207.

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of the mass of evidence which is committed to the ψ and to its subsets (if any), while plausibility of ψ, denoted by Pl, is defined as one minus the Bel committed to ψ’s contradiction, denoted by Bel(~ψ). It is more convenient to explain such definitions in terms of mathematical representation:

(7.18)

For instance, using the example discussed earlier, the Bel for hypothesis {B, F} can be represented by:

(7.19)

and it can be found that Bel(ψ) is a measure of the total amount of belief in ψ including all ψ’s subsets. When ψ is a singleton, Bel(ψ)=m(ψ). For instance, Bel({B}}=m({B}). The plausibility for hypothesis {B, F} can be represented by:

(7.20)

Bel(ψ) can thus be interpreted as the minimum amount of evidence that a pixel is properly labelled with ψ, while Pl(ψ) can be interpreted as the maximal extent to which the current evidence could allow one to belief ψ (Shafer, 1979). Note that Pl(ψ) provides another choice for us to make a decision when there is no evidence, which exactly supports hypothesis ψ.

If a pixel belongs to ψ then the probability P(ψ) may lie somewhere between the interval:

(7.21)

with the range Pl(ψ)–Bel(ψ), which in turn means that

(7.22)

That is, Pl(ψ) and Bel(ψ) provide the upper bound and lower bound of the probability of the subset. Recall from Equation (7.15) that the belief, plausibility, and interval for each class are:

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