2-Monotone outer approximations of coherent lower probabilities

Abstract

We investigate the problem of approximating a coherent lower probability on a finite space by a 2-monotone capacity that is at the same time as close as possible while not including additional information. We show that this can be tackled by means of a linear programming problem, and investigate the features of the set of undominated solutions. While our approach is based on a distance proposed by Baroni and Vicig, we also discuss a number of alternatives: quadratic programming, extensions of the total variation distance, and the Weber set from game theory. Finally, we show that our work applies to the more general problem of approximating coherent lower previsions.

Outer approximating coherent lower probabilities with belief functions

“…In [3], the authors proposed to outer approximate a coherent lower probability by means of a 2-monotone one. Following this idea, in a previous contribution [27] we studied in detail how to obtain outer approximations in terms of 2monotone lower probabilities as well as their features. These have a number of advantages over coherent lower and upper probabilities: for instance, the property of 2-monotonicity determines a unique extension as an expectation operator (the Choquet integral [5]), and their associated credal set has a neat structure in terms of its extreme points [32].…”

Outer Approximations of Coherent Lower Probabilities Using Belief Functions

“…We would like to thank Sébastien Destercke, Serafín Moral, John Klein, Anne-Laure Jousselme and the anonyous reviewers for some helpful comments. This paper presents an extension, with proofs and additional results, of a previous work [27] presented at the BELIEF'2018 conference.…”

“…Then, the p-box dened from Eq. (27) is 1 1 In Figure 2 we have depicted this p-box emphasizing its focal events. Figure 2.…”

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

“…Note that, although P K and P T V induce the same p-box, they are not the same coherent lower probability: P K P T V , as we have seen in Example 2. If we follow the notation in [4,20], we deduce from [21,Prop. 16] that P K is the unique undominated outer approximation of P T V in terms of p-boxes.…”

“…Finally, a natural next step would be to look at the distortion of imprecise probability models. This could be done in two manners: distorting each probability measure compatible with the imprecise model, and then taking the lower envelope of the union of the credal sets that result [22]; or to consider directly a distance between imprecise probability models, as in [20,21]. (1) x∈X P ({x}) = 1:…”