Decision-making with Belief Functions and pignistic probabilities

2005 ◽  
pp. 364-371 ◽  
Author(s):  
Nic Wilson
Keyword(s):  
Decision Making ◽  
Belief Functions

scholarly journals A Savage-style Utility Theory for Belief Functions

2018 ◽  
Author(s):  
Chunlai Zhou ◽  
Biao Qin ◽  
Xiaoyong Du
Keyword(s):  
Decision Making ◽  
Utility Function ◽  
State Space ◽  
Utility Theory ◽  
Preference Relation ◽  
Belief Function ◽  
The State ◽  
Independence Axiom ◽  
Belief Functions ◽  
Definition Of

In this paper, we provide an axiomatic justification for decision making with belief functions by studying the belief-function counterpart of Savage's Theorem where the state space is finite and the consequence set is a continuum [l, M] (l<M). We propose six axioms for a preference relation over acts, and then show that this axiomatization admits a definition of qualitative belief functions comparing preferences over events that guarantees the existence of a belief function on the state space. The key axioms are uniformity and an analogue of the independence axiom. The uniformity axiom is used to ensure that all acts with the same maximal and minimal consequences must be equivalent. And our independence axiom shows the existence of a utility function and implies the uniqueness of the belief function on the state space. Moreover, we prove without the independence axiom the neutrality theorem that two acts are indifferent whenever they generate the same belief functions over consequences. At the end of the paper, we compare our approach with other related decision theories for belief functions.

Imprecise and Indeterminate Probabilities

2017 ◽  
Author(s):  
Fabio Cozman
Keyword(s):  
Decision Making ◽  
Mathematical Models ◽  
Real Number ◽  
Belief Functions ◽  
Lower Upper ◽  
Credal Sets ◽  
History Of

This chapter offers a discussion of imprecision and indeterminacy in probability values; that is, there is a focus on situations where one does not attach a single real number to every possible event. There are several theories and mathematical models regarding such imprecise and indeterminate probabilities. Among these are, for instance, lower/upper or interval probabilities or expectations, Choquet capacities, belief functions and credal sets. The chapter reviews the history of some of these models and theories, and then summarizes their main technical assumptions. Some subtleties concerning conditioning and independence are also reviewed. The chapter concludes with a brief discussion of modeling and of criteria for decision-making (minimality, maximality, E-admissibility).

scholarly journals Classical probabilities and belief functions in legal cases

2020 ◽  
Vol 19 (1) ◽  
pp. 99-107
Author(s):  
Ronald Meester
Keyword(s):  
Decision Making ◽  
Criminal Law ◽  
Civil Law ◽  
Bayes Rule ◽  
Primary Concern ◽  
Belief Functions ◽  
Legal Cases ◽  
Legal Context ◽  
Classical Setting ◽  
The Individual

Abstract I critically discuss a recent suggestion in Nance (Belief Functions and Burdens of Proof. Law, Probability and Risk, 18:53–76, 2018) concerning the question which ratios of beliefs are appropriate when in criminal or civil cases one works with belief functions instead of classical probabilities. I do not call into question the use of belief functions themselves in this context, and I agree with in Nance (Belief Functions and Burdens of Proof. Law, Probability and Risk, 18:53–76, 2018) that so-called ‘uncommitted support’, possible in the framework of belief functions, should not be taken into account in a decision-theoretic framework. However, I argue against in Nance (Belief Functions and Burdens of Proof. Law, Probability and Risk, 18:53–76, 2018) in that, at least in criminal law, relative sizes of beliefs should not be used for decision-making at all. I will argue that only the individual, absolute beliefs should be considered. Since belief functions generalize classical probabilities, this position seems at first sight to conflict with the fact that odds are abundant when we use classical probabilities in a legal context. I will take the opportunity, then, to point out that also in the classical setting, odds are not our primary concern either. They are convenient since they appear, together with the likelihood ratio, in the odds form of Bayes’ rule. Apart from that, they do not have any individual significance. I also note that in civil law the conclusions might be different.

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