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...sult shows that ≤ is monotonic with respect to combination and marginalization. Proposition 4. The system (Φ, U, ⊗, ↓, ≤) satisfies the following two additional axioms of an ordered valuation algebra =-=[13]-=-. (A4) If φx ≤ ψx and φy ≤ ψy, then (φx ⊗ φy) ≤ (ψx ⊗ ψy). (A5) If φx ≤ ψx then φ ↓y x ≤ ψ ↓y x . Proof. (A4). Consider two valuations (px, ux) and (qx, vx) with scope x such that (px, ux) ≤ (qx, vx),...

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... shows that the algebra of valuations is monotonic with respect to dominance. Proposition 14. The system (Φ, U, ⊗, ↓, ≤) satisfies the following two additional axioms of an ordered valuation algebra (=-=Haenni, 2004-=-). 113Mauá, de Campos, & Zaffalon (A4) Combination is monotonic with respect to dominance, that is, if φx ≤ ψx and φy ≤ ψy then (φx ⊗ φy) ≤ (ψx ⊗ ψy) . (A5) Marginalization is monotonic with respect ...

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...mization tasks like the credal network inferences we aim at here admit a partial order ≤ over the valuations Φ (i.e., a reflexive, antisymmetric and transitive relation). An ordered valuation algebra =-=[17, 21]-=- is a system (Φ,U,×, ↓,≤), where (Φ,U,×, ↓) is a valuation algebra and ≤ is a partial order monotonic with respect to × and ↓:2 (A4) If φx ≤ ψx and φy ≤ ψy then (φy × φx) ≤ (ψy × ψx) and φ↓yx ≤ ψ↓yx ....

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...As) m such that d(m) = D. Finally, M = ∪{MD : D ⊆ V} denotes the set of all BBAs m in the evidential network such that d(m) ⊆ V. In summary, an evidential network is defined by a system (V,M, d,⊕, ↓) =-=[19]-=-. The operation of marginalisation ↓ in this system is sometimes replaced by another basic operation called variable elimination, defined as and denoted by m−x = m↓d(m)\{x} with x ∈ V. Notice that x ...

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... classical probability functions and possibility/necessity functions [27, Ch.2]. This is particularly important when the valuations need to represent domain knowledge that is expressed in the form of uncertain implication rules [28], [29]. In order to emphasise this aspect of our work, we refer to the resulting reasoning networks as evidential networks. In the paper we develop a representative model of threat in the context of air defence and implement it using an evidential network. Local computations in the network are performed using the inward propagation algorithm on the binary join tree [30]. We introduce a modified version of the standard inward propagation algorithm, which takes into account the dynamic nature of input valuations (the external evidence which drives the evidential network) by recomputing only the affected paths in the binary join tree. The paper is organised as follows. Section 2 describes valuation-based systems and the algorithms for local computation. Section 3 reviews the main concepts and tools from the theory of evidence. Section 4 develops the entities and relationships of a threat model cast in terms of an evidential network. Section 5 presents the numer...

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...el of threat in the context of air defence and implement it using an evidential network. Local computations in the network are performed using the inward propagation algorithm on the binary join tree =-=[24]-=-. We introduce a modified version of the standard inward propagation algorithm, which takes into account the dynamic nature of input valuations (the external evidence which drives the evidential netwo...