We study the algebraic varieties defined by the conditional independence statements of Bayesian networks. A complete algebraic classification is given for Bayesian networks on at most five random variables. Hidden variables are related to the geometry of higher secant varieties. 1

Given a model in algebraic statistics and data, the likelihood function is a rational function on a projective variety. Algebraic algorithms are presented for computing all critical points of this function, with the aim of identifying the local maxima in the probability simplex. Applications include models specified by rank conditions on matrices and the Jukes-Cantor models of phylogenetics. The maximum likelihood degree of a generic complete intersection is also determined. 1

Abstract: Many statistical models are algebraic in that they are defined in terms of polynomial constraints, or in terms of polynomial or rational parametrizations. The parameter spaces of such models are typically semi-algebraic subsets of the parameter space of a reference model with nice properties, such as for example a regular exponential family. This observation leads to the definition of an ‘algebraic exponential family’. This new definition provides a unified framework for the study of statistical models with algebraic structure. In this paper we review the ingredients to this definition and illustrate in examples how computational algebraic geometry can be used to solve problems arising in statistical inference in algebraic models. Key words and phrases: Algebraic statistics, computational algebraic geometry, exponential family, maximum likelihood estimation, model invariants, singularities. 1.

In graphical modelling, a bi-directed graph encodes marginal independences among random variables that are identified with the vertices of the graph. We show how to transform a bi-directed graph into a maximal ancestral graph that (i) represents the same independence structure as the original bi-directed graph, and (ii) minimizes the number of arrowheads among all ancestral graphs satisfying (i). Here the number of arrowheads of an ancestral graph is the number of directed edges plus twice the number of bi-directed edges. In Gaussian models, this construction can be used for more efficient iterative maximization of the likelihood function and to determine when maximum likelihood estimates are equal to empirical counterparts.

Semi-graphoids are combinatorial structures that arise in statistical learning theory. They are equivalent to convex rank tests and to polyhedral fans that coarsen the reflection arrangement of the symmetric group Sn. In this paper we resolve two problems on semi-graphoids posed in Studen´y’s book [19], and we answer a related question of Postnikov, Reiner, and Williams on generalized permutohedra [18]. We also study the semigroup and the toric ideal associated with semi-graphoids. 1

Conjunctive Bayesian networks (CBNs) are graphical models that describe the accumulation of events which are constrained in the order of their occurrence. A CBN is given by a partial order on a (finite) set of events. CBNs generalize the oncogenetic tree models of Desper et al. (1999) by allowing the occurrence of an event to depend on more than one predecessor event. The present paper studies the statistical and algebraic properties of CBNs. We determine the maximum likelihood parameters and present a combinatorial solution to the model selection problem. Our method performs well on two datasets where the events are HIV mutations associated with drug resistance. Concluding with a study of the algebraic properties of CBNs, we show that CBNs are toric varieties after a coordinate transformation and that their ideals possess a quadratic Gröbner basis.

Abstract: For discrete distributions in R d on a finite support D probabilities and moments are algebraically related. Intuitively, if there are n = |D | support points then there are n probabilities p(x), x ∈ D and n basic moments. By suitable interpolation of the probabilities using a Gröbner method, high order moments can be express linearly in terms of n basic moments, relative to the Gröbner basis. A main result is that high order cumulates can also be expressed as polynomial functions of n low order moments and cumulates. This means that statistical models which can be expressed via an algebraically variety for the basic probabilities and moments, such as graphical models, therefore induce a variety for the basic cumulants, which we shall call the cumulant variety. In is important to stress that the cumulant variety depends on the monomial ordering defining the original Gröbner basis.

We present algebraic methods for studying connectivity of Markov moves with margin positivity. The purpose is to develop Markov sampling methods for exact conditional inference in statistical models where a Markov basis is hard to compute. In some cases positive margins are shown to allow a set of Markov connecting moves that are much simpler than the full Markov basis. Advances in algebra have impacted in a fundamental way the study of exponential families of probability distributions. In the 1990s, computational methods of commutative algebra were brought into statistics to solve both classical and new problems in the framework of exponential family models.

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A simple graphical model for correlated defaults is proposed, with explicit formulas for the loss distribution. Algebraic geometry techniques are employed to show that this model is well posed for default dependence: it represents any given marginal distribution for single firms and pairwise correlation matrix. These techniques also provide a calibration algorithm based on maximum likelihood estimation. Finally, the model is compared with standard normal copula model in terms of tails of the loss distribution and implied correlation smile. 1