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Inferring a linear ordering over a power set

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Abstract

An observer attempts to infer the unobserved ranking of two ideal objects, A and B, from observed rankings in which these objects are `accompanied' by `noise' components, C and D. In the first ranking, A is accompanied by C and B is accompanied by D, while in the second ranking, A is accompanied by D and B is accompanied by C. In both rankings, noisy-A is ranked above noisy-B. The observer infers that ideal-A is ranked above ideal-B. This commonly used inference rule is formalized for the case in which A,B,C,D are sets. Let X be a finite set and let \(\succ\) be a linear ordering on 2X. The following condition is imposed on \(\succ\). For every quadruple (A,B,C,D)∈Y, where Y is some domain in (2X)4, if \(A \cup C \succ B \cup D\) and \(A \cup D \succ B \cup C \succ B \cup C\), then \(A \succ B\). The implications and interpretation of this condition for various domains Y are discussed.

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Spiegler, R. Inferring a linear ordering over a power set. Theory and Decision 51, 31–49 (2001). https://doi.org/10.1023/A:1012400922478

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  • DOI: https://doi.org/10.1023/A:1012400922478

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