Artificial Intelligence and Soft Computing: Behavioral ... - Arteimi.info

Formally, the set of all possible outcomes in a random experiment is called
the frame of discernment. Let n = |θ |, the cardinality of θ. Then all the
2 n subsets of theta are called the propositions in the present context. In the die
example, the proposition, “the-no-showing-i-is-even” is given by {2, 4, 6}.
In the DS theory, the probability masses are assigned to subsets of θ,
unlike Bayesian theory, where probability mass can be assigned to individual
elements (singleton subsets). When a knowledge-source of evidence assigns
probability masses to the propositions, represented by subsets of θ, the
resulting function is called a basic probability assignment (BPA).
Formally, a BPA is m
where m : 2 θ → [0,1]
where 0 ≤ m ( .) ≤ 1.0, m (φ
and ∑ m (x) = 1.0 (9.22)
x ⊆ θ
Definition 9.2: Subsets of θ, which are assigned nonzero probability
masses are called focal elements of θ.
Definition 9.3: A belief function [5-6] Bel (x), over θ, is defined by
Bel (x) = ∑ m (Y) (9.23)
Y ⊆ X
For example, if the frame of discernment θ contains mutually exclusive
subsets A, C and D, then
Bel ( { A, C, D } )
= m ({ A,C,D}) + m ({A, C}) + m ({A,D}) + m ({C,D}) + m ({a}) + m
({c}) + m ({d}).
In DS model, belief in a proposition is represented by the belief
interval. This is the unit interval [0,1], further demarcated by two points j and
k, k ≥ j. Suppose that the belief interval describes proposition A. Then the
sub-interval [o, j) is called Belief (A) and the subinterval (k,1] is called the
disbelief (A) and the remainder [j, k] is called Uncertainty (A). Belief (A) is