205.

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7.4.1 Concept development

One important aspect of the generality of the D-S theory is avoidance of the Bayesian restriction that commitment for belief to a hypothesis implies commitment of the remaining belief to its opponents, i.e. belief in event w is equivalent to P(w) so that belief in the concept ‘Not w’ is equal to 1−P(w). In many real applications, evidence partially in favour of one particular hypothesis is not necessarily evidence against alternative hypotheses. In D-S theory, complete knowledge is represented as unity, i.e. the sum of all labelling possibilities is equal to one, which includes the case of uncertainty. In other words, the measure of belief assigned to each hypothesis may be less than one. If there is some evidence in favour of hypothesis w, then the remaining belief (which is equivalent to the value of uncertainty) will be assigned to the whole hypothesis space, not simply to its opponents. A simple example may illustrate this idea.

Suppose that an analyst is trying to classify an image, which involves labelling pixels as belonging to one of three classes {B, F, P}, where B denotes bare soil, F denotes forest, and P denotes pasture. In D-S theory, this set {F, B, P} is generally called a frame of discernment, and is denoted by the symbol θ. The number of all possible subsets of the frame of discernment θ is equal to 2^|θ|, where |θ| denotes the number of one-element subsets (called ‘singletons’). In our case, there is a total of three singletons in |θ|, therefore the total number of subsets of θ is 2³=8, as shown in Figure 7.3. Note that the empty set {} is one of these subsets, but is not displayed in Figure 7.3.

D-S theory uses a number within the range of [0,1] to indicate the degree of belief in a hypothesis, given a piece of evidence. The number that expresses the degree to which the evidence supports a particular hypothesis is represented by a function called the basic probability assignment or bpa; the resulting quantity of evidence is generally called the measure of mass or mass of evidence, and is denoted by m(ψ), where ψ is any subset of θ. The sum of m(ψ), for ψ, must be equal to 1 (this is a rather important constraint). In our case, ψ can be one of the sets shown in Figure 7.3. If m(ψ)=q and no other belief is assigned to the subsets of θ, then m(θ)= 1−q, that is the remaining belief is assigned to the frame of discernment θ, rather than ψ’s opponents. This point has already been emphasised in the previous section. An example is given below.

Suppose that a source of evidence indicates that a particular pixel of interest has 30% support for the belief that the pixel belongs to class {B}, 30% support for the belief that the pixel belonging to class {F}, and 40% support for the belief that the pixel belongs to class {P}. Note again that the sum of all m(ψ) must equal 1. If one assumes that the source of evidence is fully reliable (i.e. the uncertainty is zero) then, using the theory of evidence symbolism, one obtains:

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