We provide a classification of graphical models according to their representation as exponential families. Undirected graphical models with no hidden variables are linear exponential families (LEFs), directed acyclic graphical (DAG) models and chain graphs with no hidden variables, including DAG models with several families of local distributions, are curved exponential families (CEFs) and graphical models with hidden variables are stratified exponential families (SEFs). A SEF is a finite union of CEFs of various dimensions satisfying some regularity conditions. The main results of this paper are that graphical models are SEFs and that many graphical models are not CEFs. That is, roughly speaking, graphical models when viewed as exponential families correspond to a set of smooth manifolds of various dimensions and usually not to a single smooth manifold. These results are discussed in the context of model selection. Keywords : Bayesian networks, graphical models, hidden variables, cur...

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 non-linear 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 model-fitting and testing techniques to the estimation of linear and non-linear parameters for a collaboration network between partners in a New England law firm.

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 non-independence 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 Introduction A graphical model is a family of probability distributions. The set of distributions associated with a graphical model are usually define...

Curved exponential family models are a useful generalization of exponential random graph models (ERGMs). In particular, models involving the alternating k-star, alternating k-triangle, 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 k-star statistic. Finally, this article demonstrates the use of the statnet package in R for fitting models of this sort, comparing new results on an oft-studied network dataset with results found in the literature.

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 exponential-family 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.

Entropy-based methods are widely used for solving inverse problems, especially when the solution is known to be positive. We address here the linear ill-posed 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 "information-based" 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 entropic-based inversion of linear-noisy inverse problems. This scheme was introduced by Navaza in 1985 [48] in connection with a physical modeling for crystallographic applications, and further studied by Dacunha-Castelle and Gamboa [13]. Important features of this paper are (i) a unified presentation of many well kno...

The paper gives an overview of modern likelihood asymptotics with emphasis on results and applicability. Only parametric inference in well-behaved 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 one-dimensional hypotheses the test statistic is known as r , introduced by Barndorff-Nielsen. 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 Laplace-type approximations have simultaneously developed as the technical foundation. The main results and techniques of these two lines will be reviewed, and a generalization to multi-dimensional tests is developed. In the final part of the paper further problems and ...

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

Particle positions have been observed and estimated in a series of images. The particles are assumed to perform a Brownian motion, however some of them seem to be fixed. A model is introduced with two kinds of particles, diffusing and fixed. To each particle position estimate we assume an additive normal measurement error. The parameter of the model consists of the diffusion variance, the measurement error variance, and the proportion of diffusing particles. The problem can be considered as an incomplete data problem since we do not know a priori which particles are really diffusing. The complete data is of curved exponential type and the observed data is a mixture of two normal components. The maximum likelihood estimator is computed via the EM algorithm. The estimator is shown to be strongly consistent and asymptotically normal, as the number of particles approaches infinity, under a reasonable restriction on the parameter space.

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