论文标题
詹森（Jensen）和坎特利（Cantelli）的不平等现象不精确地预防
Jensen's and Cantelli's Inequalities with Imprecise Previsions
论文作者
论文摘要
我们研究了如何将基本概率不等式扩展到不精确的框架，在这种框架中，（确切的）概率和期望被不精确的概率和下/上限预防替代。考虑到$ x $（Jensen的不等式）的凸/凹功能，我们将重点放在提供单界随机变量$ x $的信息上，或诸如$（x \ geq c）$或$（x \ leq c）$（Markov's and Cantelli's Incoralities）之类的单方面界限。至于相关不精确的不确定性措施的一致性，我们的分析认为连贯性和较弱的要求，尤其是$ 2 $ conerence，这通常足够。引入了类似詹森的不平等现象，以及最近改善詹森不平等的概括。他们的某些应用是提出的：Lyapunov的不平等和推论问题的扩展。在讨论了马尔可夫的上和下部的不平等之后，与相关的下部/上部预防相关的一致性证明了类似坎内利的不平等现象。在连贯的不准确预防的情况下，相应的坎泰利不平等现象利用了沃利的下部和上部方差，通常可以确保更好的界限。
We investigate how basic probability inequalities can be extended to an imprecise framework, where (precise) probabilities and expectations are replaced by imprecise probabilities and lower/upper previsions. We focus on inequalities giving information on a single bounded random variable $X$, considering either convex/concave functions of $X$ (Jensen's inequalities) or one-sided bounds such as $(X\geq c)$ or $(X\leq c)$ (Markov's and Cantelli's inequalities). As for the consistency of the relevant imprecise uncertainty measures, our analysis considers coherence as well as weaker requirements, notably $2$-coherence, which proves to be often sufficient. Jensen-like inequalities are introduced, as well as a generalisation of a recent improvement to Jensen's inequality. Some of their applications are proposed: extensions of Lyapunov's inequality and inferential problems. After discussing upper and lower Markov's inequalities, Cantelli-like inequalities are proven with different degrees of consistency for the related lower/upper previsions. In the case of coherent imprecise previsions, the corresponding Cantelli's inequalities make use of Walley's lower and upper variances, generally ensuring better bounds.
