A Behavioral Interpretation of Belief Functions

Abstract

Shafer's belief functions were introduced in the seventies of the previous century as a mathematical tool in order to model epistemic probability. One of the reasons that they were not picked up by mainstream probability was the lack of a behavioral interpretation. In this paper, we provide such a behavioral interpretation and re-derive Shafer's belief functions via a betting interpretation reminiscent of the classical Dutch Book Theorem for probability distributions. We relate our betting interpretation of belief functions to the existing literature.

How to assess coherent beliefs: A comparison of different notions of coherence in Dempster-Shafer theory of evidence

“…Limiting to a finite Ω and E = P(Ω), condition coherence-4 has been introduced in [31], while the case of a finite Ω and an arbitrary E has been considered in [40]. Again, limiting to a finite Ω and E = P(Ω), it is easy to show that our condition coherence-4 can be expressed in terms of the notion of B-consistency for a coherent betting function R : R Ω → {0, 1} introduced in [32]. Nevertheless, our condition coherence-4 is stronger than all conditions proposed in [31,32,40] since no assumption is made neither on Ω nor on E. We point out that a dual version of conditions coherence-1 and coherence-3 can be traced back to [3], where a more general condition dealing with a conditional plausibility assessment is considered.…”

“…Again, limiting to a finite Ω and E = P(Ω), it is easy to show that our condition coherence-4 can be expressed in terms of the notion of B-consistency for a coherent betting function R : R Ω → {0, 1} introduced in [32]. Nevertheless, our condition coherence-4 is stronger than all conditions proposed in [31,32,40] since no assumption is made neither on Ω nor on E. We point out that a dual version of conditions coherence-1 and coherence-3 can be traced back to [3], where a more general condition dealing with a conditional plausibility assessment is considered. This is due to duality between belief and plausibility functions (see, e.g., [29,46]) and the identification of E i with E i |Ω.…”

The infinite epistemic regress problem has no unique solution

“…Lack of knowledge leads to a difference between buying and selling prices. In Kerkvliet and Meester (2017) it is shown that under mild conditions, this interpretation of epistemic probability automatically leads to belief functions. So if one formulates epistemic probabilities in terms of betting odds, belief functions are natural candidates for describing them.…”

“…Several authors have formulated axioms for epistemic probability different from the classical ones: notably Cohen (1977), Walley (1991) and Shafer (1976Shafer ( , 1981Shafer ( , 2008; see also Haenni (2009), Kerkvliet and Meester (2017) and Kerkvliet and Meester (2019). Especially the approach of Shafer has been rather influential.…”

The Limits of Bayesian Thinking in Court

“…Hence, not all expressions of belief can be captured by probability theory. Epistemic uncertainty is, simply, more complicated than what probability theory allows for, and a more general setup for dealing with uncertainty might be called for; see, for instance, Cohen (1977), Kerkvliet and Meester (2017), Kerkvliet and Meester (2019), and Shafer (1976).…”