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Gibbs distributions; maximum entropy; pseudo-likelihood; Möbius formula
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.
[1] Barndorff-Nielsen O. E.: Information and Exponential Families in Statistical Theory. Wiley, New York 1978 MR 0489333 | 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 (to appear) MR 2372970 | 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 MR 1397966 | Zbl 0832.00018
[6] Janžura M.: Asymptotic results in parameter estimation for Gibbs random fields. Kybernetika 33 (1997), 2, 133–159 MR 1454275 | Zbl 0962.62092
[7] Janžura M.: A parametric model for large discrete stochastic systems. In: 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
[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
[11] Lauritzen S. L.: Graphical Models. University Press, Oxford 1006 MR 1419991 | Zbl 1055.62126
[12] Perez A.: $\varepsilon $-admissible simplifications of the dependence structure of random variables. Kybernetika 13 (1979), 439–449 MR 0472224
[13] Perez A., Studený M.: Comparison of two methods for approximation of probability distributions with prescribed marginals. Kybernetika 43 (2007), 5, 591–618 MR 2376326 | Zbl 1144.68379
[14] Winkler G.: Image Analysis, Random Fields and Dynamic Monte Carlo Methods. Springer–Verlag, Berlin 1995 MR 1316400 | Zbl 0821.68125
[15] Younes L.: Estimation and annealing for Gibbsian fields. Ann. Inst. H. Poincaré 24 (1988), 2, 269–294 MR 0953120 | Zbl 0651.62091
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