Unifying neighbourhood and distortion models: part II – new models and synthesis

Neighbourhoods of precise probabilities are instrumental to perform robustness analysis, as they rely on very few parameters. In the first part of this study, we introduced a general, unified view encompassing such neighbourhoods, and revisited some well-known models (pari mutuel, linear vacuous, constant odds-ratio). In this second part, we study models that have received little to no attention, but are induced by classical distances between probabilities, such as the total variation, the Kolmogorov and the distances. We finish by comparing those models in terms of a number of properties: precision, number of extreme points, n-monotonicity, …thus providing possible guidelines to select a neighbourhood rather than another.

[1]  Chiara Corsato,et al.  Nearly-Linear uncertainty measures , 2019, Int. J. Approx. Reason..

[2]  Didier Dubois,et al.  Unifying practical uncertainty representations - I: Generalized p-boxes , 2008, Int. J. Approx. Reason..

[3]  Enrique Miranda,et al.  Coherent updating of non-additive measures , 2015, Int. J. Approx. Reason..

[4]  Sébastien Destercke,et al.  Probability boxes on totally preordered spaces for multivariate modelling , 2011, Int. J. Approx. Reason..

[5]  Alexander Lepskiy,et al.  Imprecision indices: axiomatic, properties and applications , 2015, Int. J. Gen. Syst..

[6]  Anton Wallner Extreme points of coherent probabilities in finite spaces , 2007, Int. J. Approx. Reason..

[7]  Sébastien Destercke,et al.  On extreme points of p-boxes and belief functions , 2016, Annals of Mathematics and Artificial Intelligence.

[8]  Enrique Miranda,et al.  Outer approximating coherent lower probabilities with belief functions , 2019, Int. J. Approx. Reason..

[9]  Sébastien Destercke,et al.  Pari-mutuel probabilities as an uncertainty model , 2019, Inf. Sci..

[10]  Larry Wasserman,et al.  Dilation for Sets of Probabilities , 1993 .

[11]  Enrique Miranda,et al.  2-Monotone outer approximations of coherent lower probabilities , 2018, Int. J. Approx. Reason..

[12]  Marco Zaffalon,et al.  Inference and risk measurement with the pari-mutuel model , 2010, Int. J. Approx. Reason..

[13]  Andrey G. Bronevich On the closure of families of fuzzy measures under eventwise aggregations , 2005, Fuzzy Sets Syst..

[14]  Joaquín Abellán,et al.  Measures of divergence on credal sets , 2006, Fuzzy Sets Syst..

[15]  Larry Wasserman,et al.  Divisive Conditioning: Further Results on Dilation , 1997, Philosophy of Science.

[16]  Peter J. Huber,et al.  Robust Statistics , 2005, Wiley Series in Probability and Statistics.