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Uncertainty theory

Uncertainty theory is a branch of mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms. It was founded by Baoding Liu in 2007 and refined in 2009. Uncertainty theory is a branch of mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms. It was founded by Baoding Liu in 2007 and refined in 2009. Mathematical measures of the likelihood of an event being true include probability theory, capacity, fuzzy logic, possibility, and credibility, as well as uncertainty. Axiom 1. (Normality Axiom) M { Γ } = 1  for the universal set  Γ {displaystyle {mathcal {M}}{Gamma }=1{ ext{ for the universal set }}Gamma } . Axiom 2. (Self-Duality Axiom) M { Λ } + M { Λ c } = 1  for any event  Λ {displaystyle {mathcal {M}}{Lambda }+{mathcal {M}}{Lambda ^{c}}=1{ ext{ for any event }}Lambda } . Axiom 3. (Countable Subadditivity Axiom) For every countable sequence of events Λ1, Λ2, ..., we have Axiom 4. (Product Measure Axiom) Let ( Γ k , L k , M k ) {displaystyle (Gamma _{k},{mathcal {L}}_{k},{mathcal {M}}_{k})} be uncertainty spaces for k = 1 , 2 , ⋯ , n {displaystyle k=1,2,cdots ,n} . Then the product uncertain measure M {displaystyle {mathcal {M}}} is an uncertain measure on the product σ-algebra satisfying Principle. (Maximum Uncertainty Principle) For any event, if there are multiple reasonable values that an uncertain measure may take, then the value as close to 0.5 as possible is assigned to the event. An uncertain variable is a measurable function ξ from an uncertainty space ( Γ , L , M ) {displaystyle (Gamma ,L,M)} to the set of real numbers, i.e., for any Borel set B of real numbers, the set { ξ ∈ B } = { γ ∈ Γ | ξ ( γ ) ∈ B } {displaystyle {xi in B}={gamma in Gamma |xi (gamma )in B}} is an event.

[ "Statistics", "Machine learning", "Mathematical optimization", "Mathematical analysis", "uncertain programming", "chance theory" ]
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