Vol. 11, No. 1, 2020

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Editorial: A new beginning

Thomas Kahle and Sonja Petrović

Vol. 11 (2020), No. 1, 1–3

Algebraic statistics and Algebraic Statistics

The creation of a field that bridges two disparate areas takes both ingenuity and the ability to generate excitement about new interdisciplinary ideas. For that field to continuously evolve over two decades, expanding to include virtually every aspect of the ground fields, as well as a growing number of neighboring research areas, takes a continued and dedicated community effort. Algebraic Statistics (AStat) is being established as a journal to be run by and devoted to such a community, representing interdisciplinary researchers in the field coming from all backgrounds.

While algebra has always played a prominent role in statistics, the publication of a couple of seminal works in the late 1990s defined the new direction by connecting modern computational algebraic geometry and commutative algebra to two critical problems in statistics: sampling from discrete conditional distributions and experimental design. With the onset of the 2000s, the use of these techniques in statistics really took off, generating a large body of research papers and several textbooks. In the last decade, the field has seen a massive influx of new people bringing new ideas and perspectives to problems at the intersection of nonlinear algebra, interpreted in broadest possible sense, and statistics.

The term “algebraic statistics” has thus evolved in meaning to include an ever-expanding list of topics. We understand it as an umbrella term for using algebra (multilinear algebra, commutative algebra, and computational algebra), geometry and combinatorics to obtain insights in mathematical statistics as well as for diverse applications of these tools to data science.

The community of algebraic statisticians is quite an active one, organizing many conferences, symposia, seminars, and special sessions at regional and international meetings, and striving for involvement and representation within both nonlinear algebra and statistics. The predecessor community-run journal, which existed for a decade and published ten volumes, has now been discontinued due to a dispute in ownership with a third party interested in a profit oriented future for the journal. The core of the algebraic statistics community strongly supports the establishment of this new journal. It is a leap forward, a fresh start that takes into account historical lessons learned and seeks to grow and expand the research scope. For this endeavor, we are happy to team up with MSP as a publishing partner that is committed to support academic scholarship and to ensuring the long-term success of our research community.

The first volume

The first volume, in two issues, contains eleven papers with a mix that represents algebraic statistics well. Mathematical themes include Gröbner bases, both the standard and non-commutative versions, toric and tropical varieties, numerical nonlinear algebra, holonomic gradient descent, and algebraic combinatorics. On the side of statistics, there are models for diverse types of data, parameter estimation under the likelihood principle, covariance estimation, and time series. Applications covered include computational neuroscience, clustering analysis, engineering, material science, and geology.

  1. The paper “Maximum likelihood estimation of toric Fano varieties” showcases likelihood geometry. Its main result explains how properties of likelihood estimation depend on algebraic and geometric features of the underlying toric models.
  2. Linear covariance models are models for Gaussian random variables with linear constraints on the covariance matrix. The paper “Estimating linear covariance models with numerical nonlinear algebra” addresses the problem of maximum likelihood estimation in these models, the related complexity challenges, and introduces an accompanying package.
  3. Expected value of the one-dimensional earth mover's distance” gives explicit formulas for the expected value of a distance between a pairs of one-dimensional discrete probability distributions using algebraic combinatorics, and discusses applications of it in clustering analysis.
  4. In “Inferring properties of probability kernels from the pairs of variables they involve” the authors discuss how inference about inherently continuous and uncountable probability kernels can be encoded in discrete structures such as lattices.
  5. In computational neuroscience, neural codes model patterns of neuronal response to stimuli. The field provides many open problems for mathematics and statistics. “Minimal embedding dimensions of connected neural codes” address a problem from receptive field coding: the embedding of neural codes in low dimension.
  6. The holonomic gradient method in “Holonomic gradient method for two way contingency tables” is a numerical procedure to approximate otherwise inaccessible likelihood integrals. It is here applied in a discrete situation of contingency tables.
  7. “Algebraic analysis of rotation data” studies a well-known model for rotation data using the tools from non-commutative algebra and the holonomic gradient descent method. It also discusses applications to several areas of science and engineering.
  8. “Maximum likelihood degree of the two-dimensional linear Gaussian covariance model” provides explicit formulas for the number of solutions of likelihood equations in special cases of the same problem as in paper 2.
  9. “Tropical gaussians: a brief survey” takes a tour through the analogues of Gaussian distributions over the tropical semiring. This has applications in, for example, economics and phylogenetics.
  10. “The norm and saturation of a binomial ideal, and applications to Markov bases” connects back to the beginnings of algebraic statistics: Markov bases. Here the focus is on the complexity of Markov bases.
  11. Finally, “Compatibility of distributions in probabilistic models: An algebraic frame and some characterizations” studies the problem when and how two distributions for two sets of variables can be put together to a distribution for the union of the variables and exhibits discrete and algebraic structures in this problem.

Call for submissions

We see AStat as a primary forum serving the broad community in a focused way. As an interdisciplinary endeavor, by definition, a concerted effort will be made for AStat to serve various constituents interested in and interacting with algebraic statistics. Specifically, in our definition, AStat is devoted to algebraic aspects of statistical theory, methodology and applications, seeking to publish a wide range of research and review papers that address one of the following:

  • algebraic, geometric and combinatorial insights into statistical models or the behavior of statistical procedures;
  • development of new statistical models and methods with interesting algebraic or geometric properties;
  • novel applications of algebraic and geometric methods in statistics.

We invite the community to send their best work in algebraic statistics to be considered for publication here. This includes contributions which connect statistical theory, methodology, or application to the world of algebra, geometry, and combinatorics in ways that may not be labeled as traditional.

Keywords
algebraic statistics
Mathematical Subject Classification
Primary: 62R01
Milestones
Received: 30 June 2020
Accepted: 3 July 2020
Published: 1 October 2020
Authors
Thomas Kahle
Fakultät für Mathematik
Otto-von-Guericke Universität Magdeburg
Institut Algebra und Geometrie
39106 Magdeburg
Germany
Sonja Petrović
Department of Applied Mathematics
Illinois Institute of Technology
Chicago, IL 60616
United States