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 rst part of a two paper series where we study the sets of probabilities induced by such models, regarding them as neighbourhoods dened over specic 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 rst 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 ndings.

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

“…As with some of the models studied in the rst part of this study (i.e., the PMM and LV models), we can see that the conditional model has an increased imprecision with respect to the original one because δ B = δ P0(B) > δ, this increase being between the one we observed in the PMM (δ B = δ P P M M (B) ) and that of the LV model (δ B = δ P LV (B) ); see Equations (16) and (19) of the rst paper [19].…”

“…As we have seen, P L1 is neither 2-monotone in gambles (see Example 4) nor 2-monotone in events for cardinalities greater n ≥ 12 (see Example 6). This means that for the general case there is not a simple procedure for determining the extreme points of B δ d L 1 (P 0 ), and in particular we cannot use the procedure based in the permutations, described in [19,Eq. (2)], for computing them.…”

“…Finally, all distorting functions satisfy Ax.4 and Ax.5, meaning that the open balls are convex and continuous. From [19,Prop. 1], this implies that the closed ball coincides with the credal set associated with the coherent lower probability.…”

“…As we recalled in the Introduction, there are a number of particular cases of coherent lower probabilities that may be of interest in practice. The rst of them is 2-monotonicity: not only it guarantees that the distortion model has a unique extension to gambles as a lower prevision, but also it allows us to use a simple formula [19,Eq. (2)] to compute the extreme points of the credal set.…”

“…Our motivations for studying neighbourhood models have already been discussed in the rst part of this study [19]. Here, let us simply recall that their interest is to provide simple models relying on very few parameters, which are:…”