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Marginalization in models generated by compositional expressions. (English) Zbl 1363.05246

The paper is a contribution to the theory of compositional models, an alternative to Bayesian networks for the representation and processing of multidimensional probability distributions. The approach is based on the so called operator of composition that constructs from two low-dimensional distributions a distribution of a higher dimension. When comparing with Bayesian networks, the advantage of these models is that one can do with probability theory, one do not use graphs to represent a distribution structure.
The paper is a continuation of the author’s paper [Kybernetika 50, No. 3, 322–362 (2014; Zbl 1366.62103); erratum ibid. 51, No. 2, 387–388 (2015; Zbl 1340.05198)]. The present paper extends the previous one especially in the way that it does not restrict its consideration to expressions in which each element appears only once, and thus it reflects the requirements of applications more realistically. Moreover, the author studies general models in which the building blocks (low-dimensional distributions) are assembled (composed) in an arbitrary order predetermined by parentheses.
The author proposes several computational procedures for marginalizing general compositional expressions and studies their computational complexity. He focuses on two basic marginalization problems, which he calls single-marginal and marginal-representation problems. When solving the former problem he is interested in finding the simplest (from the computational point of view) procedure computing a given marginal distribution. As the name suggests, the goal of the later problem is to find a suitable structure making efficient computations of arbitrary marginal distributions possible.
The author considers in the paper not only probability distributions with values from interval \([0,1]\), but distributions whose values form an arbitrary semifield. This generalization enables him to apply the described apparatus, for example, to multidimensional databases, in which query answering requires to combine data stored in distinct tables.

MSC:

05C90 Applications of graph theory
05C65 Hypergraphs
05C85 Graph algorithms (graph-theoretic aspects)
65C50 Other computational problems in probability (MSC2010)
60E99 Distribution theory
68T37 Reasoning under uncertainty in the context of artificial intelligence
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