Related papers: Generalized Permutohedra from Probabilistic Graphi…
A mixed graph is a graph with some directed edges and some undirected edges. We introduce the notion of mixed matroids as a generalization of mixed graphs. A mixed matroid can be viewed as an oriented matroid in which the signs over a fixed…
Graphical models provide a framework for exploration of multivariate dependence patterns. The connection between graph and statistical model is made by identifying the vertices of the graph with the observed variables and translating the…
Motivated by modern data forms such as images and multi-view data, the multi-attribute graphical model aims to explore the conditional independence structure among vectors. Under the Gaussian assumption, the conditional independence between…
In this paper we explore mathematical tools that can be used to relate directed and undirected random graph models to each other. We identify probability spaces on which a directed and an undirected graph model are equivalent, and…
Let $G$ be an acylic directed graph. For each vertex $g \in G$, we define an involution on the independent sets of $G$. We call these involutions flips, and use them to define a new partial order on independent sets of $G$. Trim lattices…
In recent years, there has been considerable interest in estimating conditional independence graphs in the high-dimensional setting. Most prior work has assumed that the variables are multivariate Gaussian, or that the conditional means of…
We present a new family of models that is based on graphs that may have undirected, directed and bidirected edges. We name these new models marginal AMP (MAMP) chain graphs because each of them is Markov equivalent to some AMP chain graph…
The method of imsets, introduced by Studen\'y, provides a geometric and combinatorial description of conditional independence statements. Elementary conditional independence statements over a finite set of discrete random variables…
Matroids and semigraphoids are discrete structures abstracting and generalizing linear independence among vectors and conditional independence among random variables, respectively. Despite the different nature of conditional independence…
We generalize some homotopy calculation techniques such as splittings and matching trees that are introduced for the computations in the case of the independence complexes of graphs to arbitrary simplicial complexes, and exemplify their…
The Macaulay2 package GraphicalModels contains algorithms for the algebraic study of graphical models associated to undirected, directed and mixed graphs, and associated collections of conditional independence statements. Among the…
In statistical physics, the multivariate hard-core model describes a system of particles, each of which receives its own fugacity. In graph-theoretic language, the partition function of the model translates to the multivariate independence…
Compositional graphoids are fundamental discrete structures which appear in probabilistic reasoning, particularly in the area of graphical models. They are semigraphoids which satisfy the Intersection and Composition properties. These…
The conditional independence (CI) relation of a distribution in a max-linear Bayesian network depends on its weight matrix through the $C^\ast$-separation criterion. These CI~models, which we call maxoids, are compositional graphoids which…
Given data sampled from a number of variables, one is often interested in the underlying causal relationships in the form of a directed acyclic graph. In the general case, without interventions on some of the variables it is only possible…
Probabilistic Graphical Models (PGMs) encode conditional dependencies among random variables using a graph -nodes for variables, links for dependencies- and factorize the joint distribution into lower-dimensional components. This makes PGMs…
In multivariate statistics, acyclic mixed graphs with directed and bidirected edges are widely used for compact representation of dependence structures that can arise in the presence of hidden (i.e., latent or unobserved) variables. Indeed,…
The graph polynomial for the number of independent sets of size $k$ in a general undirected graph is shown to be equal to an elementary symmetric polynomial of the vertex monomials, which are determined by the edges incident at the…
This chapter of the forthcoming Handbook of Graphical Models contains an overview of basic theorems and techniques from algebraic geometry and how they can be applied to the study of conditional independence and graphical models. It also…
The imsets of Studen\'y (2005) are an algebraic method for representing conditional independence models. They have many attractive properties when applied to such models, and they are particularly nice for working with directed acyclic…