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### Project: Algorithms for forward irrelevance with choice functions

##### Description

##### Details

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.

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.

Consider the following problem: there are n uncertain variables X_1, ..., X_n, each taking values in a finite possibility space. Your (marginal) beliefs about each X_k is described using a choice function C_k. Moreover, you assess that your beliefs about X_k will not be influenced by observing the value that the previous variables X_1, ... X_k-1 take: this assessment is called 'forward irrelevance'. If the choice functions C_k are probability measures, then this forward irrelevance assessment reduces to the standard independence assumption, and will therefore be symmetric: your beliefs about X_1 won't be affected by observing the value that X_2 takes, for instance. In the imprecise setting, however, such symmetry is no longer guaranteed. Forward irrelevance is important because it is one of the two basic operations used in imprecise Bayesian networks: it is the structural assumption behind the network

X_1 -> X_2 -> ... -> X_n.

The aim of this master project is twofold: (i) to expand the definition and results "Arthur Van Camp & Enrique Miranda, Modelling epistemic irrelevance with choice functions" in about forward irrelevance for 2 variables to a finite number of them, and (ii) to find an efficient algorithm that allows to calculate inference problems related to forward irrelevance, which will be useful for later work about Bayesian networks with choice functions as local models. I expect that the mathematical results required in (i) will be rather straightforward, but they will be interesting nonetheless.

This project requires mathematical interest and algorithmic skills.

Literature

Introduction to choice functions: https://arxiv.org/pdf/1903.00336.pdf

Forward irrelevance for lower previsions: http://bellman.ciencias.uniovi.es/~emiranda/fipfine.pdf

Another study of forward irrelevance (on an infinite sequence of variables) for lower previsions: https://users.ugent.be/~jdbock/documents/JDB-2018-JMAA-paper.pdf

Forward irrelevance for two choice functions: https://arthurvancamp.github.io/journal/ijar2020a/AVC-IJAR2020a-paper-cor.pdf

- Supervisor
- Arthur van Camp
- Secondary supervisor
- Cassio de Campos
- Interested?
- Get in contact