8.5.1. Necessity and Sufficiency Measures

An evidence e can usually be in two states: absent or present when P(e) = 0 or P(e) = 1, it is of no practical interest. Either way there is nothing to observe. For h, it is the same. Therefore, we shall assume 0 < P(e) < 1 and 0 < P(h) < 1.

To study the necessity and sufficiency measures of e for h, we need to explore the influence that a state of e has on h. If the state of e makes h more plausible, we say that the state of e encourages h. If it makes h less plausible, we say that the state of e discourages h. If it neither encourages nor discourages h, then the state of e has no influence on h, or e and h are independent of each other.

For the necessity measure, we first explore how the absence of e influences h. From O(h‘ ) =  · O(h)

we define N =

0 ² N ² •

Similarly, we have

For S = • P(h‘e) = 1 , eÆ h \ e is sufficient for h

1 < S < • P(h‘e) > P(h) , e encourages h

S = 1 No influence

0 < S < 1 P(h‘e) < P(h) , e discourages h

S = 0 P(h‘e) = 0 , eÆ  ,  Æ e \ e is sufficient for  .

From the above analysis, it is clear that N and S are the measures for necessity and sufficiency, respectively. N, S and O(h) needed to evaluate O(h‘ ) and O(h‘e) are provided by domain experts. Quite often, instead of directly supply N and S domain experts may supply values of P(e‘h) and P(e‘ ). This implies that observing evidential probabilities under a certain hypothesis h or  .

N =  =

S =  .