Project: Concepts of independence for choice functions
Description
In recent years, imprecise-probabilistic choice functions have gained growing interest, primarily from a theoretical point of view. These versatile and expressive uncertainty models have demonstrated their capacity to represent decision-making scenarios that extend beyond simple pairwise comparisons of options, accommodating situations of indecision as well. They generalise many other uncertainty models, such as probability measures, and even sets of them, and as such they may potentially give a unified view of (aspects of) imprecise probability theory.
A choice function maps any (finite) set of possible decisions, called 'option set', to a subset of admissible (or non-rejected) decisions. The idea is that a choice function identifies the decisions that are not rejected from within every option set.
An important aspect of any theory of uncertainty is (stochastic) independence. For (classical) probabilities, this is necessarily a symmetric concept: under weak assumptions, if uncertain variable X is independent to uncertain variable Y, then necessarily is Y independent to X. For imprecise probabilities, this is no longer the case, and we differentiate between 'irrelevance' (which is the asymmetric version) and 'independence' (which is irrelevance assumed it both directions). Moreover, there further choices to be made about the concept of irrelevance or independence. To see this, consider a set of probability mass functions M. One choice to be made is: If M is to be called 'independent', should M only consist of probability mass functions that are independent in the standard sense, or should the lower bounds of the associated expectation operators be invariant under conditioning? This leads to the different concepts of strong independence and epistemic independence.
Recently, epistemic independence for choice functions has been studied ("Arthur Van Camp, Kevin Blackwell & Jason Konek, Independent natural extension for choice functions"), but strong independence for choice functions has not received attention so far. Due to the nature of choice function, and its rich representation theorems, different notions of "strong" independence will be exist.
The aim of this project is to do a preliminary study of the strong independence notions for choice functions, to characterise them, and compare to the extent possible. Depending on the student's interests, we may in a later stage either try to characterise the inference mechanism behind some of the independence concepts (the 'natural extension'), or try to find an efficient algorithm that calculates which options are rejected from within a given option set, under independence. This work will be important for later work about Bayesian networks with choice functions as local models.
This project requires mathematical interest and potentially algorithmic interest, depending on the student's interest.
Literature
Introduction to choice functions: https://arxiv.org/pdf/1903.00336.pdf
Overview of independence concepts for lower previsions: http://bellman.ciencias.uniovi.es/~emiranda/indnatex.pdf
Epistemic independence for choice functions: https://arthurvancamp.github.io/journal/ijar2023a/AVC-IJAR2023a-paper.pdf
Details
- Supervisor
-
Arthur van Camp
- Secondary supervisor
-
Cassio de Campos
- Interested?
- Get in contact