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Many-dimensional observables on Łukasiewicz tribe: constructions, conditioning and conditional independence

Tomáš Kroupa (2005)

Kybernetika

Probability on collections of fuzzy sets can be developed as a generalization of the classical probability on σ -algebras of sets. A Łukasiewicz tribe is a collection of fuzzy sets which is closed under the standard fuzzy complementation and under the pointwise application of the Łukasiewicz t-norm to countably many fuzzy sets. An observable is a fuzzy set-valued mapping defined on a σ -algebra of sets and satisfying some additional properties; formally, the role of an observable is in a sense analogous...

Multi-variate correlation and mixtures of product measures

Tim Austin (2020)

Kybernetika

Total correlation (‘TC’) and dual total correlation (‘DTC’) are two classical ways to quantify the correlation among an n -tuple of random variables. They both reduce to mutual information when n = 2 . The first part of this paper sets up the theory of TC and DTC for general random variables, not necessarily finite-valued. This generality has not been exposed in the literature before. The second part considers the structural implications when a joint distribution μ has small TC or DTC. If TC ( μ ) = o ( n ) , then μ is...

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