Consider a vector (X_{i})_{i ∈ I}
of random variables which follow a multivariate normal
distribution. To such a vector, we can associate the collection
of all — in the context of this vector — true
statements of the form "X_{i} is independent of
X_{j} given the vector X_{K}". These statements
are called conditional independence (CI) statements and
they are abbreviated to "X_{i} ⫫ X_{j} |
X_{K}". There are certain inference rules for CI
statements which hold universally for all gaussian distributions,
e.g. if X_{1} ⫫ X_{2} and
X_{1} ⫫ X_{2} | X_{3} hold and
the distribution is gaussian, then at least one of the statements
X_{1} ⫫ X_{3} and
X_{2} ⫫ X_{3} must also hold.

These conditional independences are encoded in the vanishing
almost-principal minors of the distribution's covariance matrix.
Determining the CI structures of gaussians is therefore equivalent
to the task of finding which subsets of a certain set of polynomials
in the entries of a positive-definite matrix can simultaneously vanish.

Since these sets are known to be fairly complicated, one tries
to approximate them with formal methods. A tractable subset of
gaussian CI inference rules are considered as axioms and
any combinatorial structure on formal triples (ij|K), representing CI
statements, which fulfills the axioms is called a gaussoid.
Gaussoids are in many ways similar to matroids, for example in their
respective notions of duality, minor and realization. The defining
axioms for gaussoids have polynomial counterparts which are analogous
to the quadratic Grassmann-Plücker relations which give rise to
oriented matroids. Similar extensions of gaussoids are possible and
lead, in turn, to natural objects in statistics, e.g. positively
orientable gaussoids correspond to MTP_{2} distributions.

Unlike matroid theory, however, the study of gaussoids has barely
left its child's shoes. The goal of this project is to develop the
theory of gaussoids and strengthen its links to matroid theory,
algebraic statistics, optimization and complexity theory.

Last Modification: 2019-02-06 - Contact Person: Sebastian Sager - Impressum