The concept of bimodal constraint/distribution
In the bimodal constraint,
R is a bimodal distribution of the form
with the understanding that Prob (X is At) is Pf. (Zadeh, 2002), that is, Pf is a granular value of Prob (X is Af), i = 1,...,n. (See next section for the definition of granular value.)
To clarify the meaning of a bimodal distribution it is expedient to start with an example. I am considering buying Ford stock. I ask my stockbroker, "What is your perception of the near-term prospects for Ford stock?" He tells me, "A moderate decline is very likely; a steep decline is unlikely; and a moderate gain is not likely." My question is: What is the probability of a large gain?
Information provided by my stockbroker may be represented as a collection of ordered pairs:
• Price: ((unlikely, steep decline), (very likely, moderate decline), (not likely, moderate gain)).
In this collection, the second element of an ordered pair is a fuzzy event or, generally, a possibility distribution, and the first element is a fuzzy probability. The expression for Price is an example of a bimodal distribution.
The importance of the concept of a bimodal distribution derives from the fact that in the context of human-centric systems most probability distributions are bimodal. Bimodal distributions can assume a variety of forms. The principal types are Type 1, Type 2 and Type 3 (Zadeh, 1979a; 1979b; 1981a). Type 1, 2 and 3 bimodal distributions have a common framework but differ in important detail (see Figure 6.12). A bimodal distribution may be viewed as an important generalization of standard probability distribution. For this reason, bimodal distributions of Type 1, 2, 3 are discussed in greater detail in the following.
• Type 1 (default): X is a random variable taking values in U
Aj,...,A„, A are events (fuzzy sets), pi = Prob(X is A), i = 1,...,n,
2i pj is unconstrained,
Pi = granular value of Pj.
BD: bimodal distribution: ((P:, A1),...,(Pn, An)) or, equivalently,
Figure 6.12 Type 1 and Type 2 bimodal distributions
Problem: What is the granular probability, P, of A? In general, this probability is fuzzy-set-valued.
A special case of bimodal distribution of Type 1 is the basic bimodal distribution (BBD). In BBD, X is a real-valued random variable, and X and P are granular (see Figure 6.13).
• Type 2 (fuzzy random set): X is a fuzzy-set-valued random variable with values
A1,...,An (fuzzy sets),
pi = Prob(X = A;), i = 1,...,n,
Pi : granular value of pi.
BD: X isrs (P1 A1 + - + PnAn),
pi = 1.
Problem: What is the granular probability, P, of A? P is not definable. What are definable are (a) the expected value of the conditional possibility of A given BD, and (b) the expected value of the conditional necessity of A given BD.
• Type 3 (Dempster-Shafer) (Dempster, 1967; Shafer, 1976; Schum, 1994): X is a random variable taking values X:,...,Xn with probabilities p1,...,pn
Xj is a random variable taking values in Ai, i = 1,...,n.
Probability distribution of Xt in Ai, i = 1,...,n, is not specified,
X isp (p1X1 + - + pnX„).
Figure 6.13 Basic bimodal distribution
Problem: What is the probability, p, that X is in A? Because probability distributions of the Xt in the At are not specified, p is interval valued. What is important to note is that the concepts of upper and lower probabilities break down when the A, are fuzzy sets (Zadeh, 1979a).
Note: In applying Dempster-Shafer theory, it is important to check on whether the data fit Type 3 model. In many cases, the correct model is Type 1 rather than Type 3.
The importance of bimodal distributions derives from the fact that in many realistic settings a bimodal distribution is the best approximation to our state of knowledge. An example is assessment of degree of relevance, since relevance is generally not well defined. If I am asked to assess the degree of relevance of a book on knowledge representation to summarization, my state of knowledge about the book may not be sufficient to justify an answer such as 0.7. A better approximation to my state of knowledge may be "likely to be high." Such an answer is an instance of a bimodal distribution.
What is the expected value of a bimodal distribution? This question is considered in the section on protoformal deduction rules.