Electronic Journal of Statistics

Estimation of high-dimensional partially-observed discrete Markov random fields

Yves F. Atchade

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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.

Article information

Electron. J. Statist., Volume 8, Number 2 (2014), 2242-2263.

First available in Project Euclid: 31 October 2014

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Zentralblatt MATH identifier

Primary: 62M40: Random fields; image analysis 62G20: Asymptotic properties

Network estimation high-dimensional inference penalized likelihood inference misspecification pseudo-likelihood Markov random fields


Atchade, Yves F. Estimation of high-dimensional partially-observed discrete Markov random fields. Electron. J. Statist. 8 (2014), no. 2, 2242--2263. doi:10.1214/14-EJS946. https://projecteuclid.org/euclid.ejs/1414761922

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  • [1] Bach, F. (2010). Self-concordant analysis for logistic regression., Electron. J. Statist. 4 384–414.
  • [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.
  • [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.
  • [4] Bickel, P. J. and Levina, E. (2008). Regularized estimation of large covariance matrices., Ann. Statist. 36 199–227.
  • [5] Bickel, P. J., Ritov, Y. and Tsybakov, A. B. (2009). Simultaneous analysis of lasso and Dantzig selector., Ann. Statist. 37 1705–1732.
  • [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.
  • [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.
  • [8] Drton, M. and Perlman, M. D. (2004). Model selection for Gaussian concentration graphs., Biometrika 91 591–602.
  • [9] Georgii, H.-O. (1988)., Gibbs measures and phase transitions, vol. 9 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin.
  • [10] Guo, J., Levina, E., Michailidis, G. and Zhu, J. (2010). Joint structure estimation for categorical markov networks. Tech. rep., Univ. of, Michigan.
  • [11] Höfling, H. and Tibshirani, R. (2009). Estimation of sparse binary pairwise Markov networks using pseudo-likelihoods., J. Mach. Learn. Res. 10 883–906.
  • [12] Lam, C. and Fan, J. (2009). Sparsistency and rates of convergence in large covariance matrix estimation., Ann. Statist. 37 4254–4278.
  • [13] Meinshausen, N. and Buhlmann, P. (2006). High-dimensional graphs with the lasso., Annals of Stat. 34 1436–1462.
  • [14] Meinshausen, N. and Yu, B. (2009). Lasso-type recovery of sparse representations for high-dimensional data., Ann. Statist. 37 246–270.
  • [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.
  • [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.
  • [17] Rothman, A. J., Bickel, P. J., Levina, E. and Zhu, J. (2008). Sparse permutation invariant covariance estimation., Electron. J. Stat. 2 494–515.
  • [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.
  • [19] Yuan, M. and Lin, Y. (2007). Model selection and estimation in the Gaussian graphical model., Biometrika 94 19–35.