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.
翻译:我们研究基本概率不平等如何扩大到一个不精确的框架,在这个框架(精确)概率和期望被不准确的概率和低/高设想所取代(精确)概率和低/高设想所取代。我们注重不平等,提供关于单一约束随机的X美元的信息,考虑的是Jensen的不平等(Jensen的不平等)或美元(X\geqc)或(X\leqc)美元(Markov的和Cantelli的不平等)等单方界限(美元或(X\leqc)美元)或(X\leqc)美元(Markov的和Cantelli的不平等)。关于相关不准确的不确定性措施的一致性,我们的分析考虑到一致性和弱要求,特别是$的趋同性,这往往证明是足够的。杰森式的不平等被引入了,并概括了Jensen的不平等最近得到的改善。他们提出的一些应用是:延长Lyapunov的不平等和推断问题。在讨论上下马尔科夫的不平等之后,Cantelli的不平等被证明,相关的低/上下预测具有不同程度的一致性。在不同的程度上证明,在相关的低/上/上/上/上调的不平等中可以保证更精确的升级。