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21
Stratified exponential families: Graphical models and model selection
 Annals of Statistics
, 2001
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Cited by 54 (6 self)
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JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.
Inference in Curved Exponential Family Models for Networks
 Journal of Computational and Graphical Statistics
, 2006
"... Network data arise in a wide variety of applications. Although descriptive statistics for networks abound in the literature, the science of fitting statistical models to complex network data is still in its infancy. The models considered in this article are based on exponential families; therefore, ..."
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Cited by 42 (9 self)
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Network data arise in a wide variety of applications. Although descriptive statistics for networks abound in the literature, the science of fitting statistical models to complex network data is still in its infancy. The models considered in this article are based on exponential families; therefore, we refer to them as exponential random graph models (ERGMs). Although ERGMs are easy to postulate, maximum likelihood estimation of parameters in these models is very difficult. In this article, we first review the method of maximum likelihood estimation using Markov chain Monte Carlo in the context of fitting linear ERGMs. We then extend this methodology to the situation where the model comes from a curved exponential family. The curved exponential family methodology is applied to new specifications of ERGMs, proposed by Snijders et al. (2004), having nonlinear parameters to represent structural properties of networks such as transitivity and heterogeneity of degrees. We review the difficult topic of implementing likelihood ratio tests for these models, then apply all these modelfitting and testing techniques to the estimation of linear and nonlinear parameters for a collaboration network between partners in a New England law firm.
Curved exponential family models for social networks
 Social Networks
, 2007
"... Curved exponential family models are a useful generalization of exponential random graph models (ERGMs). In particular, models involving the alternating kstar, alternating ktriangle, and alternating ktwopath statistics of Snijders et al. [Snijders, T.A.B., Pattison, P.E., Robins, G.L., Handcock, M ..."
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Cited by 21 (1 self)
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Curved exponential family models are a useful generalization of exponential random graph models (ERGMs). In particular, models involving the alternating kstar, alternating ktriangle, and alternating ktwopath statistics of Snijders et al. [Snijders, T.A.B., Pattison, P.E., Robins, G.L., Handcock, M.S., in press. New specifications for exponential random graph models. Sociological Methodology] may be viewed as curved exponential family models. This article unifies recent material in the literature regarding curved exponential family models for networks in general and models involving these alternating statistics in particular. It also discusses the intuition behind rewriting the three alternating statistics in terms of the degree distribution and the recently introduced shared partner distributions. This intuition suggests a redefinition of the alternating kstar statistic. Finally, this article demonstrates the use of the statnet package in R for fitting models of this sort, comparing new results on an oftstudied network dataset with results found in the literature.
Graphical models and exponential families
 In Proceedings of the 14th Annual Conference on Uncertainty in Arti cial Intelligence (UAI98
, 1998
"... We provide a classification of graphical models according to their representation as subfamilies of exponential families. Undirected graphical models with no hidden variables are linear exponential families (LEFs), directed acyclic graphical models and chain graphs with no hidden variables, includin ..."
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Cited by 19 (1 self)
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We provide a classification of graphical models according to their representation as subfamilies of exponential families. Undirected graphical models with no hidden variables are linear exponential families (LEFs), directed acyclic graphical models and chain graphs with no hidden variables, including Bayesian networks with several families of local distributions, are curved exponential families (CEFs) and graphical models with hidden variables are stratified exponential families (SEFs). An SEF is a finite union of CEFs satisfying a frontier condition. In addition, we illustrate how one can automatically generate independence and nonindependence constraints on the distributions over the observable variables implied by a Bayesian network with hidden variables. The relevance of these results for model selection is examined. 1
ergm: A Package to Fit, Simulate and Diagnose ExponentialFamily Models for Networks
 Journal of Statistical Software
, 2008
"... We describe some of the capabilities of the ergm package and the statistical theory underlying it. This package contains tools for accomplishing three important, and interrelated, tasks involving exponentialfamily random graph models (ERGMs): estimation, simulation, and goodness of fit. More precis ..."
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Cited by 19 (5 self)
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We describe some of the capabilities of the ergm package and the statistical theory underlying it. This package contains tools for accomplishing three important, and interrelated, tasks involving exponentialfamily random graph models (ERGMs): estimation, simulation, and goodness of fit. More precisely, ergm has the capability of approximating a maximum likelihood estimator for an ERGM given a network data set; simulating new network data sets from a fitted ERGM using Markov chain Monte Carlo; and assessing how well a fitted ERGM does at capturing characteristics of a particular network data set.
A New Look at the Entropy for Solving Linear Inverse Problems
 IEEE Transactions on Information Theory
, 1994
"... Entropybased methods are widely used for solving inverse problems, especially when the solution is known to be positive. We address here the linear illposed and noisy inverse problems y = Ax + n with a more general convex constraint x 2 C, where C is a convex set. Although projective methods ar ..."
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Cited by 14 (4 self)
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Entropybased methods are widely used for solving inverse problems, especially when the solution is known to be positive. We address here the linear illposed and noisy inverse problems y = Ax + n with a more general convex constraint x 2 C, where C is a convex set. Although projective methods are well adapted to this context, we study here alternative methods which rely highly on some "informationbased" criteria. Our goal is to enlight the role played by entropy in this frame, and to present a new and deeper point of view on the entropy, using general tools and results of convex analysis and large deviations theory. Then, we present a new and large scheme of entropicbased inversion of linearnoisy inverse problems. This scheme was introduced by Navaza in 1985 [48] in connection with a physical modeling for crystallographic applications, and further studied by DacunhaCastelle and Gamboa [13]. Important features of this paper are (i) a unified presentation of many well kno...
Likelihood Asymptotics
, 1998
"... The paper gives an overview of modern likelihood asymptotics with emphasis on results and applicability. Only parametric inference in wellbehaved models is considered and the theory discussed leads to highly accurate asymptotic tests for general smooth hypotheses. The tests are refinements of the u ..."
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Cited by 12 (0 self)
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The paper gives an overview of modern likelihood asymptotics with emphasis on results and applicability. Only parametric inference in wellbehaved models is considered and the theory discussed leads to highly accurate asymptotic tests for general smooth hypotheses. The tests are refinements of the usual asymptotic likelihood ratio tests, and for onedimensional hypotheses the test statistic is known as r , introduced by BarndorffNielsen. Examples illustrate the applicability and accuracy as well as the complexity of the required computations. Modern likelihood asymptotics has developed by merging two lines of research: asymptotic ancillarity is the basis of the statistical development, and saddlepoint approximations or Laplacetype approximations have simultaneously developed as the technical foundation. The main results and techniques of these two lines will be reviewed, and a generalization to multidimensional tests is developed. In the final part of the paper further problems and ...
On Multivariate Monotonic Measures Of Location With High Breakdown Point
, 1999
"... this article is to introduce a new scheme for robust multivariate ranking by making use of a not so familiar notion called monotonicity. Under this scheme, as in the case of classical outward ranking, we get an increasing sequence of regions diverging away from a central region (may be a single poin ..."
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Cited by 3 (0 self)
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this article is to introduce a new scheme for robust multivariate ranking by making use of a not so familiar notion called monotonicity. Under this scheme, as in the case of classical outward ranking, we get an increasing sequence of regions diverging away from a central region (may be a single point) as nucleus. The nuclear region may be defined as the median region. 1 Introduction
New probabilistic inference algorithms that harness the strengths of variational and Monte Carlo methods
, 2009
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ON THE COMPUTATIONAL COMPLEXITY OF MCMCBASED ESTIMATORS IN LARGE SAMPLES
"... In this paper we examine the implications of the statistical large sample theory for the computational complexity of Bayesian and quasiBayesian estimation carried out using Metropolis random walks. Our analysis is motivated by the LaplaceBernsteinVon Mises central limit theorem, which states that ..."
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Cited by 2 (1 self)
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In this paper we examine the implications of the statistical large sample theory for the computational complexity of Bayesian and quasiBayesian estimation carried out using Metropolis random walks. Our analysis is motivated by the LaplaceBernsteinVon Mises central limit theorem, which states that in large samples the posterior or quasiposterior approaches a normal density. Using this observation, we establish polynomial bounds on the computational complexity of general Metropolis random walks methods in large samples. Our analysis covers cases, where the underlying loglikelihood or extremum criterion function is possibly nonconcave, discontinuous, and of increasing dimension. However, the central limit theorem restricts the deviations from continuity and logconcavity of the loglikelihood or extremum criterion function in a very specific manner. Under minimal assumptions for the central limit theorem framework to hold, we show that the Metropolis algorithm is theoretically