Sherlock Holmes Doesn't Play Dice: The mathematics of uncertain reasoning when something may happen, that one is not even able to figure out

TL;DR

The paper addresses the inadequacy of traditional Probability Theory to express radical uncertainty about events not yet imagined. It advances Evidence Theory (ET), augmented by the Transferable Belief Model (TBM), to represent unknown unknowns with $m(\emptyset)>0$ and to model suspension of judgement with $m(\Theta)\ge0$, while maintaining a flexible frame of discernment $\Theta$ and overlapping hypotheses $A_i$. It then clarifies the relationships between ET, Probability Theory, and Information Theory, showing how ET encompasses Bayes' rule in the singleton, zero-unknowns limit and how imprecise probabilities and sub-additivity extend its applicability. Finally, it surveys coherence-based decision frameworks (CSNs, ENs, VNs, OWNs) and argues that open-world networks provide a robust path for integrating novel evidence into decision making, with potential for human–machine collaboration in uncertain environments.

Abstract

While Evidence Theory (also known as Dempster-Shafer Theory, or Belief Functions Theory) is being increasingly used in data fusion, its potentialities in the Social and Life Sciences are often obscured by lack of awareness of its distinctive features. In particular, with this paper I stress that an extended version of Evidence Theory can express the uncertainty deriving from the fear that events may materialize, that one is not even able to figure out. By contrast, Probability Theory must limit itself to the possibilities that a decision-maker is currently envisaging. I compare this extended version of Evidence Theory to sophisticated extensions of Probability Theory, such as imprecise and sub-additive probabilities, as well as unconventional versions of Information Theory that are employed in data fusion and transmission of cultural information. A further extension to multi-agent interaction is outlined.

Figures (3)

• Figure 1: Transformation of imprecise probabilities defined on singletons into single-valued probabilities defined on intervals. Step-wise cumulative lower probability function $F_*$ and cumulative upper probability function $F^*$ identify intervals $A_i$ with probability $p(A_i) = p^* - p_*$. Notably, for any $(i, i\,')$ it may happen that $A_i \cap A_{i\:'} \neq \emptyset$.
• Figure 2: In (a), the classical IT framework where single characters $\{A_i\}$ are first coded into sets $A_i$, then transmitted through a noisy channel which may generate intersections between these sets, and finally decoded. In (b), the fusion of partially overlapping information originating from different sources. The original overlap may be enhanced by coding and further enhanced by transmission through a noisy channel.
• Figure 3: The possibility sets corresponding to a cyclic hypergraph (on the left) and an acyclic hypergraph (on the right). For each hypergraph, the vertices of hyperedges are the elements entailed by the possibility sets, denoted by Greek letters. For instance, the possibility set $A_1 = \{\alpha, \eta\}$ corresponds to a hyperedge (a segment) of vertices $\alpha$ and $\eta$. The cyclic hypergraph on the left is made of hyperedges $A_1 = \{\alpha, \eta\}$, $A_2 = \{\beta, \zeta\}$, $A_3 = \{\delta, \theta\}$, $A_4 = \{\epsilon, \zeta\}$, $A_5 = \{\beta, \gamma, \delta\}$, $A_6 = \{\delta, \epsilon, \eta\}$. The acyclic hypergraph on the right can be obtained by removing $A_2$ and $A_4$ and adding $A_6 = \{\beta, \delta, \epsilon\}$ and $A_7 = \{\beta, \epsilon, \zeta\}$. Thus, the acyclic hypergraph has been obtained by coarsening the FoD. Freely redrawn from shafer-shenoy-90XIV.