We are pleased to announce that our paper:
Neighbourhood models induced by the Euclidean distance and the Kullback-Leibler divergence
has been accepted for publication in Fuzzy Sets and Systems.
This paper, authored by Ignacio Montes, continues the research on distortion or neighbourhood models, a class of robust probabilistic models obtained by creating a neighbourhood of probability measures centred on a given initial probability, with a given radius and with respect to a distorting function used to compare probability measures. In particular, this paper investigates the imprecise probability models that arise when considering the Euclidean distance or the Kullback–Leibler divergence as the distorting function.
To this end, after introducing some preliminaries in Section 2, Section 3 is devoted to the distortion model determined by the Euclidean distance. A closed formula for the lower prevision associated with this model is obtained, and some interesting properties are presented. For example, while the lower prevision is not 2-monotone on gambles, it is 2-monotone on events.
Section 4 performs a similar study for the distortion model induced by the Kullback–Leibler divergence. Not surprisingly, this model is more difficult to handle from an analytical point of view. Nevertheless, it is possible to prove that, like the Euclidean model, it is 2-monotone on events but not on gambles.
Section 5 presents a comparison of the Euclidean and Kullback–Leibler models with some well-known distortion models, such as the linear vacuous model (also known as epsilon-contamination), the pari-mutuel model, the total variation model, and the constant odds ratio model. This comparative study shows that, at least from a practical point of view, the Euclidean and Kullback–Leibler models do not possess particularly attractive properties, even though their utility and interpretation as distortion models are unquestionable.