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Convergence of the groups posterior distribution in latent or stochastic block models. (English) Zbl 1329.62285

The paper deals with latent block models and, particularly, with stochastic block models in order to facilitate the clustering analysis. The authors focus on a procedure able to cluster both objects and variables (i.e., to simultaneously cluster rows and columns of a data matrix). They propose a unified framework to establish sufficient conditions for the convergence of the groups posterior distribution to a Dirac mass located at the actual random groups configuration.

MSC:

62H30 Classification and discrimination; cluster analysis (statistical aspects)

Software:

Mixnet
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References:

[1] Airoldi, E., Blei, D., Fienberg, S. and Xing, E. (2008). Mixed-membership stochastic blockmodels. J. Mach. Learn. Res. 9 1981-2014. · Zbl 1225.68143
[2] Allman, E.S., Matias, C. and Rhodes, J.A. (2009). Identifiability of parameters in latent structure models with many observed variables. Ann. Statist. 37 3099-3132. · Zbl 1191.62003
[3] Allman, E.S., Matias, C. and Rhodes, J.A. (2011). Parameter identifiability in a class of random graph mixture models. J. Statist. Plann. Inference 141 1719-1736. · Zbl 1207.62010
[4] Ambroise, C. and Matias, C. (2012). New consistent and asymptotically normal parameter estimates for random-graph mixture models. J. R. Stat. Soc. Ser. B Stat. Methodol. 74 3-35.
[5] Bickel, P. and Chen, A. (2009). A nonparametric view of network models and Newman-Girvan and other modularities. Proc. Natl. Acad. Sci. USA 106 21068-21073. · Zbl 1359.62411
[6] Celisse, A., Daudin, J.-J. and Pierre, L. (2012). Consistency of maximum-likelihood and variational estimators in the stochastic block model. Electron. J. Stat. 6 1847-1899. · Zbl 1295.62028
[7] Channarond, A., Daudin, J.-J. and Robin, S. (2012). Classification and estimation in the Stochastic Blockmodel based on the empirical degrees. Electron. J. Stat. 6 2574-2601. · Zbl 1295.62065
[8] Choi, D.S., Wolfe, P.J. and Airoldi, E.M. (2012). Stochastic blockmodels with a growing number of classes. Biometrika 99 273-284. · Zbl 1318.62207
[9] Daudin, J.-J., Picard, F. and Robin, S. (2008). A mixture model for random graphs. Stat. Comput. 18 173-183.
[10] DeSarbo, W.S., Fong, D.K.H., Liechty, J. and Saxton, M.K. (2004). A hierarchical Bayesian procedure for two-mode cluster analysis. Psychometrika 69 547-572. · Zbl 1306.62403
[11] Flynn, C. and Perry, P. (2013). Consistent biclustering. Technical report, . arXiv:1206.6927
[12] Frank, O. and Harary, F. (1982). Cluster inference by using transitivity indices in empirical graphs. J. Amer. Statist. Assoc. 77 835-840. · Zbl 0505.62043
[13] Gazal, S., Daudin, J.-J. and Robin, S. (2012). Accuracy of variational estimates for random graph mixture models. J. Stat. Comput. Simul. 82 849-862. · Zbl 1431.62232
[14] Govaert, G. and Nadif, M. (2003). Clustering with block mixture models. Pattern Recognition 36 463-473.
[15] Govaert, G. and Nadif, M. (2008). Block clustering with Bernoulli mixture models: Comparison of different approaches. Comput. Statist. Data Anal. 52 3233-3245. · Zbl 1452.62444
[16] Govaert, G. and Nadif, M. (2010). Latent block model for contingency table. Comm. Statist. Theory Methods 39 416-425. · Zbl 1187.62117
[17] Hartigan, J.A. (1972). Direct clustering of a data matrix. J. Amer. Statist. Assoc. 67 123-129.
[18] Holland, P.W., Laskey, K.B. and Leinhardt, S. (1983). Stochastic blockmodels: First steps. Social Networks 5 109-137.
[19] Latouche, P., Birmelé, E. and Ambroise, C. (2011). Overlapping stochastic block models with application to the French political blogosphere. Ann. Appl. Stat. 5 309-336. · Zbl 1220.62083
[20] Latouche, P., Birmelé, E. and Ambroise, C. (2012). Variational Bayesian inference and complexity control for stochastic block models. Stat. Model. 12 93-115.
[21] Mariadassou, M., Robin, S. and Vacher, C. (2010). Uncovering latent structure in valued graphs: A variational approach. Ann. Appl. Stat. 4 715-742. · Zbl 1194.62125
[22] Massart, P. (2007). Concentration Inequalities and Model Selection. Lecture Notes in Math. 1896 . Berlin: Springer. Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour, July 6-23, 2003, With a foreword by Jean Picard.
[23] Nowicki, K. and Snijders, T.A.B. (2001). Estimation and prediction for stochastic blockstructures. J. Amer. Statist. Assoc. 96 1077-1087. · Zbl 1072.62542
[24] Picard, F., Miele, V., Daudin, J.-J., Cottret, L. and Robin, S. (2009). Deciphering the connectivity structure of biological networks using MixNet. BMC Bioinformatics 10 1-11.
[25] Rohe, K., Chatterjee, S. and Yu, B. (2011). Spectral clustering and the high-dimensional stochastic blockmodel. Ann. Statist. 39 1878-1915. · Zbl 1227.62042
[26] Rohe, K. and Yu, B. (2012). Co-clustering for directed graphs: the stochastic co-blockmodel and a spectral algorithm. Technical report, . arXiv:1204.2296
[27] Snijders, T.A.B. and Nowicki, K. (1997). Estimation and prediction for stochastic blockmodels for graphs with latent block structure. J. Classification 14 75-100. · Zbl 0896.62063
[28] Wyse, J. and Friel, N. (2012). Block clustering with collapsed latent block models. Stat. Comput. 22 415-428. · Zbl 1322.62046
[29] Zanghi, H., Ambroise, C. and Miele, V. (2008). Fast online graph clustering via Erdős Rényi mixture. Pattern Recognition 41 3592-3599. · Zbl 1151.68623
[30] Zanghi, H., Picard, F., Miele, V. and Ambroise, C. (2010). Strategies for online inference of model-based clustering in large and growing networks. Ann. Appl. Stat. 4 687-714. · Zbl 1194.62096
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