A survey of credibility theory.

*(English)*Zbl 1133.90426Summary: This paper provides a survey of credibility theory that is a new branch of mathematics for studying the behavior of fuzzy phenomena. Some basic concepts and fundamental theorems are introduced, including credibility measure, fuzzy variable, membership function, credibility distribution, expected value, variance, critical value, entropy, distance, credibility subadditivity theorem, credibility extension theorem, credibility semicontinuity law, product credibility theorem, and credibility inversion theorem. Recent developments and applications of credibility theory are summarized. A new idea on chance space and hybrid variable is also documented.

##### MSC:

90C70 | Fuzzy and other nonstochastic uncertainty mathematical programming |

03E72 | Theory of fuzzy sets, etc. |

60A05 | Axioms; other general questions in probability |

62A01 | Foundations and philosophical topics in statistics |

68T37 | Reasoning under uncertainty in the context of artificial intelligence |

##### Keywords:

Fuzzy variable; Credibility measure; Credibility theory; Fuzzy random variable; Random fuzzy variable; Conditional credibility
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\textit{B. Liu}, Fuzzy Optim. Decis. Mak. 5, No. 4, 387--408 (2006; Zbl 1133.90426)

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