Related papers: Differential cumulants, hierachical models and mon…
We study a class of determinantal ideals that are related to conditional independence (CI) statements with hidden variables. Such CI statements correspond to determinantal conditions on a matrix whose entries are probabilities of events…
When a monomial ideal has linear quotients with respect to an admissible order of increasing support-degree, we provide two proofs of different flavors to show that it is componentwise support-linear. We also introduce the variable…
We study the Macaulay coefficients induced by the ideal and quotient segments of a degree-$\delta$ monomial in $n$ variables. We give explicit formulas for these coefficients and establish a duality between the two theories. Our main result…
The set of all m-tuples of compatible full conditional distributions on discrete random variables is an algebraic set whose defining ideal is a unimodular toric ideal. We identify the defining polynomials of these ideals with closed walks…
There is developed a differential-algebraic approach to studying the representations of commuting differentiations in functional differential rings under nonlinear differential constraints. An example of the differential ideal with the only…
We define higher order infinitesimal noncommutative probability space and infinitesimal non-crossing cumulant functionals. In this framework, we generalize to higher order the notion of infinitesimal freeness, via a vanishing of mixed…
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…
In this paper, basic properties of monomial difference ideals are studied. We prove the finitely generated property of well-mixed difference ideals generated by monomials. Furthermore, a finite prime decomposition of radical well-mixed…
We obtain a necessary and sufficient condition under which random-coefficient discrete choice models, such as mixed-logit models, are rich enough to approximate any nonparametric random utility models arbitrarily well across choice sets.…
We consider ideals arising in the context of conditional independence models that generalize the class of ideals considered by Fink [7] in a way distinct from the generalizations of Herzog-Hibi-Hreinsdottir-Kahle-Rauh [13] and Ay-Rauh [1].…
Many random combinatorial objects have a component structure whose joint distribution is equal to that of a process of mutually independent random variables, conditioned on the value of a weighted sum of the variables. It is interesting to…
We generalize the differential dimension polynomial from prime differential ideals to characterizable differential ideals. Its computation is algorithmic, its degree and leading coefficient remain differential birational invariants, and it…
While there is substantial need for dependence models in higher dimensions, most existing models quickly become rather restrictive and barely balance parsimony and flexibility. Hierarchical constructions may improve on that by grouping…
Explicit expressions are proven for derivatives of the ratio of a determinant or Pfaffian determinant and a Vandermonde determinant. Such ratios appear for example in general group integrals of Harish-Chandra--Itzykson--Zuber type and in…
For every natural number $m$, the existentially closed models of the theory of fields with $m$ commuting derivations can be given a first-order geometric characterization in several ways. In particular, the theory of these differential…
Let $\mathbf{X}(n) \in \mathbb{R}^d$ be a sequence of random vectors, where $n\in\mathbb{N}$ and $d = d(n)$. Under certain weakly dependence conditions, we prove that the distribution of the maximal component of $\mathbf{X}$ and the…
The difference variational bicomplex, which is the natural setting for systems of difference equations, is constructed and used to examine the geometric and algebraic properties of various systems. Exactness of the bicomplex gives a…
We construct several pairwise-incomparable bounds on the projective dimensions of edge ideals. Our bounds use combinatorial properties of the associated graphs; in particular we draw heavily from the topic of dominating sets. Through…
We study the multifractal analysis of self-similar measures arising from random homogeneous iterated function systems. Under the assumption of the uniform strong separation condition, we see that this analysis parallels that of the…
We deal with the complete-intersection property of maximally differential ideals. Also, we connect the Gorenstein homology of derivations to the Gorenstein property of the base rings. These equipped with some applications.