arXiv subjects

Tobias Boege

Publications and source records attributed to Tobias Boege.

25 records · Page 2Linked to original sources

arXiv2022

Marginal Independence Models

We impose rank one constraints on marginalizations of a tensor, given by a simplicial complex. Following work of Kirkup and Sullivant, such marginal independence models can be made toric by a linear change of coordinates. We study their toric ideals, with emphasis on random graph models and independent set polytopes of matroids. We develop the numerical algebra of parameter estimation, using both Euclidean distance and maximum likelihood, and we present a comprehensive database of small models.

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arXiv2021

Gaussoids are two-antecedental approximations of Gaussian conditional independence structures

The gaussoid axioms are conditional independence inference rules which characterize regular Gaussian CI structures over a three-element ground set. It is known that no finite set of inference rules completely describes regular Gaussian CI as the ground set grows. In this article we show that the gaussoid axioms logically imply every inference rule of at most two antecedents which is valid for regular Gaussians over any ground set. The proof is accomplished by exhibiting for each inclusion-minimal gaussoid extension of at most two CI statements a regular Gaussian realization. Moreover we prove that all those gaussoids have rational positive-definite realizations inside every $\varepsilon$-ball around the identity matrix. For the proof we introduce the concept of algebraic Gaussians over arbitrary fields and of positive Gaussians over ordered fields and obtain the same two-antecedental completeness of the gaussoid axioms for algebraic and positive Gaussians over all fields of characteristic zero as a byproduct.

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arXiv2021

Reciprocal Maximum Likelihood Degrees of Brownian Motion Tree Models

We give an explicit formula for the reciprocal maximum likelihood degree of Brownian motion tree models. To achieve this, we connect them to certain toric (or log-linear) models, and express the Brownian motion tree model of an arbitrary tree as a toric fiber product of star tree models.

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arXiv2021

Incidence geometry in the projective plane via almost-principal minors of symmetric matrices

We present an encoding of a polynomial system into vanishing and non-vanishing constraints on almost-principal minors of a symmetric, principally regular matrix, such that the solvability of the system over some field is equivalent to the satisfiability of the constraints over that field. This implies two complexity results about Gaussian conditional independence structures. First, all real algebraic numbers are necessary to construct inhabitants of non-empty Gaussian statistical models defined by conditional independence and dependence constraints. This gives a negative answer to a question of Petr Šimeček. Second, we prove that the implication problem for Gaussian CI is polynomial-time equivalent to the existential theory of the reals.

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arXiv2019

Construction Methods for Gaussoids

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.

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arXiv2018

The Geometry of Gaussoids

A gaussoid is a combinatorial structure that encodes independence in probability and statistics, just like matroids encode independence in linear algebra. The gaussoid axioms of Lnenicka and Matús are equivalent to compatibility with certain quadratic relations among principal and almost-principal minors of a symmetric matrix. We develop the geometric theory of gaussoids, based on the Lagrangian Grassmannian and its symmetries. We introduce oriented gaussoids and valuated gaussoids, thus connecting to real and tropical geometry. We classify small realizable and non-realizable gaussoids. Positive gaussoids are as nice as positroids: they are all realizable via graphical models.

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arXiv2016

On permutations with decidable cycles

Recursive permutations whose cycles are the classes of a decidable equivalence relation are studied; the set of these permutations is called $\mathrm{Perm}$, the group of all recursive permutations $\mathcal{G}$. Multiple equivalent computable representations of decidable equivalence relations are provided. $\mathcal{G}$-conjugacy in $\mathrm{Perm}$ is characterised by computable isomorphy of cycle equivalence relations. This result parallels the equivalence of cycle type equality and conjugacy in the full symmetric group of the natural numbers. Conditions are presented for a permutation $f \in \mathcal{G}$ to be in $\mathrm{Perm}$ and for a decidable equivalence relation to appear as the cycle relation of a member of $\mathcal{G}$. In particular, two normal forms for the cycle structure of permutations are defined and it is shown that conjugacy to a permutation in the first normal form is equivalent to membership in $\mathrm{Perm}$. $\mathrm{Perm}$ is further characterised as the set of maximal permutations in a family of preordered subsets of automorphism groups of decidable equivalences. Conjugacy to a permutation in the second normal form corresponds to decidable cycles plus decidable cycle finiteness problem. Cycle decidability and cycle finiteness are both shown to have the maximal one-one degree of the Halting Problem. Cycle finiteness is used to prove that conjugacy in $\mathrm{Perm}$ cannot be decided and that it is impossible to compute cycle deciders for products of members of $\mathrm{Perm}$ and finitary permutations. It is also shown that $\mathrm{Perm}$ is not recursively enumerable and that it is not a group.

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