Editing Dempster–Shafer theory (section)

===Belief and plausibility===
Shafer's formalism starts from a set of ''possibilities'' under consideration, for instance numerical values of a variable, or pairs of linguistic variables like "date and place of origin of a relic" (asking whether it is antique or a recent fake). A hypothesis is represented by a subset of this ''frame of discernment'', like "(Ming dynasty, China)", or "(19th century, Germany)".<ref name="SH76"/>{{rp|p.35f.}}

Shafer's framework allows for belief about such propositions to be represented as intervals, bounded by two values, ''belief'' (or ''support'') and ''plausibility'':

:''belief'' ≤ ''plausibility''.

In a first step, subjective probabilities (''masses'') are assigned to all subsets of the frame; usually, only a restricted number of sets will have non-zero mass (''focal elements'').<ref name="SH76"/>{{rp|39f.}} ''Belief'' in a hypothesis is constituted by the sum of the masses of all subsets of the hypothesis-set. It is the amount of belief that directly supports either the given hypothesis or a more specific one, thus forming a lower bound on its probability.  Belief (usually denoted ''Bel'') measures the strength of the evidence in favor of a proposition ''p''.  It ranges from 0 (indicating no evidence) to 1 (denoting certainty).  ''Plausibility'' is 1 minus the sum of the masses of all sets whose intersection with the hypothesis is empty. Or, it can be obtained as the sum of the masses of all sets whose intersection with the hypothesis is not empty. It is an upper bound on the possibility that the hypothesis could be true, because there is only so much evidence that contradicts that hypothesis.  Plausibility (denoted by Pl) is thus related to Bel by Pl(''p'')&nbsp;=&nbsp;1&nbsp;&minus;&nbsp;Bel(~''p'').  It also ranges from 0 to 1 and measures the extent to which evidence in favor of ~''p'' leaves room for belief in ''p''.

For example, suppose we have a belief of 0.5 for a proposition, say "the cat in the box is dead." This means that we have evidence that allows us to state strongly that the proposition is true with a confidence of 0.5. However, the evidence contrary to that hypothesis (i.e. "the cat is alive") only has a confidence of 0.2. The remaining mass of 0.3 (the gap between the 0.5 supporting evidence on the one hand, and the 0.2 contrary evidence on the other) is "indeterminate," meaning that the cat could either be dead or alive.  This interval represents the level of uncertainty based on the evidence in the system.

{| class="wikitable"
! Hypothesis !! Mass !! Belief!! Plausibility
|-
| Neither (alive nor dead) || 0 || 0 || 0
|-
| Alive || 0.2 || 0.2 || 0.5
|-
| Dead || 0.5 || 0.5 || 0.8
|-
| Either (alive or dead) || 0.3 || 1.0 || 1.0
|}

The "neither" hypothesis is set to zero by definition (it corresponds to "no solution"). The orthogonal hypotheses "Alive" and "Dead" have probabilities of 0.2 and 0.5, respectively. This could correspond to "Live/Dead Cat Detector" signals, which have respective reliabilities of 0.2 and 0.5. Finally, the all-encompassing "Either" hypothesis (which simply acknowledges there is a cat in the box) picks up the slack so that the sum of the masses is 1. The belief for the "Alive" and "Dead" hypotheses matches their corresponding masses because they have no subsets; belief for "Either" consists of the sum of all three masses (Either, Alive, and Dead) because "Alive" and "Dead" are each subsets of "Either". The "Alive" plausibility is 1&nbsp;−&nbsp;''m'' (Dead): 0.5 and the "Dead" plausibility is 1&nbsp;−&nbsp;''m'' (Alive): 0.8.  In other way, the "Alive" plausibility is ''m''(Alive) + ''m''(Either) and the "Dead" plausibility is ''m''(Dead) + ''m''(Either). Finally, the "Either" plausibility sums ''m''(Alive)&nbsp;+&nbsp;''m''(Dead)&nbsp;+&nbsp;''m''(Either). The universal hypothesis ("Either") will always have 100% belief and plausibility—it acts as a [[checksum]] of sorts.

Here is a somewhat more elaborate example where the behavior of belief and plausibility begins to emerge. We're looking through a variety of detector systems at a single faraway signal light, which can only be coloured in one of three colours (red, yellow, or green):

{| class="wikitable"
! Hypothesis !! Mass !! Belief !! Plausibility
|-
| None || 0 || 0 || 0
|-
| Red || 0.35 || 0.35 || 0.56
|-
| Yellow || 0.25 || 0.25 || 0.45
|-
| Green || 0.15 || 0.15 || 0.34
|-
| Red or Yellow || 0.06 || 0.66 || 0.85
|-
| Red or Green || 0.05 || 0.55 || 0.75
|-
| Yellow or Green || 0.04 || 0.44 || 0.65
|-
| Any || 0.1 || 1.0 || 1.0
|}

Events of this kind would not be modeled as distinct entities in probability space as they are here in mass assignment space. Rather the event "Red or Yellow" would be considered as the union of the events "Red" and "Yellow", and (see [[probability axioms]]) ''P''(Red or Yellow) ≥ ''P''(Yellow), and ''P''(Any)&nbsp;=&nbsp;1, where ''Any'' refers to ''Red'' or ''Yellow'' or ''Green''. In DST the mass assigned to ''Any'' refers to the proportion of evidence that can not be assigned to any of the other states, which here means evidence that says there is a light but does not say anything about what color it is. In this example, the proportion of evidence that shows the light is either ''Red'' or ''Green'' is given a mass of 0.05. Such evidence might, for example, be obtained from a R/G color blind person. DST lets us extract the value of this sensor's evidence. Also, in DST the empty set is considered to have zero mass, meaning here that the signal light system exists and we are examining its possible states, not speculating as to whether it exists at all.