How is mass calculated in example 2: Signal-light problem. I understand the mass of the subsets are added, i.e.
m(red or yellow)=mass(red)+mass(yellow)=0.35+0.25=0.6
m(red or green)=mass(red)+mass(green)=0.35+0.15=0.5
m(yellow or green)=mass(yellow)+mass(green)=0.25+0.15=0.4
But in each case, the combined masses are divided by 10, please help me understand why?
Dempster–Shafer theory
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). Belief in a hypothesis is constituted by the sum of the masses of all sets enclosed by it. It is the amount of belief that directly supports a given hypothesis or a more specific one, forming a lower bound. 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, i.e. it “could possibly be the true state of the system” up to that value, because there is only so much evidence that contradicts that hypothesis. Plausibility (denoted by Pl) is defined to be Pl(p) = 1 − 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 and a plausibility of 0.8 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 your system.
The null 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 − m (Dead) and the “Dead” plausibility is 1 − m (Alive). 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) + m(Dead) + 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):
Events of this kind would not be modeled as disjoint sets 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) = 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't be assigned to any of the other states, which here means evidence that says there is a light but doesn't 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 Null 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.
