Uncertainty theory (Liu)

The uncertainty theory invented by Baoding Liu[1] is a branch of mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms.

[clarification needed] Mathematical measures of the likelihood of an event being true include probability theory, capacity, fuzzy logic, possibility, and credibility, as well as uncertainty.

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

Uncertainty distribution is inducted to describe uncertain variables.

Theorem (Peng and Iwamura, Sufficient and Necessary Condition for Uncertainty Distribution): A function

be uncertainty distributions of independent uncertain variables

the joint uncertainty distribution of uncertain vector

is defined by provided that at least one of the two integrals is finite.

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

be independent uncertain variables with finite expected values.

be an uncertain variable with finite expected value

be an uncertain variable with finite expected value,

be an uncertain variable with regular uncertainty distribution

be an uncertain variable with regular uncertainty distribution

be an uncertain variable whose uncertainty distribution is arbitrary but the expected value

Then Theorem 1(Liu, Markov Inequality): Let

, we have Theorem 3 (Liu, Holder's Inequality) Let

Then we have Theorem 4:(Liu [127], Minkowski Inequality) Let

are uncertain variables defined on the uncertainty space

are uncertain variables with finite expected values.

Then the conditional uncertain measure of A given B is defined by Theorem 1: Let

Then M{·|B} defined by Definition 1 is an uncertain measure, and

to the set of real numbers such that Definition 3: The conditional uncertainty distribution

be an uncertain variable with regular uncertainty distribution

be an uncertain variable with regular uncertainty distribution

given B is defined by provided that at least one of the two integrals is finite.