Uncertain Measure And Its Applications

Randomness is a fundamental type of uncertainty present in the real world, andprobability theory is a branch of mathematics for studying random phenomena. Asan improvement of our understanding about uncertain phenomena, researchers havechallenged the"countable additivity"condition, for example, in capacity theory, fuzzymeasure theory, possibility theory, and so on. However, since this class of non-additivemeasures does not consider the self-duality property, all of those measures are incon-sistent with the law of contradiction and law of excluded middle, which are basic rulesin mathematics. In order to solve this important problem, Liu (2007) founded an un-certainty theory that is a branch of mathematics based on normality, monotonicity,self-duality, countable subadditivity, and product measure axioms.This dissertation first proposes the judgement conditions of continuous uncertainmeasures, then studies some important concepts about uncertain variables when the un-certain measure is continuous; more specifically, uncertainty distribution, critical value,expected value are studied, among others. Furthermore, this dissertation discusses themaximum entropy principle for uncertain variables and proves some maximum entropytheorems with moment constraints. Also defined and studied are cross-entropy of un-certain variables, generalized cross-entropy and generalized entropy. Finally, based ontools such as conditional uncertainty and identification functions, some expressions ofLiu's inference rule for uncertain systems are derived.In conclusion, this dissertation contributes to the research area of uncertainty the-ory in the following aspects: 1. continuous uncertain measure is proposed and somebasic properties for uncertain variables in uncertainty space with continuous uncertainmeasures are studied; 2. maximum entropy principle for uncertain variables is investi-gated; 3. Liu's inference rule with multiple antecedents and with multiple if-then rulesis considered.