Unifying neighbourhood and distortion models: part I – new results on old models

ABSTRACT

Neighbourhoods of precise probabilities are instrumental to perform robustness analysis, as they rely on very few parameters. Many such models, sometimes referred to as distortion models, have been proposed in the literature, such as the pari mutuel model, the linear vacuous mixtures or the constant odds ratio model. This paper is the first part of a two paper series where we study the sets of probabilities induced by such models, regarding them as neighbourhoods defined over specific metrics or premetrics. We also compare them in terms of a number of properties: precision, number of extreme points, n-monotonicity, behaviour under conditioning, etc. This first part tackles this study on some of the most popular distortion models in the literature, while the second part studies less known neighbourhood models and summarises our findings.

KEYWORDS:

• Neighbourhood models
• distorted probabilities
• pari mutuel model
• linear vacuous mixtures
• constant odds ratio

Acknowledgments

We would like to thank the reviewers for their interesting and insightful comments.

Disclosure statement

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

Notes

1 Walley's theory deals only with bounded random variables on $\mathcal{X}$, also called gambles; since in this paper we work with finite spaces, the restriction of boundedness is trivially satisfied.

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 (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 and project IDI/2018/000176 from the Department of Employment, Industry and Tourism of 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 Ph.D., 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 Compiégne. His main research interests are in the fields of decision making and uncertainty reasoning (modeling, propagating, learning) with imprecise probabilistic models.