  • (Pre)Doc Course (7. May – 25. July 2007) – A summer course “Integer Points in Polyhedra” (R Thomas, R Hemmecke, M Beck, C Haase).
  • Summer School (17. June – 21 June 2006) – A summer school “Algebraic statistics, tropical geometry and computational biology” (B Sturmfels, S Sullivant)

Algebraic Statistics (video)

December 15, 2008 to December 18, 2008

Serkan Hosten (SFSU), Lior Pachter (UCB), Bernd Sturmfels (UCB)

Sumio Watanabe, Ph.D.

Professor Huzihiro Araki (Mathematical Physics) Professor Araki is famous for his contribution to noncommutative operator theory and quantum field theory.
Hal Tasaki (Theoretical Physics)
Takashi Hara (Mathematical Physics)
David Mumford (Algebraic geometry, Pattern Theory)
Stephan E. Feinberg (Mathematical Statistics)
Bernd Sturmfels (Algebraic Statistics)
Nobuki Takayama (Algebraic Analysis)
Akimichi Takemura (Mathematical Statistics)
Giovanni Pistone (Algebraic Statistics)
Lior Pachter (Mathematical Biology)
Seth Sullivant (Algebraic Statistics)
Russell Steele (Mathematical Statistics)
Mathias Drton (Algebraic Statistics)
Ruriko Yoshida (Algebraic Statistics)
Luis David Garcia Puente (Algebraic Statistics)
Jason Morton (Algebraic Statistics)
Shaowei Lin (Algebraic Statistics)
Piotr Zwiernik (Algebraic Statistics)
Dan Geiger (Computer Science)
Dmitry Rusakov (Computer Science)
Miki Aoyagi (Complex Manifold Theory)
Keisuke Yamazaki (Statistical Learning Theory)
Shinichi Nakajima (Variational Learning Theory)
Kazuho Watanabe (Variational Learning Theory)
Kenji Nagata (Information Physics)