# 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 *A*_{t}) is *P _{f}.* (Zadeh, 2002), that is,

*P*is a granular value of Prob (X is A

_{f }_{f}),

*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), *p _{i} =* Prob(X is

*A), i = 1,...,n,*

2* _{i} pj* is unconstrained,

*P _{i} =* granular value of

*P*

_{j}.BD: bimodal distribution: ((P_{:}, *A _{1}),...,(P_{n}, A_{n}))* 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

*A _{1},...,A_{n}* (fuzzy sets),

*p _{i} =* Prob(X = A

_{;}),

*i = 1,...,n,*

*P _{i}* : granular value of

*p*

_{i}.BD: X isrs (P_{1} A_{1} + - + *P _{n}A_{n}),*

*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_{:},...,X_{n} with probabilities *p*1*,...,pn*

*Xj* is a random variable taking values in *A _{i}, i =* 1,...,n.

Probability distribution of *X _{t}* in

*A*1,...,n, is not specified,

_{i}, i =X isp (p_{1}X_{1} + - + *pnX„).*

*Figure 6.13* Basic bimodal distribution

Problem: What is the probability, *p,* that *X* is in *A?* Because probability distributions of the *X _{t}* in the

*A*are not specified,

_{t}*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.