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On Weakness of Evidential Networks
Vejnarová, Jiřina
In evidence theory several counterparts of Bayesian networks based on different paradigms have been proposed. We will present, through simple examples, problems appearing in two kinds of these models caused either by the conditional independence concept (or its misinterpretation) or by the use of a conditioning rule. The latter kind of problems can be avoided if undirected models are used instead.
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Polymatroids and polyquantoids
Matúš, František
When studying entropy functions of multivariate probability distributions, polymatroids and matroids emerge. Entropy functions of pure multiparty quantum states give rise to analogous notions, called here polyquantoids and quantoids. Polymatroids and polyquantoids are related via linear mappings and duality. Quantum secret sharing schemes that are ideal are described by selfdual matroids. Expansions of integer polyquantoids to quantoids are studied and linked to that of polymatroids.
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LP relaxations and pruning for characteristic imsets
Studený, Milan
The geometric approach to learning BN structure is to represent it by a certain vector; a suitable such zero-one vector is the characteristic imset, which allows to reformulate the task of finding global maximum of a score over BN structures as an integer linear programming problem. The main contribution of this report is an LP relaxation of the corresponding polytope, that is, a polyhedral description of the domain of the respective integer linear programming problem.
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On Three Conditioning Rules in Evidence Theory
Vejnarová, Jiřina
In evidence theory various rules were proposed to define conditional beliefs and/or plausibilities (or basic assignments). However, there exist no generally accepted criteria along which these rules can be compared. In this paper we concentrate to three of them (Dempster's conditioning rule, focusing and the approach based on lower and upper envelopes of sets of conditional probabilities) to study their mutual relationship. A new conditional rule for variables is presented afterwards and its correctness is proven.
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On polyhedral approximations of polytopes for learning Bayes nets
Studený, Milan ; Haws, D.
We review three vector encodings of Bayesian network structures. The first one has recently been applied by Jaakkola et al., the other two use special integral vectors, called imsets. The central topic is the comparison of outer polyhedral approximations of the corresponding polytopes. We show how to transform the inequalities suggested by Jaakkola et al. to the framework of imsets. The result of our comparison is the observation that the implicit polyhedral approximation of the standard imset polytope suggested in (Studený Vomlel 2010) gives a closer approximation than the (transformed) explicit polyhedral approximation from (Jaakkola et al. 2010). Finally, we confirm a conjecture from (Studený Vomlel 2010) that the above-mentioned implicit polyhedral approximation of the standard imset polytope is an LP relaxation of the polytope.
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Conditional probability spaces and closures of exponential families
Matúš, František
A set of conditional probabilities is introduced by conditioning in the probability measures from an exponential family. A closure of the set is found, using previous results on the closure of another exponential family in the variational distance. The conditioning in the exponential family of all positive probabilities on a finite space is discussed and related to the permutahedra.
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