Finite approximations to coherent choice

TL;DR

This paper investigates how approximating loss functions and credal sets impacts choice functions, providing bounds on these effects even when credal sets are non-convex and non-closed, with specific focus on pairwise choice.

Contribution

It introduces bounds on approximation effects for non-convex, non-closed credal sets and simplifies pairwise choice approximation by focusing on extreme points.

Findings

Bounds on approximation effects for non-convex credal sets

Approximation of extreme points suffices in pairwise choice

Practical limitations on credal set approximation

Abstract

This paper studies and bounds the effects of approximating loss functions and credal sets on choice functions, under very weak assumptions. In particular, the credal set is assumed to be neither convex nor closed. The main result is that the effects of approximation can be bounded, although in general, approximation of the credal set may not always be practically possible. In case of pairwise choice, I demonstrate how the situation can be improved by showing that only approximations of the extreme points of the closure of the convex hull of the credal set need to be taken into account, as expected.