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Characteristic properties of equivalent structures in compositional models. (English) Zbl 1214.68400
Summary: Compositional model theory serves as an alternative approach to multidimensional probability distribution representation and processing. Every compositional model over a finite non-empty set of variables $$N$$ is uniquely defined by its generating sequence – an ordered set of low-dimensional probability distributions. A generating sequence structure induces a system of conditional independence statements over $$N$$ valid for every multidimensional distribution represented by a compositional model with this structure.
The equivalence problem is how to characterise whether all independence statements induced by structure $$\mathcal P$$ are induced by a second structure $$\mathcal P^{\prime}$$ and vice versa. This problem can be solved in several ways. A partial solution of the so-called direct characterisation of an equivalence problem is represented here. We deduce and describe three properties of equivalent structures necessary for equivalence of the respective structures. We call them characteristic properties of classes of equivalent structures.

##### MSC:
 68T37 Reasoning under uncertainty in the context of artificial intelligence
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##### References:
 [1] R. Jiroušek, Multidimensional Compositional Models. Part 1: Introduction, Internal Publication 2006/4ÚTIA AVČR, Prague, 2006. [2] R. Jiroušek, Persegrams of compositional models revisited: conditional independence, in: Proceedings of the 12th International Conference on Information Processing and Management of Uncertainty in Knowledge-based Systems, Malaga, 2008. [3] T. Kočka, R.R. Bouckaert, M. Studený, On the Inclusion Problem, Research Report 2010, ÚTIA AVČR, Prague, 2001. [4] Bína, V.; Jiroušek, R., Marginalization in multidimensional compositional models, Kybernetika, 42, 4, 405-422, (2006) · Zbl 1249.65010 [5] V. Kratochvíl, Equivalence problem in compositional models, in: T. Kroupa, J. Vejnarová, (Eds.), Proceedings of the 8th Workshop on Uncertainty Processing, Prague, 2009. [6] V. Kratochvíl, Motivation for different characterization of equivalent Persegrams, in: V. Novák, V. Pavliska, M. Štěpnička (Eds.), Proceedings of the 12th Czech-Japan seminar on Data Analysis and Decision Making under Uncertainty, Ostrava, 2009. [7] V. Kratochvíl, Different approaches of study direct equivalence characterization, in: P. Ambroř, Z. Masáková (Eds.), Doktorandské dny 2009, Prague, 2009.
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