Conference Agenda

Algebraic statistics studies statistical models through the lens of algebra, geometry, and combinatorics. From model selection to inference, this interdisciplinary field has seen applications in a wide range of statistical procedures. This session will focus broadly on new developments in algebraic statistics, both on the theoretical side and the applied side.

This talk brings many areas together: discrete geometry, statistics, algebraic geometry, invariant theory, geometric modeling, symbolic and numerical computations. We study the algebraic relations among moments of uniform probability distributions on polytopes. This is already a non-trivial matter for quadrangles in the plane. In fact, we need to combine invariant theory of the affine group with numerical algebraic geometry to compute first relevant relations. Moreover, the numerator of the generating function of all moments of a fixed polytope is the adjoint of the polytope, which is known from geometric modeling. We prove the conjecture that the adjoint is the unique polynomial of minimal degree which vanishes on the non-faces of a simple polytope. This talk is based on joint work with Kristian Ranestad, Boris Shapiro and Bernd Sturmfels.

The lattice points of a lattice polytope give rise to a family of toric varieties when we allow complex coefficients in the monomial parametrization of the "usual" toric variety associated to the polytope. The maximum likelihood degree (ML degree) of any member of this family is at most the normalized volume of the polytope. The set of coefficient vectors associated to ML degrees smaller than the volume is parametrized by Gelfand-Kapranov-Zelevinsky's principal A-determinant. Not much is known about how the ML degree changes as one moves in the parameter space. We will discuss what we know starting with toric surfaces.

Directed graphical models specify noisy functional relationships among a collection of random variables. In the Gaussian case, each such model corresponds to a semi-algebraic set of positive definite covariance matrices. The set is given via parametrization, and much work has gone into obtaining an implicit description in terms of polynomial (in-)equalities. Implicit descriptions shed light on problems such as parameter identification, model equivalence, and constraint-based statistical inference. For models given by directed acyclic graphs, which represent settings where all relevant variables are observed, there is a complete theory: All conditional independence relations can be found via graphical d-separation and are sufficient for an implicit description. The situation is far more complicated, however, when some of the variables are hidden. We consider models associated to mixed graphs that capture the effects of hidden variables through correlated error terms. The notion of trek separation explains when the covariance matrix in such a model has submatrices of low rank and generalizes d-separation. However, in many cases, such as the infamous Verma graph, the polynomials defining the graphical model are not determinantal, and hence cannot be explained by d-separation or trek-separation. We show that these constraints often correspond to the vanishing of nested determinants and can be graphically explained by a notion of restricted trek separation.