Characteristic properties of equivalent structures in compositional models

Kratochvíl, Václav

doi:10.1016/j.ijar.2010.12.005

Cited by 6 publications

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“…An efficient way of accomplishing this for compositional models is an open question. In this regard, a result that characterizes equivalent structures of decomposable models [19] seems to be promising.…”

Section: Discussionmentioning

confidence: 99%

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Compositional models in valuation-based systems

Jiroušek

Shenoy

2014

International Journal of Approximate Reasoning

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Compositional models were initially described for discrete probability theory, and later extended for possibility theory and for belief functions in Dempster-Shafer (D-S) theory of evidence. Valuation-based system (VBS) is an unifying theoretical framework generalizing some of the well known and frequently used uncertainty calculi. This generalization enables us to not only highlight the most important theoretical properties necessary for efficient inference (analogous to Bayesian inference in the framework of Bayesian network), but also to design efficient computational procedures. Some of the specific calculi covered by VBS are probability theory, a version of possibility theory where combination is the product t-norm, Spohn's epistemic belief theory, and D-S belief function theory. In this paper, we describe compositional models in the general framework of VBS using the semantics of no-double counting, which is central to the VBS framework. Also, we show that conditioning can be expressed using the composition operator. We define a special case of compositional models called decomposable models, again in the VBS framework, and demonstrate that for the class of decomposable compositional models, conditioning can be done using local computation. As all results are obtained for the VBS framework, they hold in all calculi that fit in the VBS framework. For the D-S theory of belief functions, the compositional model defined here differs from the one studied by Jiroušek, Vejnarová, and Daniel. The latter model can also be described in the VBS framework, but with a combination operator that is different from Dempster's rule of combination. For the version of possibility theory in which combination is the product t-norm, the compositional model defined here reduces to the one studied by Vejnarová. IntroductionThe framework of valuation-based systems (VBS) was introduced in [28,32,34] . The main idea behind VBS is to capture the common features of various uncertainty calculi and other domains such as optimization, decision-making theories, database systems, and solving systems of equations. Briefly, knowledge about a set of variables is represented by a set of functions called valuations. Each valuation is associated with a subset of variables. There are two operators called combination and marginalization. Combination allows us to aggregate knowledge, and marginalization allows us to coarsen knowledge to a smaller set of variables. The combination of all valuations, called the joint valuation, represents the joint knowledge of all variables. Making inferences can be described as finding marginals of the joint valuation for variables of interest. The VBS framework can be used to describe various uncertainty theories such as probability theory, a version of possibility theory where combination is the product t-norm [43], Spohn's epistemic belief theory [37,30], and Dempster-Shafer (D-S) belief function theory [26]. It can also be used to describe, e.g., propositional logic [29], solving systems of equations [25...

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Section: Discussionmentioning

confidence: 99%

“…Notice that the constant c appearing in Equation (19) comes from normalization constants in Equations (16) and (18). However, since both ρ σ defined in Section 9 and ρ T p σ defined by Vejnarová are normalized possibility distributions, c = 1.…”

Section: Compositional Models For Possibility Theorymentioning

confidence: 99%

Compositional models in valuation-based systems

Jiroušek

Shenoy

2014

International Journal of Approximate Reasoning

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“…Note that a sequential compositional expression is uniquely determined by its base sequence; accordingly, as in [16] and [17] the base sequence of a sequential compositional expression may be called its "structure". A further property of a sequential compositional expression is that, by Fact 4.3, each intermediate table is always an extension of the distribution function f 1 in f on X 1 (which is the key of θ).…”

Section: Sequential Compositional Expressionsmentioning

confidence: 99%

“…Next, we define the notion of equivalence between compositional expressions having the same base scheme and the same key. (Note that our notion of equivalence is stronger than that studied in [16,17].) Thus, Theorem 1.1 can be re-stated as follows: If X 2 ∩ X 3 ⊆ X 1 , then the four compositional expressions…”

Section: Introductionmentioning

confidence: 99%

Equivalence of compositional expressions and independence relations in compositional models

Malvestuto¹

2014

Kybernetika

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“…In some cases, the composition scheme has a closed form and a typical case is the composition scheme associated with a sequential expression (see "formal ratios" introduced by Kratochvíl [13,14]). …”

Section: Introductionmentioning

confidence: 99%

Compositional models, Bayesian models and recursive factorization models

Malvestuto¹

2016

Kybernetika

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Characteristic properties of equivalent structures in compositional models

Cited by 6 publications

References 0 publications

Compositional models in valuation-based systems

Compositional models in valuation-based systems

Equivalence of compositional expressions and independence relations in compositional models

Compositional models, Bayesian models and recursive factorization models

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