×

Estimation of high-dimensional partially-observed discrete Markov random fields. (English) Zbl 1302.62206

Summary: We consider the problem of estimating the parameters of discrete Markov random fields from partially observed data in a high-dimensional setting. Using a \(\ell^{1}\)-penalized pseudo-likelihood approach, we fit a misspecified model obtained by ignoring the missing data problem. We derive an estimation error bound that highlights the effect of the misspecification. We report some simulation results that illustrate the theoretical findings.

MSC:

62M40 Random fields; image analysis
62G20 Asymptotic properties of nonparametric inference
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] Bach, F. (2010). Self-concordant analysis for logistic regression., Electron. J. Statist. 4 384-414. · Zbl 1329.62324 · doi:10.1214/09-EJS521
[2] Banerjee, O., El Ghaoui, L. and d’Aspremont, A. (2008). Model selection through sparse maximum likelihood estimation for multivariate Gaussian or binary data., J. Mach. Learn. Res. 9 485-516. · Zbl 1225.68149
[3] Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems., J. Roy. Statist. Soc. Ser. B 36 192-236. With discussion by D. R. Cox, A. G. Hawkes, P. Clifford, P. Whittle, K. Ord, R. Mead, J. M. Hammersley, and M. S. Bartlett and with a reply by the author. · Zbl 0327.60067
[4] Bickel, P. J. and Levina, E. (2008). Regularized estimation of large covariance matrices., Ann. Statist. 36 199-227. · Zbl 1132.62040 · doi:10.1214/009053607000000758
[5] Bickel, P. J., Ritov, Y. and Tsybakov, A. B. (2009). Simultaneous analysis of lasso and Dantzig selector., Ann. Statist. 37 1705-1732. · Zbl 1173.62022 · doi:10.1214/08-AOS620
[6] Chandrasekaran, V., Parrilo, P. A. and Willsky, A. S. (2012). Latent variable graphical model selection via convex optimization., The Annals of Statistics 40 1935-1967. · Zbl 1288.62085 · doi:10.1214/12-AOS1020
[7] d’Aspremont, A., Banerjee, O. and El Ghaoui, L. (2008). First-order methods for sparse covariance selection., SIAM J. Matrix Anal. Appl. 30 56-66. · Zbl 1156.90423 · doi:10.1137/060670985
[8] Drton, M. and Perlman, M. D. (2004). Model selection for Gaussian concentration graphs., Biometrika 91 591-602. · Zbl 1108.62098 · doi:10.1093/biomet/91.3.591
[9] Georgii, H.-O. (1988)., Gibbs measures and phase transitions , vol. 9 of de Gruyter Studies in Mathematics . Walter de Gruyter & Co., Berlin. · Zbl 0657.60122 · doi:10.1515/9783110850147
[10] Guo, J., Levina, E., Michailidis, G. and Zhu, J. (2010). Joint structure estimation for categorical markov networks. Tech. rep., Univ. of, Michigan. · Zbl 1203.62190
[11] Höfling, H. and Tibshirani, R. (2009). Estimation of sparse binary pairwise Markov networks using pseudo-likelihoods., J. Mach. Learn. Res. 10 883-906. · Zbl 1245.62121
[12] Lam, C. and Fan, J. (2009). Sparsistency and rates of convergence in large covariance matrix estimation., Ann. Statist. 37 4254-4278. · Zbl 1191.62101 · doi:10.1214/09-AOS720
[13] Meinshausen, N. and Buhlmann, P. (2006). High-dimensional graphs with the lasso., Annals of Stat. 34 1436-1462. · Zbl 1113.62082 · doi:10.1214/009053606000000281
[14] Meinshausen, N. and Yu, B. (2009). Lasso-type recovery of sparse representations for high-dimensional data., Ann. Statist. 37 246-270. · Zbl 1155.62050 · doi:10.1214/07-AOS582
[15] Negahban, S. N., Ravikumar, P., Wainwright, M. J. and Yu, B. (2012). A unified framework for high-dimensional analysis of \(m\)-estimators with decomposable regularizers., Statistical Science 27 538-557. · Zbl 1331.62350 · doi:10.1214/12-STS400
[16] Ravikumar, P., Wainwright, M. J. and Lafferty, J. D. (2010). High-dimensional Ising model selection using \(\ell_1\)-regularized logistic regression., Ann. Statist. 38 1287-1319. · Zbl 1189.62115 · doi:10.1214/09-AOS691
[17] Rothman, A. J., Bickel, P. J., Levina, E. and Zhu, J. (2008). Sparse permutation invariant covariance estimation., Electron. J. Stat. 2 494-515. · Zbl 1320.62135 · doi:10.1214/08-EJS176
[18] Xue, L., Zou, H. and Cai, T. (2012). Nonconcave penalized composite conditional likelihood estimation of sparse ising models., The Annals of Statistics 40 1403-1429. · Zbl 1284.62451 · doi:10.1214/12-AOS1017
[19] Yuan, M. and Lin, Y. (2007). Model selection and estimation in the Gaussian graphical model., Biometrika 94 19-35. · Zbl 1142.62408 · doi:10.1093/biomet/asm018
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.