zbMATH — the first resource for mathematics

Tree and local computations in a cross-entropy minimization problem with marginal constraints. (English) Zbl 1204.93113
Summary: In probability theory, Bayesian statistics, artificial intelligence and database theory the minimum cross-entropy principle is often used to estimate a distribution with a given set \(P\) of marginal distributions under the proportionality assumption with respect to a given “prior” distribution \(q\). Such an estimation problem admits a solution if and only if there exists an extension of \(P\) that is dominated by \(q\). In this paper we consider the case that \(q\) is not given explicitly, but is specified as the maximum-entropy extension of an auxiliary set \(Q\) of distributions. There are three problems that naturally arise: (1) the existence of an extension of a distribution set (such as \(P\) and \(Q\)), (2) the existence of an extension of \(P\) that is dominated by the maximum entropy extension of \(Q\), (3) the numeric computation of the minimum cross-entropy extension of \(P\) with respect to the maximum entropy extension of \(Q\). In the spirit of a divide-and-conquer approach, we prove that, for each of the three above-mentioned problems, the global solution can be easily obtained by combining the solutions to subproblems defined at node level of a suitable tree.

93E10 Estimation and detection in stochastic control theory
62E10 Characterization and structure theory of statistical distributions
Full Text: Link EuDML
[1] Asmussen, S., Edwards, D.: Collapsibility and response variables in contingency tables. Biometrika 70 (1983), 367-378. · Zbl 0549.62041
[2] Bacharach, M.: Biproportional Matrices and Input-Output Change. Cambridge University Press, Cambridge 1970. · Zbl 0195.49705
[3] Badsberg, J.-H., Malvestuto, F. M.: An implementation of the iterative proportional fitting procedure by propagation trees. Comput. Statist. Data Analysis 37 (2001), 297-322. · Zbl 1061.65500
[4] Beeri, C., Vardi, M.: On the Properties of Full Join Dependencies. Adv. Database Theory I, Plenum Press, New York 1981. · Zbl 0462.68022
[5] Beeri, C., Fagin, R., Maier, D., Yannakakis, M.: On the desirability of acyclic database schemes. J. Assoc. Comput. Mach. 30 (1983), 479-513. · Zbl 0624.68087
[6] Berge, C.: Hypergraphs. North-Holland, Amsterdam 1989. · Zbl 0674.05001
[7] Berge, C.: Discrete Multivariate Analysis. MIT Press, Cambridge 1975.
[8] Csiszár, I.: I-divergence geometry of probability distributions and minimization problems. Ann. Probab. 3 (1975), 146-158. · Zbl 0318.60013
[9] Csiszár, I.: Maxent, mathematics, and information theory. Proc. Internat. Workshop on “Maximum entropy and Bayesian methods”, 1995, pp. 35-50. · Zbl 0886.62007
[10] Dall’Aglio, G., Kotz, K., Salinetti, G.: Advances in Probability Distributions with Given Marginals. Kluwer Academic Pub., Dordrecht, Boston, London 1991. · Zbl 0722.00031
[11] Darroch, J. N., Ratcliff, D.: Generalized iterative scaling for log-linear models. Ann. Math. Statist. 43 (1972), 1470-1480. · Zbl 0251.62020
[12] Deming, W. E.: Statistical Adjustment of Data. Dover Pub., New York 1943. · Zbl 0060.31504
[13] Deming, W. E., Stephan, F. F.: On a least squares adjustment of a sampled frequency table when the expected marginal totals are known. Ann. Math. Statist. 11 (1940), 427-444. · Zbl 0024.05502
[14] Endo, Y., Takemura, A. I.: Iterative proportional scaling via decomposable submodels for contingency tables. Comput. Statist. Data Analysis 53 (2009), 966-978. · Zbl 1452.62393
[15] Fienberg, S. E.: An iterative procedure for estimation in contingency tables. Ann. Math. Statist. 41 (1970), 907-917. · Zbl 0198.23401
[16] Fienberg, S. E., Meyer, M. M.: Iterative proportional fitting. Encyclopedia of Statistical Sciences (S. Kotz, N. L. Johnson, and C. B. Read, 4, John Wiley and Sons, New York, pp. 275-279.
[17] Haberman, S. J.: Log-linear Models for Contingency Tables. University of Chicago Press, Chicago 1974. · Zbl 0294.62026
[18] Ireland, C. T., Kullback, S.: Contingency tables with given marginals. Biometrika 55 (1968), 179-188. · Zbl 0155.26701
[19] Jiroušek, R.: Composition of probability measures on finite spaces. Proc. XIII Internat. Conf. Uncertainty in Artificial Intelligence 1997, pp. 274-281.
[20] Jiroušek, R., Přeučil, S.: On the effective implementation of the iterative proportional fitting procedure. Comput. Statist. Data Analysis 19 (1995), 177-189. · Zbl 0875.65122
[21] Johnson, R. W.: Axiomatic characterization of the directed divergences and their linear combinations. IEEE Trans. Inform. Theory 25 (1979), 709-716. · Zbl 0422.60016
[22] Kellerer, H. G.: Verteilungsfunktionen mit gegebenen marginalverteilungen. Zeitschrift Wahrscheinlichkeitstheorie und Verw. Gebiete 3 (1964), 247-270. · Zbl 0126.34003
[23] Kellerer, H. G.: Masstheoretische marginalprobleme. Math. Annalen 153 (1964), 168-198. · Zbl 0118.05003
[24] Kern-Isberner, G.: Characterizing the principle of minimum-cross entropy within a conditional-logical framework. Artificial Intelligence 98 (1998), 169-208. · Zbl 0903.68181
[25] Ku, H. H., Kullback, S.: Interaction in multidimensional contingency tables: an information-theoretic approach. J. Res. Nat. Bur. Standards - Math. Sci. 72 B (1968), 159-199. · Zbl 0274.62036
[26] Lauritzen, S. L.: Graphical Models. Oxford Science Pub., Clarendom Press, Oxford 1996. · Zbl 0907.62001
[27] Lauritzen, S. L., Speed, M. P., Vijayan, K.: Decomposable graphs and hypergraphs. J. Austral. Math. Soc. Ser. A 36 (1984), 12-29. · Zbl 0533.05046
[28] Lauritzen, S. L., Spiegelhalter, D. J.: Local computations with probabilities on graphical structures and their application to expert systems. J. Roy. Stat. Soc. Ser. B 50 (1988), 157-224. · Zbl 0684.68106
[29] Leimer, G.: Optimal decomposition by clique separators. Discrete Math. 113 (1993), 99-123. · Zbl 0793.05128
[30] Leontief, W. W.: The Structure of American Economy 1919-1929. Oxford University Press, New York 1941.
[31] Leontief, W. W., Strout, A.: Multiregional input-output analysis. Structural Interdependence and Economic Development, 1963, pp. 119-169.
[32] Madigan, D., Mosurski, K.: An extension of the results of Asmussen and Edwards on collapsibility in contingency tables. Biometrika 77 (1990), 315-319. · Zbl 0731.62113
[33] Madigan, D., Mosurski, K.: Errata: An extension of the results of Asmussen and Edwards on collapsibility in contingency tables. Biometrika 86 (1999) 973. · Zbl 1156.62334
[34] Maier, D.: The Theory of Relational Databases. Computer Science Press, 1983. ( · Zbl 0519.68082
[35] Maier, D., Ullman, J. D.: Connections in acyclic hypergraphs. Theoret. Comput. Sci. 32 (1984), 185-199. · Zbl 0557.05054
[36] Malvestuto, F. M.: Answering queries in categorical data bases. Proc. VI ACM Symp. Principles of Database Systems 1987, pp. 87-96.
[37] Malvestuto, F. M.: Existence of extensions and product extensions for discrete probability distributions. Discrete Math. 69 (1988), 61-77. · Zbl 0637.60021
[38] Malvestuto, F. M.: Computing the maximum-entropy extension of given discrete probability distributions. Computat. Statist. Data Anal. 8 (1989), 299-311. · Zbl 0726.62012
[39] Malvestuto, F. M.: Testing implication of hierarchical loglinear models for discrete probability distributions. Statist. Computing 6 (1996), 169-176.
[40] Malvestuto, F. M.: A hypergraph-theoretic analysis of collapsibility and decomposability for extended loglinear models. Statist. Computing 11 (2001), 155-169.
[41] Malvestuto, F. M.: From conditional independences to factorization constraints with discrete random variables. Ann. Math. Artificial Intelligence 35 (2002), 253-285. · Zbl 1001.68033
[42] Malvestuto, F. M.: Canonical and monophonic convexities in hypergraphs. Discrete Math. 309 (2009), 4287-4298. · Zbl 1211.05093
[43] Malvestuto, F. M., Moscarini, M.: A fast algorithm for query optimization in universal-relation databases. J. Comput. System Sci. 56 (1998), 299-309. · Zbl 0913.68060
[44] Malvestuto, F. M., Moscarini, M.: Decomposition of a hypergraph by partial-edge separators. Theoret. Comput. Sci. 237 (2000), 57-79. · Zbl 0939.68089
[45] Malvestuto, F. M., Pourabbas, E.: Customized answers to summary queries via aggregate views. Proc. XVI Intl. Conf. Scientific & Statistical Database Management 2004, pp. 193-202.
[46] Malvestuto, F. M., Pourabbas, E.: Local computation of answers to table queries on summary databases. Proc. XVII Intl. Conf. Scientific & Statistical Database Management 2005, pp. 263-272.
[47] Matúš, F.: Discrete marginal problem for complex measures. Kybernetika 24 (1988), 36-46. · Zbl 0636.60002
[48] Matúš, F.: On the maximum-entropy extensions of probability measures over undirected graphs. Proc. III Workshop Uncertainty Processing in Expert Systems 1994, pp. 181-198.
[49] Matúš, F., Flusser, J.: Image representations via a finite Radon transform. IEEE Trans. Pattern Analysis and Machine Intelligence 15 (1993), 996-1006. · Zbl 05112528
[50] Purcell, N. J., Kish, L.: Estimation for small domains. Biometrics 35 (1979), 365-384. · Zbl 0419.62092
[51] Purcell, N. J., Kish, L.: Postcensal estimates for local areas (or domains). Internat. Statist. Rev. 48 (1980), 3-18. · Zbl 0433.62080
[52] Rüschendorf, L.: Convergence of the iterative proportional fitting procedure. Ann. Statist. 23 (1995), 1160-1174. · Zbl 0851.62038
[53] Shore, J. E., Johnson, R. W.: Properties of cross-entropy minimization. IEEE Trans. Inform. Theory 27 (1981), 472-482. · Zbl 0459.94008
[54] Stephan, F. F.: An iterative method of adjusting sample frequencies tables when expected marginal totals are known. Ann. Math. Statist. 13 (1942), 166-178. · Zbl 0060.31505
[55] Stone, R., Brown, A.: A Computable Model for Economic Growth: A Programme for Growth, No. 1. Chapman Hall, London 1962.
[56] Tarjan, R. E., Yannakakis, M.: Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce hypergraphs. SIAM J. Comput. 13 (1984), 566-579. · Zbl 0545.68062
[57] Vomlel, J.: Integrating inconsistent data in a probabilistic model. J. Appl. Non-Classical Logics 14 (2004), 365-386. · Zbl 1185.68699
[58] Vorob’ev, N. N.: Consistent families of measures and their extensions. Theor. Prob. Appl. 7 (1962), 147-163. · Zbl 0201.49102
[59] Vorob’ev, N. N.: Markov measures and Markov extensions. Theor. Prob. Appl. 8 (1963), 420-429. · Zbl 0126.13701
[60] Yannakakis, M.: Computing the minimum fill-in is NP-complete. SIAM J. Algebraic Discrete Mathematics 2 (1981), 77-79. · Zbl 0496.68033
[61] Yannakakis, M.: Algorithms for acyclic database schemes. Proc. VII Internat. Conf. Very Large Data Bases 1981, pp. 82-94.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.