Smoothing and filtering with a class of outer measures

Filtering and smoothing with a generalised representation of uncertainty is considered. Here, uncertainty is represented using a class of outer measures. It is shown how this representation of uncertainty can be propagated using outer-measure-type versions of Markov kernels and generalised Bayesian-like update equations. This leads to a system of generalised smoothing and filtering equations where integrals are replaced by supremums and probability density functions are replaced by positive functions with supremum equal to one. Interestingly, these equations retain most of the structure found in the classical Bayesian filtering framework. It is additionally shown that the Kalman filter recursion can be recovered from weaker assumptions on the available information on the corresponding hidden Markov model

Jensen's and Cantelli's Inequalities with Imprecise Previsions

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.Comment: Published in Fuzzy Sets and Systems - https://dx.doi.org/10.1016/j.fss.2022.06.02