Exact Goodness‐of‐Fit Testing for the Ising Model

Campo, Abraham Martín del; Cepeda, Sarah; Uhler, Caroline

doi:10.1111/sjos.12251

Cited by 6 publications

(8 citation statements)

References 32 publications

(51 reference statements)

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“…To our knowledge, no similar results had been obtained so far for Ising models even though there has been recent intensive work on Ising models in statistics (Foygel Barber & Drton, ; Martín del Campo et al., ; Bhattacharya & Mukherjee, ) and in machine learning (Murphy, ; Bresler, ; Johnson et al., ).…”

Section: Introductionmentioning

confidence: 61%

Explicit, identical maximum likelihood estimates for some cyclic Gaussian and cyclic Ising models

Marchetti

Wermuth

2017

Stat

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Cyclic models are a subclass of graphical Markov models with simple, undirected probability graphs that are chordless cycles. In general, all currently known distributions require iterative procedures to obtain maximum likelihood estimates in such cyclic models. For exponential families, the relevant conditional independence constraint for a variable pair is given all remaining variables, and it is captured by vanishing canonical parameters involving this pair. For Gaussian models, the canonical parameter is a concentration, that is, an off-diagonal element in the inverse covariance matrix, while for Ising models, it is a conditional log-linear, two-factor interaction. We give conditions under which the two different likelihood functions, that is, one for continuous and one for binary variables, permit nevertheless explicit maximum likelihood estimates, and we show that their estimated correlation matrices are identical, provided the relevant starting correlation matrices coincide.

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Section: Introductionmentioning

confidence: 61%

Explicit, identical maximum likelihood estimates for some cyclic Gaussian and cyclic Ising models

Marchetti

Wermuth

2017

Stat

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“…For example, [Dob12] proposes a general dynamic approach to constructing applicable local moves for marginal table models and proves that they can be utilized to cover the entire fiber. Meanwhile, [HAT12] use a Poisson-size combination of the smallest possible set of moves to explore fibers of discrete logistic regression models, and [MdCCU17] use another small subset of a Markov basis for testing the Ising model on a large biological dataset. Each of these methods extends the use of Markov bases to larger and larger datasets: the approach in [Dob12] was demonstrated on tables up to 256 cells, [HAT12] show their method approaches its limits on 10 × 10 × 10 tables, and [MdCCU17] are able to use their method on a biological dataset of size 800 × 800.…”

Section: Discussionmentioning

confidence: 99%

“…Meanwhile, [HAT12] use a Poisson-size combination of the smallest possible set of moves to explore fibers of discrete logistic regression models, and [MdCCU17] use another small subset of a Markov basis for testing the Ising model on a large biological dataset. Each of these methods extends the use of Markov bases to larger and larger datasets: the approach in [Dob12] was demonstrated on tables up to 256 cells, [HAT12] show their method approaches its limits on 10 × 10 × 10 tables, and [MdCCU17] are able to use their method on a biological dataset of size 800 × 800. In this paper, we are able to obtain good results for tables up of size 2661 × 2661 × 2 × 2 (networks with 2661 vertices) and show that our method slows down due to a goodness-of-fit statistic computation on tables of size 4334 × 4334 × 2 × 2 (networks with 4334 vertices).…”

Section: Discussionmentioning

confidence: 99%

Goodness of fit for log-linear ERGMs

Gross¹,

Petrović²,

Stasi³

2021

Preprint

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Many popular models from the networks literature can be viewed through a common lens. We describe it here and call the class of models log-linear ERGMs. It includes degree-based models, stochastic blockmodels, and combinations of these. Given the interest in combined node and block effects in network formation mechanisms, we introduce a general directed relative of the degree-corrected stochastic blockmodel: an exponential family model we call p1-SBM. It is a generalization of several well-known variants of the blockmodel.We study the problem of testing model fit for the log-linear ERGM class. The model fitting approach we take, through the use of quick estimation algorithms borrowed from the contingency table literature and effective sampling methods rooted in graph theory and algebraic statistics, results in an exact test whose p-value can be approximated efficiently in networks of moderate sizes.We showcase the performance of the method on two data sets from biology: the connectome of C. elegans and the interactome of Arabidopsis thaliana. These two networks, a neuronal network and a protein-protein interaction network, have been popular examples in the network science literature, but a model-based approach to studying them has been missing thus far.

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“…Recent results, e.g., [56,32], show that for some very popular models of random graphs, Markov bases can be constructed by appropriately composing generators of the edge subring of a bipartite graph. The problem of the computational difficulty of Markov bases has been addressed for various models recently [59,34,32,10] by considering subsets of generators that can be applied to given data. Another direction of interest is how well Markov chains based on various types of bases behave; the interested readers should see, e.g., [55, Section 3.1] for references to the literature on mixing times.…”

Section: Substituting This Into the Previous Equation Givesmentioning

confidence: 99%

Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas

et al. 2016

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We give a combinatorial proof of the Dedekind-Mertens formula by computing the initial ideal of the content ideal of the product of two generic polynomials. As a side effect we obtain a complete classification of the rank 1 Cohen-Macaulay modules over the determinantal rings K[X]/I 2 (X). Let f, g be polynomials in one indeterminate over a commutative ring A. The Dedekind-Mertens formula relates the content ideals of f , g, and their product f g: one has c(f g)c(f) d = c(g)c(f) d+1 , d= deg g. It is the best universally valid variant of Gauß' classical formula c(f g) = c(f)c(g) for polynomials over a principal ideal domain. (The content ideal of f ∈ A[T ] is the ideal generated by the coefficients of f in A.) Content ideals and the Dedekind-Mertens formula have recently received much attention; see Glaz and Vasconcelos [8], Corso, Vasconcelos, and Villarreal [6] and Heinzer and Huneke [9], [10]. For detailed historical information about the Dedekind-Mertens formula, see [9]. The main objective of this paper is a combinatorial proof of the formula based on a Gröbner basis approach to the ideal c(f g) for polynomials with indeterminate coefficients; in fact we will determine the initial ideal of c(f g) with respect to a suitable term order. (For information on term orders and Gröbner bases we refer the reader to Eisenbud [7].) A side effect of our approach is very precise numerical information about the rank one Cohen-Macaulay modules over the determinantal ring S = K[X]/I 2 (X) where X is an m × n matrix of indeterminates and I 2 (X) the ideal generated by its 2-minors. This connection extends the ideas of [6] and was in fact suggested by them. The actual motive for our work was the need for some explicit computation modulo c(f g) in Boffi, Bruns, and Guerrieri [2], or, more precisely, modulo an ideal generalizing c(f g) slightly. Theorem 1. Let K be a field, R = K[

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Exact Goodness‐of‐Fit Testing for the Ising Model

Cited by 6 publications

References 32 publications

Explicit, identical maximum likelihood estimates for some cyclic Gaussian and cyclic Ising models

Explicit, identical maximum likelihood estimates for some cyclic Gaussian and cyclic Ising models

Goodness of fit for log-linear ERGMs

Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas

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