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

Abstract

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 ${\text{L}}_1$ 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.

Keywords:

• Neighbourhood models
• distorted probabilities
• total variation distance
• Kolmogorov distance
• $L_1$ distance

Acknowledgments

We want to thank the reviewers for the careful reading of the paper. Their many valuable suggestions and comments helped us to improve the quality of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 We refer the reader to the first part for a more detailed introduction to these notions.

2 In Montes, Miranda, and Destercke (2019, Proposition 2), it is stated that the maximal number of extreme points induced by a PMM is $n/2(\genfrac{}{}{0ex}{}{n}{n/2})$, when n even, and $((n+1)/2)(\genfrac{}{}{0ex}{}{n}{(n+1)/2})$, when n is odd. The expression given in Table 4 is equivalent.

3 As for some models, we need to consider lower previsions.

Additional information

Funding

This work was carried out in the framework of the Labex MS2T, funded by the French Government, through the National Agency for Research – Agence Nationale de la Recherche (Reference ANR-11-IDEX-0004-02). We also acknowledge the financial support of project PGC2018-098623-B-I00 from the Ministry of Science, Innovation and Universities of Spain – Ministerio de Ciencia e Innovación and project GRUPIN/IDI/2018/000176 from the Department of Employment, Industry and Tourism of Asturias – Gobierno del Principado de Asturias.

Notes on contributors

Ignacio Montes

Ignacio Montes received the B.Sc. degree in mathematics in 2009 and the M.Sc. degree in 2010 both from the University of Oviedo, Spain. In 2014, he earned a Ph.D. degree in Mathematics and Statistics. After finishing his PhD, he spent some months at the HEUSIASYC Research Unit of the Technologic University of Compiègne as a postdoctoral researcher. After this short stay in France, he worked at the Carlos III University of Madrid as an Assistant Professor. Currently, he is an Associate Professor at the Department of Statistics and Operations Research at the University of Oviedo. Dr. Montes belongs to the UNIMODE (UNcertainty and Imprecision MOdelling in Decision making) research unit, and his main research interests include preference modelling with imprecise probabilities, stochastic orderings and dependence modelling with copulas.

Enrique Miranda

Enrique Miranda is an Associate Professor in the Department of Statistics and Operations Research at the University of Oviedo, Spain. He was formerly at Rey Juan Carlos University in Madrid. He currently serves as an Area Editor at the International Journal of Approximate Reasoning, and has been a member of the Executive Committee of the Society for Imprecise Probability: Theories and Applications (SIPTA) since 2009. He is the author of 55 publications included in the Journal Citation Reports. His research interests include foundations of uncertainty modelling, and in particular models of non-additive measures, random sets and imprecise probabilities.

Sébastien Destercke

Sébastien Destercke graduated in 2004 as an engineer from the Faculté Polytechnique de Mons in Belgium. In 2008, he earned a Ph.D. degree in computer science from Université Paul Sabatier, in Toulouse (France). He briefly worked in the French agricultural research centre working for international development, before becoming a CNRS researcher in the Heudiasyc Laboratory, in Compiàgne. His main research interests are in the fields of decision making and uncertainty reasoning (modeling, propagating, learning) with imprecise probabilistic models.