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*Ignacio Montes*

## Introduction to imprecise probability

#### Abstract

Imprecise Probabilities is the generic term used to refer to all the mathematical models that serve as an alternative and extension of the usual Probability Theory when we deal with imprecise or incomplete information. Among these flexible models, we could mention coherent lower previsions, 2-monotone capacities or belief function.

This tutorial reviews the basic concepts of the theory of imprecise probabilities as well as their interpretations. We start with coherent lower previsions, one of the key notions in Walleyâ€™s behavioural theory of imprecise probabilities. Later, we explain the equivalent representations in terms of credal sets and coherent sets of desirable gambles. Finally, we introduce some other imprecise models that, while slightly less expressive, are easier to use in practice, such as 2-monotone lower probabilities, belief functions or p-boxes.

#### Syllabus

Part 1 |
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1) Summary about probability |

2) Coherent lower probabilities |

2.1) Definition |

2.2) Credal Set |

2.3) ASL/Coherence |

2.4) Conjugate upper probability |

3) Coherent lower previsions |

3.1) Definition |

3.2) Credal Set |

3.3) ASL/Coherence |

3.4) Conjugate upper probability |

4) Natural extension |

5) Interpretation of CLP |

5.1) Behavioural |

5.2) Epistemic |

6) Coherent lower previsions |

Part 2 |
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7) Monotone LP |

8) Belief Functions |

9) P-boxes |

10) Possibility measures |