We study maximum likelihood estimation for the statistical model for undirected random graphs, known as the β-model, in which the degree sequences are minimal sufficient statistics. We derive ...necessary and sufficient conditions, based on the polytope of degree sequences, for the existence of the maximum likelihood estimator (MLE) of the model parameters. We characterize in a combinatorial fashion sample points leading to a nonexistent MLE, and nonestimability of the probability parameters under a nonexistent MLE. We formulate conditions that guarantee that the MLE exists with probability tending to one as the number of nodes increases.
Hypergraph encodings of arbitrary toric ideals Petrović, Sonja; Thoma, Apostolos; Vladoiu, Marius
Journal of combinatorial theory. Series A,
August 2019, 2019-08-00, Letnik:
166
Journal Article
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Relying on the combinatorial classification of toric ideals using their bouquet structure, we focus on toric ideals of hypergraphs and study how they relate to general toric ideals. We show that ...hypergraphs exhibit a surprisingly general behavior: the toric ideal associated to any general matrix can be encoded by that of a 0/1 matrix, while preserving the essential combinatorics of the original ideal. We provide two universality results about the unboundedness of degrees of various generating sets: minimal, Graver, universal Gröbner bases, and indispensable binomials. Finally, we provide a polarization-type operation for arbitrary positively graded toric ideals, which preserves all the combinatorial signatures and the homological properties of the original toric ideal.
Markov Bases: A 25 Year Update Almendra-Hernández, Félix; De Loera, Jesús A.; Petrović, Sonja
Journal of the American Statistical Association,
04/2024, Letnik:
119, Številka:
546
Journal Article
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In this article, we evaluate the challenges and best practices associated with the Markov bases approach to sampling from conditional distributions. We provide insights and clarifications after 25 ...years of the publication of the Fundamental theorem for Markov bases by Diaconis and Sturmfels. In addition to a literature review, we prove three new results on the complexity of Markov bases in hierarchical models, relaxations of the fibers in log-linear models, and limitations of partial sets of moves in providing an irreducible Markov chain.
Supplementary materials
for this article are available online.
Due to the health problems caused by the malnutrition of the world’s population, the focus of wheat breeding is turning to the improvement of the nutritional quality of wheat grain. Recently, the ...consumption of wheatgrass has become increasingly popular. The aim of this study was to determine the variability of total Mg, Fe, and Zn concentration, protein content, and phytic acid in wheat grains for a total of 93 genotypes. In addition, the variability of total and in vitro bioavailable concentrations of Mg, Fe, and Zn and protein content in the fresh juice of wheatgrass was investigated for the same 93 genotypes. The results obtained indicated significant variation in the phytate and nutrient compounds among examined wheat genotypes. In the grain, all examined traits significantly varied except Fe; the largest variability was found in phytate concentration (45.6%). In wheatgrass juice, the greatest variability was found for the in vitro bioavailable Zn (38.3%). Within wheat genotypes, outstanding values for some traits were detected, which could be used in breeding programs. The development of mineral-rich wheat genotypes depends on the identification of genetic resources with high levels of essential micronutrients and a better understanding of genotypic and environmental interactions.
Abstract We construct Bayesian and frequentist finite-sample goodness-of-fit tests for three different variants of the stochastic blockmodel for network data. Since all of the stochastic blockmodel ...variants are log-linear in form when block assignments are known, the tests for the latent block model versions combine a block membership estimator with the algebraic statistics machinery for testing goodness-of-fit in log-linear models. We describe Markov bases and marginal polytopes of the variants of the stochastic blockmodel and discuss how both facilitate the development of goodness-of-fit tests and understanding of model behaviour. The general testing methodology developed here extends to any finite mixture of log-linear models on discrete data, and as such is the first application of the algebraic statistics machinery for latent-variable models.
We investigate Rees algebras and special fiber rings obtained by blowing up specialized Ferrers ideals. This class of monomial ideals includes strongly stable monomial ideals generated in degree two ...and edge ideals of prominent classes of graphs. We identify the equations of these blow-up algebras. They generate determinantal ideals associated to subregions of a generic symmetric matrix, which may have holes. Exhibiting Gröbner bases for these ideals and using methods from Gorenstein liaison theory, we show that these determinantal rings are normal Cohen–Macaulay domains that are Koszul, that the initial ideals correspond to vertex decomposable simplicial complexes, and we determine their Hilbert functions and Castelnuovo–Mumford regularities. As a consequence, we find explicit minimal reductions for all Ferrers and many specialized Ferrers ideals, as well as their reduction numbers. These results can be viewed as extensions of the classical Dedekind–Mertens formula for the content of the product of two polynomials.
Exponential random graph models, or ERGMs, are a flexible and general class of models for modeling dependent data. While the early literature has shown them to be powerful in capturing many network ...features of interest, recent work highlights difficulties related to the models’ ill behavior, such as most of the probability mass being concentrated on a very small subset of the parameter space. This behavior limits both the applicability of an ERGM as a model for real data and inference and parameter estimation via the usual Markov chain Monte Carlo algorithms. To address this problem, we propose a new exponential family of models for random graphs that build on the standard ERGM framework. Specifically, we solve the problem of computational intractability and “degenerate” model behavior by an interpretable support restriction. We introduce a new parameter based on the graph-theoretic notion of degeneracy, a measure of sparsity whose value is commonly low in real-world networks. The new model family is supported on the sample space of graphs with bounded degeneracy and is called degeneracy-restricted ERGMs, or DERGMs for short. Since DERGMs generalize ERGMs—the latter is obtained from the former by setting the degeneracy parameter to be maximal—they inherit good theoretical properties, while at the same time place their mass more uniformly over realistic graphs. The support restriction allows the use of new (and fast) Monte Carlo methods for inference, thus making the models scalable and computationally tractable. We study various theoretical properties of DERGMs and illustrate how the support restriction improves the model behavior. We also present a fast Monte Carlo algorithm for parameter estimation that avoids many issues faced by Markov Chain Monte Carlo algorithms used for inference in ERGMs.
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