## Lower Desirability Functions: A Convenient Imprecise Hierarchical Uncertainty Model (1999)

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### BibTeX

@MISC{Cooman99lowerdesirability,

author = {Gert de Cooman},

title = {Lower Desirability Functions: A Convenient Imprecise Hierarchical Uncertainty Model},

year = {1999}

}

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### Abstract

I introduce and study a fairly general imprecise secondorder uncertainty model, in terms of lower desirability. A modeller's lower desirability for a gamble is defined as her lower probability for the event that a given subject will find the gamble (at least marginally) desirable. For lower desirability assessments, rationality criteria are introduced that go back to the criteria of avoiding sure loss and coherence in the theory of (first-order) imprecise probabilities. I also introduce a notion of natural extension that allows the least committal coherent extension of lower desirability assessments to larger domains, as well as to a first-order model, which can be used in statistical reasoning and decision making. The main result of the paper is what I call Precision--Imprecision Equivalence: as far as certain behavioural implications of this model are concerned, it does not matter whether the subject's underlying first-order model is assumed to be precise or imprecise. Keywords. Hie...

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Citation Context ...ituations, can be found in [7]. By far the most common hierarchical model is the Bayesian one, where both the first and the second-order model are (precise) probability measures, or linear previsions =-=[1, 10, 11, 15]-=-. Other models allow imprecision in the second-order model, but still assume that the first-order model is precise; examples are the robust Bayesian models [2], models involving second-order possibili... |

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Citation Context ...call this result Precision–Imprecision Equivalence. It generalises a number of results known in the literature: the close formal analogy between Walley’s behavioural theory of imprecise probabilities =-=[13]-=- and Bayesian sensitivity analysis [2], the results concerning second-order possibility distributions in [14] and the representation theorems in [7]. The paper is organised as follows. Section 2 gives... |

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Citation Context ...ituations, can be found in [7]. By far the most common hierarchical model is the Bayesian one, where both the first and the second-order model are (precise) probability measures, or linear previsions =-=[1, 10, 11, 15]-=-. Other models allow imprecision in the second-order model, but still assume that the first-order model is precise; examples are the robust Bayesian models [2], models involving second-order possibili... |

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Citation Context ...ituations, can be found in [7]. By far the most common hierarchical model is the Bayesian one, where both the first and the second-order model are (precise) probability measures, or linear previsions =-=[1, 10, 11, 15]-=-. Other models allow imprecision in the second-order model, but still assume that the first-order model is precise; examples are the robust Bayesian models [2], models involving second-order possibili... |

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Citation Context ...ving second-order possibility distribu∗ Postdoctoral Fellow of the Fund for Scientific Research – Flanders (Belgium) (FWO). tions [4, 7, 14], and the Gärdenfors and Sahlin epistemic reliability model =-=[9]-=-. I know of no detailed analysis where imprecision is explicitly allowed at both levels, but see [13, Section 5.10.5] for a brief discussion. In this paper, I introduce and study a particular type of ... |

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Citation Context ... sup[X] then gX(x) = 0. The p–representation of d is the lower probability (P, Dp(KX), P p ) that is defined by P p (Dp(X − x)) = gX(x). Note that Dp(KX) is a chain of sets. We can use the results in =-=[5]-=- to arrive at the following conclusions. If (g0)–(g3) hold, P p is coherent — it is a finitely additive probability on Dp(K) — and d is representable. The natural extension Ep of P p to all events is ... |

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Citation Context ...near previsions Mα corresponds to a lower prevision P α , defined by P α (Y ) = inf{P (Y ): P ∈ Mα}. The first-order natural extension is a uniform average of the P α: E 1 (Y ) = ∫ 1 0 P α(Y )dα (see =-=[14]-=- for a proof). The present discussion focuses on one gamble X. We may follow the same approach for a number of gambles X in a collection KS. This leads to a lower desirability function d defined on th... |

19 | A possibilistic hierarchical model for behavior under uncertainty - DeCooman, Walley - 2002 |

15 |
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Citation Context ...se where every gamble X has a ‘fair price’, meaning that the supremum acceptable buying price agrees with the infimum acceptable selling price, we obtain the theory of linear previsions of de Finetti =-=[8]-=-. A linear prevision P on a set of gambles K is a map taking K to the set of real numbers R, such that for all m ≥ 0 and n ≥ 0, and for any X1, . . . , Xn and Y1, . . . , Ym in K, [ ∑n m∑ ] sup G(Xk)(... |

13 |
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Citation Context ...) < 0}.For the first-order natural extension E 1 of d we find, since a necessity measure is 2–monotone and the natural extension of a 2–monotone lower probability can be found by Choquet integration =-=[12, 13]-=-: E 1 (Y ) = Ep (Y ∗ ∫ sup[Y ] ) = inf[Y ] + e(Y − y)dy inf[Y ] ∫ sup[Y ] = inf[Y ] + inf{g inf[Y ] + X (P (X)): P (Y ) < y}dy There is another way of writing this first-order natural extension. Let M... |

8 |
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Citation Context ...initely additive probability on Dp(K) — and d is representable. The natural extension Ep of P p to all events is a necessity measure, that is, the conjugate lower probability of a possibility measure =-=[3, 5]-=- with possibility distribution π : P → [0, 1] given by π(P ) = 1−g + X (P (X)), where g+ X : R → [0, 1] is the right-continuous non-decreasing mapping defined by g + X (x) = gX(x+) = sup{gX(x + ɛ): ɛ ... |

7 |
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Citation Context ... is precise; examples are the robust Bayesian models [2], models involving second-order possibility distribu∗ Postdoctoral Fellow of the Fund for Scientific Research – Flanders (Belgium) (FWO). tions =-=[4, 7, 14]-=-, and the Gärdenfors and Sahlin epistemic reliability model [9]. I know of no detailed analysis where imprecision is explicitly allowed at both levels, but see [13, Section 5.10.5] for a brief discuss... |

4 |
2002) An imprecise hierarchical model for behaviour under uncertainty, Theory and Decision
- Cooman, Walley
(Show Context)
Citation Context ... is uncertain about what it is. The modeller’s uncertainty is then called second-order uncertainty. A list of examples showing that second-order uncertainty occurs in many situations, can be found in =-=[7]-=-. By far the most common hierarchical model is the Bayesian one, where both the first and the second-order model are (precise) probability measures, or linear previsions [1, 10, 11, 15]. Other models ... |

2 |
A behavioural theory of fuzzy probability
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- 1999
(Show Context)
Citation Context ...s relating the present model to Walley’s theory of first-order imprecise probabilities [13], Bayesian sensitivity analysis [2], and to the theory of fuzzy probability and buying functions explored in =-=[4, 6, 7]-=-. Section 8 concludes the paper. 2 Preliminary Notions In this section, I discuss a number of aspects of the precise and imprecise uncertainty models that will serve as a basis for the development of ... |