The Markov Bases Database


Welcome to the Markov Bases Database!

This database tries to bring together all available information about Markov bases. Markov bases are used in algebraic statistics. From a statistical point of view they can be used to estimate the goodness of a fit of empirical data to a statistical model. In algebraic geometry Markov bases are equivalent to generating set of toric ideals. These ideas are explained in the seminal 1998 paper from P. Diaconis and B. Sturmfels[1]. If you want to know how Markov bases are computed please have a look at the Theory section of the 4ti2-homepage.

At the moment the database is far from complete. We invite everybody to submit their Markov bases computations. Feel free to write us us with either of the following:

If you wish to download the whole database to run extensive statistics or your own searches, please also e-mail us to arrange the download.


Click on DATABASE or SEARCH at the top of this page to access the database. At the top of the page the database entries are shown. At the bottom a form can be used to search the database.

A click on any name in the database leads to a page that summarizes the data of any specific Markov base. On this page there is also a list of files that contain further information. Most important, for every Markov basis in the database there is a corresponding file with ending .mar that contains the Markov basis itself.

All Markov bases have been computed by 4ti2. Thus the format of the files is just the 4ti2-format (cf. the Data Structure section of the 4ti2-homepage): Basically every file contains a matrix. The first line contains its width and its height. If the file contains elements of a Markov basis (e.g. a file of type .mar or .mar.rep), then the Markov basis elements are written as row vectors.

For those models created with the genmodel script provided by 4ti2 the corresponding .mod file is also provided. In this case, the .mod file indicates how the probabilities in the .mar file are ordered.

Normality of the corresponding semigroup was checked using Normaliz.


See here for further references.

  1. P. Diaconis, B. Sturmfels. Algebraic algorithms for sampling from conditional distributions. Annals of Statistics, 26 (1998), 363-397.
  2. 4ti2 team. 4ti2 -- A software package for algebraic, geometric and combinatorial problems on linear spaces. Available at www.4ti2.de.


Thomas Kahle - thomas-kahle(at)gmx.de
Johannes Rauh - jarauh(at)gmx.net

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page last update on March 29 2016. ->Contact