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Marginal problem, statistical estimation, and Möbius formula. (English) Zbl 1138.93059
Summary: A solution to the marginal problem is obtained in a form of parametric exponential (Gibbs-Markov) distribution, where the unknown parameters are obtained by an optimization procedure that agrees with the maximum likelihood (ML) estimate. With respect to a difficult performance of the method we propose also an alternative approach, providing the original basis of marginals can be appropriately extended. Then the (numerically feasible) solution can be obtained either by the maximum pseudo-likelihood (MPL) estimate, or directly by Möbius formula.
##### MSC:
 93E12 Identification in stochastic control theory 62H12 Estimation in multivariate analysis 60G60 Random fields
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##### References:
 [1] Barndorff-Nielsen O. E.: Information and Exponential Families in Statistical Theory. Wiley, New York 1978 · Zbl 0387.62011 [2] Besag J.: Statistical analysis of non-lattice data. The Statistician 24 (1975), 179-195 [3] Csiszár I., Matúš F.: Generalized maximum likelihood estimates for exponential families. Probability Theory and Related Fields · Zbl 1133.62039 [4] Dobrushin R. L.: Prescribing a system of random variables by conditional distributions. Theor. Probab. Appl. 15 (1970), 458-486 · Zbl 0264.60037 [5] Gilks W. R., Richardson, S., (eds.) D. J. Spiegelhalter: Markov Chain Monte Carlo in Practice. Chapman and Hall, London 1996 · Zbl 0832.00018 [6] Janžura M.: Asymptotic results in parameter estimation for Gibbs random fields. Kybernetika 33 (1997), 2, 133-159 · Zbl 0962.62092 [7] Janžura M.: A parametric model for large discrete stochastic systems. Second European Conference on Highly Structured Stochastic Systems, Pavia 1999, pp. 148-150 [8] Janžura M., Boček P.: A method for knowledge integration. Kybernetika 34 (1988), 1, 41-55 · Zbl 1274.65004 [9] Jaynes E. T.: On the rationale of the maximum entropy methods. Proc. IEEE 70 (1982), 939-952 [10] Jiroušek R., Vejnarová J.: Construction of multidimensional model by operators of composition: Current state of art. Soft Computing 7 (2003), 328-335 · Zbl 1088.68799 [11] Lauritzen S. L.: Graphical Models. University Press, Oxford 1006 · Zbl 1055.62126 [12] Perez A.: $$\varepsilon$$-admissible simplifications of the dependence structure of random variables. Kybernetika 13 (1979), 439-449 · Zbl 0382.62003 [13] Perez A., Studený M.: Comparison of two methods for approximation of probability distributions with prescribed marginals. Kybernetika 43 (2007), 5, 591-618 · Zbl 1144.68379 [14] Winkler G.: Image Analysis, Random Fields and Dynamic Monte Carlo Methods. Springer-Verlag, Berlin 1995 · Zbl 0821.68125 [15] Younes L.: Estimation and annealing for Gibbsian fields. Ann. Inst. H. Poincaré 24 (1988), 2, 269-294 · Zbl 0651.62091
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