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Analytical Expressions of the Posterior Operator 129

Figure 6.1. Sub-intervals of Xaoft-

have a volume in common in the three-dimensional vector space of \ sojt. The
collection of all combinations of intervals R q (q = 1,...,4) constitutes the
domain of the veector

COMMENT 6.1: Proposition 6. 1 is a special case of Proposition 6.2. Indeed,
since the interval (soft) data of Equation 3.82 (p. 85) implies

dx aoftU (U denotes uniform pdf), Equation 6.3 reduces to Equation 6.1.
In some practical applications, the probability knowledge (Eq. 3.33) may

w

be available for a subset of x soft > hile the interval knowledge (Eq. 3.32)

is available for the remaining soft data points. In such a case one can

obtain the posterior pdf by combining Propositions 6.1 and 6.2. If knowledge

is provided about each element of the vector Xsoft > independentlyand identi-

cally distributed, then w e have
We already mentioned that the probability models above should always
be understood to apply in the appropriate context, which defines the current
state of knowledge. In several cases, it may not be possible to articulate an
expert's intuition, belief, or evidence S propositionally. Then, instead of trying
to describe the quality of S itself, one may describe the effects of 5 on the
expert by saying that after examination of the situation, the expert suggests a
specific probability law.