TY - JOUR
AB - The concepts of faithfulness and strong-faithfulness are important for statistical learning of graphical models. Graphs are not sufficient for describing the association structure of a discrete distribution. Hypergraphs representing hierarchical log-linear models are considered instead, and the concept of parametric (strong-) faithfulness with respect to a hypergraph is introduced. Strong-faithfulness ensures the existence of uniformly consistent parameter estimators and enables building uniformly consistent procedures for a hypergraph search. The strength of association in a discrete distribution can be quantified with various measures, leading to different concepts of strong-faithfulness. Lower and upper bounds for the proportions of distributions that do not satisfy strong-faithfulness are computed for different parameterizations and measures of association.
AU - Klimova, Anna
AU - Uhler, Caroline
AU - Rudas, Tamás
ID - 2014
IS - 7
JF - Computational Statistics & Data Analysis
TI - Faithfulness and learning hypergraphs from discrete distributions
VL - 87
ER -
TY - JOUR
AB - We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. This constitutes the largest family of enumerative problems whose Galois groups have been largely determined. Using a criterion of Vakil and a special position argument due to Schubert, our result follows from a particular inequality among Kostka numbers of two-rowed tableaux. In most cases, a combinatorial injection proves the inequality. For the remaining cases, we use the Weyl integral formulas to obtain an integral formula for these Kostka numbers. This rewrites the inequality as an integral, which we estimate to establish the inequality.
AU - Brooks, Christopher
AU - Martin Del Campo Sanchez, Abraham
AU - Sottile, Frank
ID - 1579
IS - 6
JF - Transactions of the American Mathematical Society
TI - Galois groups of Schubert problems of lines are at least alternating
VL - 367
ER -
TY - JOUR
AB - We prove that the three-state toric homogeneous Markov chain model has Markov degree two. In algebraic terminology this means, that a certain class of toric ideals is generated by quadratic binomials. This was conjectured by Haws, Martin del Campo, Takemura and Yoshida, who proved that they are generated by degree six binomials.
AU - Noren, Patrik
ID - 1997
IS - May-June
JF - Journal of Symbolic Computation
TI - The three-state toric homogeneous Markov chain model has Markov degree two
VL - 68/Part 2
ER -
TY - JOUR
AB - The paper describes a generalized iterative proportional fitting procedure that can be used for maximum likelihood estimation in a special class of the general log-linear model. The models in this class, called relational, apply to multivariate discrete sample spaces that do not necessarily have a Cartesian product structure and may not contain an overall effect. When applied to the cell probabilities, the models without the overall effect are curved exponential families and the values of the sufficient statistics are reproduced by the MLE only up to a constant of proportionality. The paper shows that Iterative Proportional Fitting, Generalized Iterative Scaling, and Improved Iterative Scaling fail to work for such models. The algorithm proposed here is based on iterated Bregman projections. As a by-product, estimates of the multiplicative parameters are also obtained. An implementation of the algorithm is available as an R-package.
AU - Klimova, Anna
AU - Rudas, Tamás
ID - 2008
IS - 3
JF - Scandinavian Journal of Statistics
TI - Iterative scaling in curved exponential families
VL - 42
ER -
TY - JOUR
AB - An asymptotic theory is developed for computing volumes of regions in the parameter space of a directed Gaussian graphical model that are obtained by bounding partial correlations. We study these volumes using the method of real log canonical thresholds from algebraic geometry. Our analysis involves the computation of the singular loci of correlation hypersurfaces. Statistical applications include the strong-faithfulness assumption for the PC algorithm and the quantification of confounder bias in causal inference. A detailed analysis is presented for trees, bow ties, tripartite graphs, and complete graphs.
AU - Lin, Shaowei
AU - Uhler, Caroline
AU - Sturmfels, Bernd
AU - Bühlmann, Peter
ID - 2013
IS - 5
JF - Foundations of Computational Mathematics
TI - Hypersurfaces and their singularities in partial correlation testing
VL - 14
ER -