Conditional probabilities via line arrangements and point configurations

@article{Clarke2020ConditionalPV,
  title={Conditional probabilities via line arrangements and point configurations},
  author={Oliver Clarke and Fatemeh Mohammadi and Harshit J. Motwani},
  journal={arXiv: Commutative Algebra},
  year={2020}
}
We study the connection between probability distributions satisfying certain conditional independence (CI) constraints, and point and line arrangements in incidence geometry. To a family of CI statements, we associate a polynomial ideal whose algebraic invariants are encoded in a hypergraph. The primary decompositions of these ideals give a characterisation of the distributions satisfying the original CI statements. Classically, these ideals are generated by 2-minors of a matrix of variables… 

Figures from this paper

Incidence geometry in the projective plane via almost-principal minors of symmetric matrices
  • T. Boege
  • Mathematics, Computer Science
    ArXiv
  • 2021
TLDR
It is proved that the implication problem for Gaussian CI is polynomial-time equivalent to the existential theory of the reals, and two complexity results about Gaussian conditional independence structures are proved.
Generalized Cohen-Macaulay binomial edge ideals
Let G be a simple graph on n vertices and let JG,m be the generalized binomial edge ideal associated to G in the polynomial ring K[xij, 1 ≤ i ≤ m, 1 ≤ j ≤ n]. We classify the Cohen-Macaulay
Matroid stratifications of hypergraph varieties, their realization spaces, and discrete conditional independence models
We study conditional independence (CI) models in statistical theory, in the case of discrete random variables, from the point of view of algebraic geometry and matroid theory. Any CI model with

References

SHOWING 1-10 OF 54 REFERENCES
Binomial edge ideals and conditional independence statements
TLDR
It follows that all binomial edge ideals are radical ideals, and the results apply for the class of conditional independence ideals where a fixed binary variable is independent of a collection of other variables, given the remaining ones.
Minimal primes of ideals arising from conditional independence statements
Abstract We study ideals whose primary decomposition specifies the relevant structural zeros of certain conditional independence models. The ideals we study generalize the class of ideals considered
Conditional independence ideals with hidden variables
TLDR
This work studies a class of determinantal ideals that are related to conditional independence (CI) statements with hidden variables, and focuses on an example that generalizes the CI ideals of the intersection axiom.
Prime splittings of determinantal ideals
ABSTRACT We consider determinantal ideals, where the generating minors are encoded in a hypergraph. We study when the generating minors form a Gröbner basis. In this case, the ideal is radical, and
On the primary decomposition of some determinantal hyperedge ideal
TLDR
The method is based on the algorithm for primary decomposition by Gianni, Trager and Zacharias, but it was not able to decompose the ideal using the standard form of that algorithm, nor by any other method known to us.
Generalized binomial edge ideals
  • J. Rauh
  • Mathematics, Computer Science
    Adv. Appl. Math.
  • 2013
TLDR
A class of binomial ideals associated to graphs with finite vertex sets that generalize the binomial edge ideals, and they arise in the study of conditional independence ideals, are studied.
On the Matroid Stratification of Grassmann Varieties, Specialization of Coordinates, and a Problem of N. White
Abstract We resolve a problem of N. White [J. Combin. Theory Ser. B29 (1980), 168–175] by constructing representable matroids F and G with F ⩽ G in the weak map order such that no coordinatization of
Information-theoretic inference of common ancestors
TLDR
This work proves an information-theoretic inequality that allows for the inference of common ancestors of observed parts in any DAG representing some unknown larger system and shows that a large amount of dependence in terms of mutual information among the observations implies the existence of a common ancestor that distributes this information.
Geometric equations for matroid varieties
TLDR
The ideals of matroid varieties, the Zariski closures of these strata are studied and the Grassmann-Cayley algebra may be used to derive non-trivial elements of $I_x$ geometrically when the combinatorics of the matroid is sufficiently rich.
On Invariant Theory
Here we develop a technique of computing the invariants of n−ary forms and systems of forms using the discriminants of corresponding multilinear forms buit of their partial derivatives, which should
...
1
2
3
4
5
...