Construction methods for gaussoids

Abstract

The number of n-gaussoids is shown to be a double exponential function in n. The necessary bounds are achieved by studying construction methods for gaussoids that rely on prescribing 3-minors and encoding the resulting combinatorial constraints in a suitable transitive graph. Various special classes of gaussoids arise from restricting the allowed 3-minors.

The geometry of Gaussian double Markovian distributions

“…For example, for Markov relations this is true, as proved in [26,Proposition 2]. A CI structure is a Markov relation if and only if it is an upward-stable (or ascending) gaussoid [5], or, equivalently, a positively orientable gaussoid [4]. Summarizing the mostly well-known facts collected so far, we have:…”

Gaussoids are two-antecedental approximations of Gaussian conditional independence structures

“…Both, isomorphy and duality, are special cases of a larger symmetry group, the hyperoctahedral group B N , which is generated by the reflection symmetries of the N -dimensional cube. The combinatorial motivation for considering this group is explained in [4], but it is not important for the present work. As an abstract group, B N equals the semidirect product (Z/2) N S N , i.e., each of its elements can be uniquely written as a composition of a swap from (Z/2) N and a permutation from S N .…”

“…It is the technical condition which allows the formation of all Schur complements of the matrix, which correspond to conditional distributions in the positive-definite setting. The property is inherited by the inverse matrix, by principal submatrices and Schur complements and therefore is precisely what is required to salvage the theory of minors of regular Gaussians; see [4]. Definition 3.1 A gaussoid G is algebraically realizable (over K) if there is a principally regular matrix Γ over K such that G = Γ .…”

“…Proof B not being a gaussoid requires that (ij |N) and (kl|M) are distinct and form the antecedent set of a gaussoid axiom. Thus the two CI statements lie in a 3-face of the ambient ij klN M-cube; see [4]. We can therefore reduce the study of gaussoid extensions of B to this 3-face and hence, after conditioning, to a 3-element ground set.…”

“…Lemma 6. 4 The gaussoid {(ij |), (ij |M)} on the ground set ij M is rationally realizable near all hyperoctahedral images of the identity.…”