Integral Representation of Coherent Lower Previsions by Super-Additive Integrals

Serena Doria,Radko Mesiar,Adam Šeliga

doi:10.3390/axioms9020043

Serena Doria, Radko Mesiar + Show 1 more

Open Access

https://doi.org/10.3390/axioms9020043

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Abstract

Coherent lower previsions generalize the expected values and they are defined on the class of all real random variables on a finite non-empty set. Well known construction of coherent lower previsions by means of lower probabilities, or by means of super-modular capacities-based Choquet integrals, do not cover this important class of functionals on real random variables. In this paper, a new approach to the construction of coherent lower previsions acting on a finite space is proposed, exemplified and studied. It is based on special decomposition integrals recently introduced by Even and Lehrer, in our case the considered decomposition systems being single collections and thus called collection integrals. In special case when these integrals, defined for non-negative random variables only, are shift-invariant, we extend them to the class of all real random variables, thus obtaining so called super-additive integrals. Our proposed construction can be seen then as a normalized super-additive integral. We discuss and exemplify several particular cases, for example, when collections determine a coherent lower prevision for any monotone set function. For some particular collections, only particular set functions can be considered for our construction. Conjugated coherent upper previsions are also considered.

Highlights

One of the major tools supporting the standard probability theory is the notion of expectation of a random variable
A construction method for coherent lower previsions is proposed; to do that we start by proving a super-additivity property of the collection integral and we need to restrict to a special class of collection integrals that allow one to extend their domain to all functions while preserving super-additivity
In this paper we have introduced, discussed and exemplified a new construction method for coherent lower previsions acting on finite universe Ω, and, by duality, for coherent upper previsions

Summary

Introduction

One of the major tools supporting the standard probability theory is the notion of expectation of a random variable. As another non-linear integral recall Shilkret integral [5], which is well defined for any of the above mentioned monotone set functions, but it coincides with the Lebesgue integral only in the case when Dirac measures are considered Both Choquet and Shilkret integrals are based on the standard real operations of addition and multiplication (what is not the case of Sugeno integral [6], for example). If collections C are classes of disjoint subsets of Ω, the super-additive integral can be defined for any monotone set function μ since the underlying collection integral is shift-invariant and so it can be used to construct a coherent lower prevision.

Preliminaries

Super-Additive Integral Defined by Shift-Invariant Collection Integral

Construction of Coherent Lower and Upper Previsions And Examples

Conclusions and Discussion