accuracy of a crop inventory database. The case- 
dependent property determines that the assignment of 
values to A in the database case could only be 
conducted on a case-by-case basis. 
In the case where an uncertainty value is generated 
from experts' statements, however, it is possible to 
provide a generic method for quantifying A, since 
experts' judgment is the dominant factor affecting the 
uncertainty value. Examining the judgment-making 
process, we can find that experts' confidence in making 
a judgment is a key factor that contributes to the UIU 
problem included in the judgment, and essentially, 
this confidence is mainly based on the sample size 
used by the expert in making the judgment. This 
analysis allows us to directly apply the approach 
addressed by Neapolitan (1990) for obtaining the 
uncertainty in probabilities . 
Consider first the case where a propositional variable, 
say D, has precisely two alternatives, d; is the presence 
of a particular event, and P(d;) is the probability of d's 
occurrence. If we let x be a variable which represents a 
possible value of P(d;), then the general formula for 
the beta distribution is given by 
(a * b 4 1)! 
B(a,b) = ---------—------- x à (1-x)b 
a! b! 
(9) 
where a, b >= 0. This function is a probability density 
function p(x) for the possible values of P(d;). For all a,b 
>= 0, it is proven that 
if u(x) = B(a, b) (10) 
then fi do =1 (11) 
0 
and P(d;) = Jx u(x) d(x) = (a + 1) / (a +b + 2) (12) 
0 
Neapolitan (1990) points out that the values of a and b 
are based on the expert's confidence with the estimate 
of probability P(d;), and a+b can be liken to the sample 
size Se essentially used by the expert in the estimation, 
namely: 
a + b =Se (13) 
Using equations (12) and (13), we can solve for a and b. 
Replacing the values of a, b and P(d;) in equation (9), 
we thus obtain a probability value that indicates the 
certainty of the estimated probability by the expert. The 
obtained probability value can be regarded as the 
accuracy of expert's knowledge in the case where the 
propositional variable has two alternatives. 
Furthermore, Neapolitan (1990) extends the case of two 
alternatives of the propositional variable to t 
alternatives, and concludes that the density function 
942 
for the possible value of P(di) is the Dirichlet 
distribution: 
(b; + a; +t-1)! 
Dir; (a1, a2, …, at) = —--------------------- x ai(1-x)®i+t-2) (14) 
a;!(b; + t -2)! 
where 
t 
bi = S aj = aj. (15) 
j=1 
Similar to the two alternative case, if u(x)- Dir; (a, a», 
..., at), then 
f dx=1 (16) 
0 
and 
"un sana dx = (aj+1)/(aı+a2+..+at+t) (17) 
0 
To estimate the value of Dir; , the expert needs to 
specify values, aj, ay, ..., and a;, which are all >= 0, such 
that his experience is approximately equivalent to 
having seen d; occuring a; times, dy occuring a; times, 
..., and dt occuring a; times in a; + a; + ... + a;, total 
occurrences. Thus, the estimate of Dir; can be used as 
the accuracy value being addressed in the case where 
the propositional variable has multiple possible 
values. 
Neapolitan's approach to the estimation of the UIU 
value included in experts’ knowledge can be applied to 
the situation where time dimension is not involved in 
the estimation. When this is not the case, a better 
approach is to include the variable of time periods in 
the estimation, thus, the Neapolitan's approach needs 
modification, or a new approach needs to be devised. 
This is a topic remained for further research. 
Determination of constant C. C is the constant 
included in function (3). To determine the value of 
the constant, we can suppose an ideal condition where 
T=1,5=1,A=1,and Sd = 0. Based on the meaning of 
the variables involved, there should be CIU(T=1, S=1, 
A=1, Sd=0) = 1, which means that the certainty in an 
uncertainty value elicited from a database or from an 
expert reaches a maximum in the ideal situation. 
Replacing the values of CIU, T, S, A, and Sd to in 
equation (3), the constant of C can then be obtained. 
Adjustment of Certainty Values Using CIU 
Once the value of CIU is obtained, the next step is to 
adjust the certainty value generated from a database or 
provided by experts using the CIU value. Let the 
certainty value be CV, and the adjusted certainty value 
be ACV, then ACV can be obtained using the following 
simple function: 
ACV = CV * CIU (18)