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Uncertainty theory is a branch of mathematics based on normality, monotonicity, selfduality, countable subadditivity, and product measure axioms.^{[clarification needed]} It was founded by Baoding Liu ^{[1]} in 2007 and refined in 2009.^{[2]}
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) .
Axiom 2. (Monotonicity Axiom) .
Axiom 3. (SelfDuality Axiom) .
Axiom 4. (Countable Subadditivity Axiom) For every countable sequence of events Λ_{1}, Λ_{2}, ..., we have
Axiom 5. (Product Measure Axiom) Let be uncertainty spaces for . Then the product uncertain measure is an uncertain measure on the product σalgebra 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 to the set of real numbers, i.e., for any Borel set B of real numbers, the set is an event.
Uncertainty distribution is inducted to describe uncertain variables.
Definition:The uncertainty distribution of an uncertain variable ξ is defined by .
Theorem(Peng and Iwamura, Sufficient and Necessary Condition for Uncertainty Distribution) A function is an uncertain distribution if and only if it is an increasing function except and .
Definition: The uncertain variables are said to be independent if
for any Borel sets of real numbers.
Theorem 1: The uncertain variables are independent if
for any Borel sets of real numbers.
Theorem 2: Let be independent uncertain variables, and measurable functions. Then are independent uncertain variables.
Theorem 3: Let be uncertainty distributions of independent uncertain variables respectively, and the joint uncertainty distribution of uncertain vector . If are independent, then we have
for any real numbers .
Theorem: Let be independent uncertain variables, and a measurable function. Then is an uncertain variable such that
where are Borel sets, and means for any.
Definition: Let be an uncertain variable. Then the expected value of is defined by
provided that at least one of the two integrals is finite.
Theorem 1: Let be an uncertain variable with uncertainty distribution . If the expected value exists, then
Theorem 2: Let be an uncertain variable with regular uncertainty distribution . If the expected value exists, then
Theorem 3: Let and be independent uncertain variables with finite expected values. Then for any real numbers and , we have
Definition: Let be an uncertain variable with finite expected value . Then the variance of is defined by
Theorem: If be an uncertain variable with finite expected value, and are real numbers, then
Definition: Let be an uncertain variable, and . Then
is called the αoptimistic value to , and
is called the αpessimistic value to .
Theorem 1: Let be an uncertain variable with regular uncertainty distribution . Then its αoptimistic value and αpessimistic value are
Theorem 2: Let be an uncertain variable, and . Then we have
Theorem 3: Suppose that and are independent uncertain variables, and . Then we have
,
,
,
,
,
.
Definition: Let be an uncertain variable with uncertainty distribution . Then its entropy is defined by
where .
Theorem 1(Dai and Chen): Let be an uncertain variable with regular uncertainty distribution . Then
Theorem 2: Let and be independent uncertain variables. Then for any real numbers and , we have
Theorem 3: Let be an uncertain variable whose uncertainty distribution is arbitrary but the expected value and variance . Then
Theorem 1(Liu, Markov Inequality): Let be an uncertain variable. Then for any given numbers and , we have
Theorem 2 (Liu, Chebyshev Inequality) Let be an uncertain variable whose variance exists. Then for any given number, we have
Theorem 3 (Liu, Holder’s Inequality) Let and be positive numbers with , and let and be independent uncertain variables with and . Then we have
Theorem 4:(Liu [127], Minkowski Inequality) Let be a real number with , and let and be independent uncertain variables with and . Then we have
Definition 1: Suppose that are uncertain variables defined on the uncertainty space . The sequence is said to be convergent a.s. to if there exists an event with such that
for every . In that case we write ,a.s.
Definition 2: Suppose that are uncertain variables. We say that the sequence converges in measure to if
for every .
Definition 3: Suppose that are uncertain variables with finite expected values. We say that the sequence converges in mean to if
Definition 4: Suppose that are uncertainty distributions of uncertain variables , respectively. We say that the sequence converges in distribution to if at any continuity point of .
Theorem 1: Convergence in Mean Convergence in Measure Convergence in Distribution. However, Convergence in Mean Convergence Almost Surely Convergence in Distribution.
Definition 1: Let be an uncertainty space, and . Then the conditional uncertain measure of A given B is defined by
Theorem 1: Let be an uncertainty space, and B an event with . Then M{·B} defined by Definition 1 is an uncertain measure, and is an uncertainty space.
Definition 2: Let be an uncertain variable on . A conditional uncertain variable of given B is a measurable function from the conditional uncertainty space to the set of real numbers such that
Definition 3: The conditional uncertainty distribution of an uncertain variable given B is defined by
provided that .
Theorem 2: Let be an uncertain variable with regular uncertainty distribution , and a real number with . Then the conditional uncertainty distribution of given is
Theorem 3: Let be an uncertain variable with regular uncertainty distribution , and a real number with . Then the conditional uncertainty distribution of given is
Definition 4: Let be an uncertain variable. Then the conditional expected value of given B is defined by
provided that at least one of the two integrals is finite.