Electronic Journal of Statistics

Smoothness of marginal log-linear parameterizations

Robin J. Evans

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We provide results demonstrating the smoothness of some marginal log-linear parameterizations for distributions on multi-way contingency tables. First we give an analytical relationship between log-linear parameters defined within different margins, and use this to prove that some parameterizations are equivalent to ones already known to be smooth. Second we construct an iterative method for recovering joint probability distributions from marginal log-linear pieces, and prove its correctness in particular cases. Finally we use Markov chain theory to prove that certain cyclic conditional parameterizations are also smooth. These results are applied to show that certain conditional independence models are curved exponential families.

Article information

Electron. J. Statist., Volume 9, Number 1 (2015), 475-491.

First available in Project Euclid: 24 March 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H17: Contingency tables
Secondary: 62H20: Measures of association (correlation, canonical correlation, etc.)

Conditional independence contingency table curved exponential family log-linear parameter marginal parameterization


Evans, Robin J. Smoothness of marginal log-linear parameterizations. Electron. J. Statist. 9 (2015), no. 1, 475--491. doi:10.1214/15-EJS1009. https://projecteuclid.org/euclid.ejs/1427203126

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  • [1] Bergsma, W., Croon, M. A., and Hagenaars, J. A., Marginal Models: For Dependent, Clustered, and Longitudinal Categorical Data. Springer Science & Business Media, 2009.
  • [2] Bergsma, W. P. and Rudas, T., Marginal models for categorical data., Ann. Stat., 30(1):140–159, 2002.
  • [3] Colombi, R. and Forcina, A., A class of smooth models satisfying marginal and context specific conditional independencies., Journal of Multivariate Analysis, 126:75–85, 2014.
  • [4] Drton, M., Discrete chain graph models., Bernoulli, 15(3):736–753, 2009.
  • [5] Evans, R. J. and Richardson, T. S., Marginal log-linear parameterizations for graphical Markov models., Journal of Royal Statistical Society, Series B, 75:743–768, 2013.
  • [6] Forcina, A., Smoothness of conditional independence models for discrete data., Journal of Multivariate Analysis, 106:49–56, 2012.
  • [7] Forcina, A., Lupparelli, M., and Marchetti, G. M., Marginal parameterizations of discrete models defined by a set of conditional independencies., Journal of Multivariate Analysis, 101 :2519–2527, 2010.
  • [8] Lang, J. B. and Agresti, A., Simultaneously modeling joint and marginal distributions of multivariate categorical responses., Journal of the American Statistical Association, 89(426):625–632, 1994.
  • [9] Lauritzen, S. L., Graphical Models. Clarendon Press, Oxford, UK, 1996.
  • [10] Norris, J. R., Markov Chains. Cambridge University Press, 1997.
  • [11] Rudas, T., Bergsma, W. P., and Németh, R., Marginal log-linear parameterization of conditional independence models., Biometrika, 94 :1006–1012, 2010.