×

On the geometry of discrete exponential families with application to exponential random graph models. (English) Zbl 1326.62071

Summary: There has been an explosion of interest in statistical models for analyzing network data, and considerable interest in the class of exponential random graph (ERG) models, especially in connection with difficulties in computing maximum likelihood estimates. The issues associated with these difficulties relate to the broader structure of discrete exponential families. This paper re-examines the issues in two parts. First we consider the closure of \(k\)-dimensional exponential families of distribution with discrete base measure and polyhedral convex support \(P\). We show that the normal fan of \(P\) is a geometric object that plays a fundamental role in deriving the statistical and geometric properties of the corresponding extended exponential families. We discuss its relevance to maximum likelihood estimation, both from a theoretical and computational standpoint. Second, we apply our results to the analysis of ERG models. By means of a detailed example, we provide some characterization of the properties of ERG models, and, in particular, of certain behaviors of ERG models known as degeneracy.

MSC:

62F99 Parametric inference
62F10 Point estimation
62A09 Graphical methods in statistics
05C80 Random graphs (graph-theoretic aspects)

Software:

statnet
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] Bardorff-Nielsen, O. (1978)., Information and Exponential Families in Statistical Theory, New York: John Wiley & Sons, New York. · Zbl 0387.62011
[2] Bardorff-Nielsen, O. (2006). Koopman-Darmois-Pitman Families, Encycploedia of Statistical Sciences.
[3] Bishop, Y.M.M., Fienberg, S.E., and Holland, P.W. (1975)., Discrete Multivariate Analysis: Theory and Practice . MIT Press, Cambridge, MA, 1975. Reprinted by Springer-Verlag, New York, 2007. · Zbl 0332.62039
[4] Brown, L. (1986)., Fundamentals of Statistical Exponential Families , IMS Lecture Notes-Monograph Series, Vol.9, Hayward, CA. · Zbl 0685.62002
[5] Csiszár, I. and Matúš, F. (2008). Generalized maximum likelihood estimates for exponential families, Probability Theory and Related Fields , 141, 213-246. · Zbl 0830.62031 · doi:10.1093/biomet/82.3.639
[6] Csiszár, I. and Matúš, F. (2005). Closure of exponential families, Annals of Probability , 33 (2), 582-600. · Zbl 1068.60008 · doi:10.1214/009117904000000766
[7] Csiszár, I. and Matúš, F. (2003). Information projection revisited, IEEE Transaction of Information Theory , 49 (6), 1474-1490. · Zbl 1063.94016 · doi:10.1109/TIT.2003.810633
[8] Csiszár, I. and Matúš, F. (2001). Convex cores of measures, Studia Scientiarum Mathematicarum Hungarica , 38, 177-190. · Zbl 0997.28002 · doi:10.1556/SScMath.38.2001.1-4.12
[9] Eriksson, N., Fienberg, S.E., Rinaldo, A. and Sullivant S. (2006). Polyhedral conditions for the nonexistence of the MLE for hierarchical log-linear models, Journal of Symbolic Computation , 41, 222-233. · Zbl 1120.62015 · doi:10.1016/j.jsc.2005.04.003
[10] Fienberg, S.E. and Wasserman, S. (1981). An exponential family of probability distributions for directed graphs: comment, Journal of the American Statistical Association, 76(373), 54-57. · Zbl 0457.62090 · doi:10.2307/2287037
[11] Frank, O. and Strauss, D. (1986). Markov Graphs, Journal of the American Statistical Association , 81, 832-842. · Zbl 0607.05057 · doi:10.2307/2289017
[12] Geiger, A., Meek, C. and Sturmfels, B. (2006). On the toric algebra of graphical models, Annals of Statistics , 34(3), 1463-1492. · Zbl 1104.60007 · doi:10.1214/009053606000000263
[13] Geyer, C. (2008). Likelihood Inference in Exponential Families and Directions of Recession, Technical Report TR 672, Department of Statistics, University of, Minnesota. · Zbl 1326.62070 · doi:10.1214/08-EJS349
[14] Geyer, C.J. and Thompson, E.A. (1992). Constrained Monte Carlo maximum likelihood calculations (with discussion), Journal of the Royal Statistical Society, Series B, 54, 657-699.
[15] Goldenberg, A., Zheng, A.X., Fienberg, S.E., and Airoldi, E.M. (2009). A survey of statistical network models, Foundations and Trends in Machine Learning, · Zbl 1184.68030 · doi:10.1561/2200000005
[16] Goodreau, S. (2007). Advances in exponential random graph, ( p * ) models applied to a large social network, Social Networks, 29, 231-248.
[17] Haberman, S.J. (1981). An exponential family of probability distributions for directed graphs: comment, Journal of the American Statistical Association, 76(373), 60-61. · Zbl 0457.62090 · doi:10.2307/2287037
[18] Handcock, M.S., Hunter, D., Butts, C., Goodreau, S. and Morris, M. (2006). Statnet: An R Package for the Statistical Analysis and Simulation of Social Networks. Manual. University of Washington, http://www.csde., washington.edu/statnet.
[19] Handcock, M.S. (2003). Assessing degeneracy in statistical models for social networks. Working paper 39, Center for Statistics and the Social Sciences, University of, Washington.
[20] Holland, P.W. and Leinhardt, S. (1981). An exponential family of probability distributions for directed graphs (with discussion)., Journal of the American Statistical Association 76 (373), 33-65. · Zbl 0457.62090 · doi:10.2307/2287037
[21] Hunter, D.R., Goodreau, S.M. and Handcock, M.S. (2008). Goodness of fit of social network models, Journal of the American Statistical Association, 103(481), 248-258. · Zbl 1471.62390 · doi:10.1198/016214507000000446
[22] Lee, Y. and Nelder, J.A. (1996). Hierarchical generalized linear models (with discussion), Journal of the Royal Statistical Society, Series B , 58, 619-678. · Zbl 0880.62076
[23] Letac, G. (1992)., Lectures on Natural Exponential Families and Their Variance Functions , Monografias de Matemática 50. Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil. · Zbl 0983.62501
[24] Moreno, J.L. (1934)., Who Shall Survive? Nervous and Mental Disease Publishing Company, Washington, DC.
[25] Rinaldo, A. (2006a). On maximum likelihood estimation in log-linear models, Technical Report 833, Department of Statistics, Carnegie Mellon, University.
[26] Rinaldo, A. (2006b). Computing Maximum Likelihood Estimates in Log-Linear Models, Technical Report 835, Department of Statistics, Carnegie Mellon, University.
[27] Robins, G., Pattison, P., Kalish, Y. and Lusher, D. (2007). An introduction to exponential random graph, ( p * ) models for social networks, Social Networks, 29, 171-191.
[28] Robins, G, Snijers, T., Wang, P., Handcock, M. and Pattison, P. (2007). Recent developments in exponential random graph, ( p * ) models for social networks, Social Networks, 29, 192-215.
[29] Rockafellar, R.T. (1970)., Convex Analysis , Princeton University Press, Princeton. · Zbl 0193.18401
[30] Schrijver, A. (1998)., Theory of Linear and Integer Programming , Wiley & Sons, New York. · Zbl 0970.90052
[31] Snijders, T.A.B. (2002). Markov Chain Monte Carlo estimation of exponential random graph models, Journal of Social Structure, 3.
[32] Strauss, D. and M. Ikeda (1990). Pseudolikelihood estimation for social networks, Journal of the American Statistical Association, 85, 204-212. · doi:10.1080/01621459.1990.10475327
[33] Sturmfels, B. (1995)., Gröbner Bases and Convex Polytopes , American Mathematical Society, Providence, RI. · Zbl 0856.13020
[34] Wasserman, S. and Pattison, P.E. (1996). Logit models and logistic regressions for social networks: I. An introduction to Markov graphs and, p * , Psychometrika 61, 401-425. · Zbl 0866.92029 · doi:10.1007/BF02294547
[35] Wasserman, S.S. and Robins, G.L. (2004). An Introduction to Random Graphs, Dependence Graphs, and, p * , in Models and Methods in Social Network Analysis , eds. P. Carrington, J.S. and Wasserman, S.S., Cambridge: Cambridge University Press, New York.
[36] Ziegler, G.M. (2001). Lectures on 0/1 polytopes, in Polytopes-Combinatorics and Computations, DMS Seminar, Band 29, G. Kalai and G.M. Ziegler editors, Birkhäuser. · Zbl 0966.52012 · doi:10.1007/978-3-0348-8438-9_1
[37] Ziegler, G.M. (1995)., Lectures on Polytopes , Springer-Verlag, New York. · Zbl 0823.52002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.