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 everexpanding 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
communityrun 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 longterm 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 noncommutative 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.
 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.
 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.
 “Expected value
of the onedimensional earth mover's distance” gives
explicit formulas for the expected value of a distance
between a pairs of onedimensional discrete probability
distributions using algebraic combinatorics, and discusses
applications of it in clustering analysis.
 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.
 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.
 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.
 “Algebraic analysis of rotation data” studies a
wellknown model for rotation data using the tools from
noncommutative algebra and the holonomic gradient descent
method. It also discusses applications to several areas of
science and engineering.
 “Maximum likelihood degree of the twodimensional
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.
 “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.
 “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.
 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.
