Evidential reasoning in large partially ordered sets

Abstract

The Dempster-Shafer theory of belief functions has proved to be a powerful formalism for uncertain reasoning. However, belief functions on a finite frame of discernment are usually defined in the power set 2 , resulting in exponential complexity of the operations involved in this framework, such as combination rules. When is linearly ordered, a usual trick is to work only with intervals, which drastically reduces the complexity of calculations. In this paper, we show that this trick can be extrapolated to frames endowed with an arbitrary lattice structure, not necessarily a linear order. This principle makes it possible to apply the Dempster-Shafer framework to very large frames such as the power set, the set of partitions, or the set of preorders of a finite set. Applications to multi-label classification, ensemble clustering and preference aggregation are demonstrated.

Introducing spatial neighbourhood in Evidential C-Means for segmentation of multi-source images: Application to prostate multi-parametric MRI

“…These authors could establish that the evidential approach was less restrictive than the MAP decision. We believe multi-sensor segmen- 30 tation methods should rely on models that take into account redundancy and conflicts between data sources. Evidence theory, also called Dempster-Shafer theory [8,7], is widely used in data fusion and pattern recognition [9] as it provides strong and native modelling of imprecision, data fusion, eventual conflictual sources and outliers rejection [10,11,12,13,14,15].…”

Preference fusion and Condorcet's paradox under uncertainty

“…Belief function theory is a mathematical framework for representing and modeling uncertainty [10]. In [11], [12], the authors proposed a belief-functionbased model for preference fusion, allowing the expression of uncertainty over the lattice order (i.e. preference structure).…”

“…However, this modeling approach does not constitute an optimal representation of preferences in the presence of uncertain and voluminous information. Based on [11], the model of uncertainty proposed by Masson, et al in [13] allows the expression of uncertainty on binary relations (i.e. preference relations).…”

“…preference fusion under uncertainty). Furthermore, most of the work considering preference fusion under uncertainty [5], [11], [13] does not address the problem of Condorcet Paradox.…”

Generalized evidence theory

“…Since first proposed by Dempster [1] and then developed by Shafer [2], has been paid much attentions for a long time and continually attracted growing interests [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30].…”