Algebraic statistics is a branch of mathematical statistics that focuses on the use of algebraic, geometric, and combinatorial methods in statistics. While the use of these methods has a long history in statistics, algebraic statistics is continuously forging new interdisciplinary connections.

This growing field has established itself squarely at the intersection of several areas of mathematics, including, for instance, multilinear algebra, commutative algebra, algebraic geometry, convex geometry, combinatorics, theoretical problems in statistics, and their practical applications. For example, algebraic statistics has been useful for experimental design, parameter estimation, and hypothesis testing.

History

Algebraic statistics can be traced back to Karl Pearson, who used polynomial algebra to study Gaussian mixture models. Subsequently, Ronald A. Fisher, Henry B. Mann, and Rosemary A. Bailey applied Abelian groups to the design of experiments. Experimental designs were also studied with affine geometry over finite fields and then with the introduction of association schemes by R. C. Bose. Orthogonal arrays were introduced by C. R. Rao also for experimental designs.

The field experienced a major revitalization in the 1990s. In 1998, Diaconis and Sturmfels introduced Gröbner bases for constructing Markov chain Monte Carlo algorithms for conditional sampling from discrete exponential families. Pistone and Wynn, in 1996, applied computational commutative algebra to the design and analysis of experiments, providing new tools for understanding confounding and identifiability in complex experimental settings. These works, along with the monograph by Giovanni Pistone, Eva Riccomagno, and Henry P. Wynn, in which the term “algebraic statistics” was first used, played a pivotal role in establishing this field as a unified area of research.

Modern researchers in algebraic statistics explore a wide range of topics, including computational biology, graphical models, and statistical learning.

Active Research Areas

• Phylogenetics

  Maximum likelihood estimation

  Method of moments

  Graphical models

  Tropical statistics

  Statistical learning theory. Algebraic geometry has also recently found applications to statistical learning theory, including a generalization of the Akaike information criterion to singular statistical models.

  Algebraic analysis and abstract statistical inference. Invariant measures on locally compact groups have long been used in statistical theory, particularly in multivariate analysis. Beurling's factorization theorem and much of the work on (abstract) harmonic analysis sought better understanding of the Wold decomposition of stationary stochastic processes, which is important in time series statistics. Encompassing previous results on probability theory on algebraic structures, Ulf Grenander developed a theory of "abstract inference". Grenander's abstract inference and his theory of patterns are useful for spatial statistics and image analysis; these theories rely on lattice theory.

  Partially ordered sets and lattices. Partially ordered vector spacess and vector lattices are used throughout statistical theory. Garrett Birkhoff metrized the positive cone using Hilbert's projective metric and proved Jentsch's theorem using the contraction mapping theorem. Birkhoff's results have been used for maximum entropy estimation (which can be viewed as linear programming in infinite dimensions) by Jonathan Borwein and colleagues. Vector latticess and conical measuress were introduced into statistical decision theory by Lucien Le Cam.

Introductory Example

Consider a random variable. Such a variable is completely characterized by the three probabilities

and these numbers satisfy

.

is a binomial random variable. Then

which arise in this way are precisely the ones satisfying

The latter is a polynomial equation, and this variety, when intersected with the simplex given by

yields a piece of an algebraic curve is Bernoulli amounts to testing whether a certain point lies on that curve or not.

References

• R. A. Bailey. , Cambridge University Press, Cambridge, 2004. 387pp. . (Chapters from preliminary draft are available on-line)

• H. B. Mann. 1949. Analysis and Design of Experiments: Analysis of Variance and Analysis-of-Variance Designs. Dover.

• L. Pachter and B. Sturmfels. Algebraic Statistics for Computational Biology. Cambridge University Press 2005.

• G. Pistone, E. Riccomango, H. P. Wynn. Algebraic Statistics. CRC Press, 2001.

• Drton, Mathias, Sturmfels, Bernd, Sullivant, Seth. Lectures on Algebraic Statistics, Springer 2009.

• Watanabe, Sumio. Algebraic Geometry and Statistical Learning Theory, Cambridge University Press 2009.

• Paolo Gibilisco, Eva Riccomagno, Maria-Piera Rogantin, Henry P. Wynn. Algebraic and Geometric Methods in Statistics, Cambridge 2009.

External links

• Algebraic Statistics Community Page

• Algebraic Statistics Journal

• Archives of Journal of Algebraic Statistics