2-相干和2-凸条件下预测
2-coherent and 2-convex Conditional Lower Previsions
摘要：
在本文中，我们探讨了（Williams）相干和凸条件预测的松弛，形成家庭的$n$ -相干和$n$ -凸条件预测，在$n$的变化。我们调查这样的预测是最普遍的一个可以合理地考虑，建议（中心）$2$ -凸，或者，如果需要积极的同质性和共轭，$2$ -连贯的较低的预测。这些预测的基本性质进行了研究。特别是，我们证明，他们满足广义贝叶斯规则，总是有一个$2$凸或，分别为$2$ -连贯的自然扩张。这些延伸的作用类似于用于连贯的下视的自然扩张的作用。相反，$n$ -凸和$n$ -连贯预见与$n\geq 3$要么是凸的或连贯的本身，或没有相同类型的扩展足够大的集。其中的不确定性概念，可以建模的$2$ -凸性，我们讨论了概括的能力和niveloid的条件框架，并表明，著名的风险措施的风险价值只保证集中$2$ -凸。在最后一部分中，我们确定的合理性要求的$2$ -凸性和$2$ -一致性的愿望的角度来看，强调他们如何削弱（威廉姆斯）的连贯性。
英文摘要：
In this paper we explore relaxations of (Williams) coherent and convex conditional previsions that form the families of $n$-coherent and $n$-convex conditional previsions, at the varying of $n$. We investigate which such previsions are the most general one may reasonably consider, suggesting (centered) $2$-convex or, if positive homogeneity and conjugacy is needed, $2$-coherent lower previsions. Basic properties of these previsions are studied. In particular, we prove that they satisfy the Generalized Bayes Rule and always have a $2$-convex or, respectively, $2$-coherent natural extension. The role of these extensions is analogous to that of the natural extension for coherent lower previsions. On the contrary, $n$-convex and $n$-coherent previsions with $n\geq 3$ either are convex or coherent themselves or have no extension of the same type on large enough sets. Among the uncertainty concepts that can be modelled by $2$-convexity, we discuss generalizations of capacities and niveloids to a conditional framework and show that the well-known risk measure Value-at-Risk only guarantees to be centered $2$-convex. In the final part, we determine the rationality requirements of $2$-convexity and $2$-coherence from a desirability perspective, emphasising how they weaken those of (Williams) coherence.
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