A Methodology for Uncertainty in Knowledge-Based Systems (Lecture Notes in Computer Science)

then solutions exist for the System (4.2) so that x++, x÷_, x_+, x__ are in the interval [0;1]. The set of all solutions is the following:

(4.3)

70

plai-px++ (x++, x+_-

Pl-P

P2a2-px÷+

, x_.

p2(1-W2)

(l-p2)~Z-Plai+px++

, x__

pl(1-Wl)

1-~-~;

Max[O;

P2-P

1 ~ ;

1-pl-P2+p

P2W2-(1-pl)Xl

p

:

] <x,+ (4.4a)

Pill P z w 2 Min [1;

P ;

P ; 1+

(1-pl)(1-~l) - p2(1-~2) } P ]

which leads to

Max

O;

pitol-p2t#2 PwP

; 1-

(l-p2) (l-~z) pi(l-oJl)] < x+_ < PwP ; 1 - Pl-P (4.4b)

_<Min [ 1 ; PwPP'~'; (1-p2)~2 ;pl_p

Max [ O; P2°)2-1DI~lp2-P ; 1-

p2(1-~v2) ;p2_p

1+ ( l - p , ) ( 1 - ~ ,] ) - (p1 - ,p 2 _) (1--~2) p

1 - (1-pl)p2_p(1-xl)]j _<x_, _< (4.4c)

-

I.

Pz-P ;

P2-P

P2-P

J

Max [ O; (1-p2)~2-plwl (1-p2)(1-~2) (1-pl)(1-~1)] 1-pl-p2+p ; 1k 1-pl-P2+p ; 1 1-pwp2+p J -<x__<(4.4d) -<Min

(1-p2)~, (1-p2)~2 1; 1-pt-p2+p ; 1-pl-p2+p

1 + p,(1-~,)-(1-p2)(1-~2)] 1-pwp2+p J

Proof of Theorem (4.1): With the abbreviations: r i =pjwj , rj = (1-Pi)X j , j = 1 , 2 and y++=px++, y+.= (pl-p)x+-, y-+: ( p 2 - p ) x - , , we maintain the following system of equations:

y - - : (1-pl-p2+p)x--

O0 1 1 y,_ ~1 1 0 1 0 y_+ = r2 0101 y__ F2 If we take into account that all probabilities are to be between 0 and !, i.e.: a) O
0 < y+_ < PwP 0 < y_+ _
D

7] I)

Max[O;r,-~lax(y+.), r2-Max(y.+)] < y++ < Min[p; r,-~lin(y+_) ; r2-Min(y_+)]

II)

Max[O; rl-Max(y++), r2-Max(y..)] < y+_ < Min[pl-p; rwMin(y++) ; F2-Min(y__)]

III)

Max[O; rl-Ilax(y__), r2-Max(y++)] < y.+ < Ilin[p2-p; F1-Min(y__); r2-Min(y++)]

IV)

Max[O; ~l-Max(y_+), r2-Max(y÷_)] < y__ < Min[1-p,-p2+p; ~,-Min(y_+) ; F2-Min(y+_)]

To solve this system by iteration we start with the trivial limits given by a) - d) and achieve: a') lllax[0; rl-pl+p; r2-p2+p] _
b')

Max[0; rwp; F2-1+pt+p2-p] _
c')

Max[0; Fl-l+pi+p2-p; r2-p] < y_+ < Min[p2-p; ~1; r2]

d')

Max[0; ~Fp2+p; F2-pl+p] _
By continuing this procedure, we use the Maxima and Minima of a') - d ' ) and find, considering rl + r~ = r2 + ~2, which is a version of (4.1): a")

Max[O; rl-p~+p; rwF2; r2-p2+p] _
b")

Max[0; r~-p; rl-r2; r2-1+pi+p2-p] _ y+_ _<Min[pl-p; rl; ~2; rl-r2+p2-p]

c")

Max[0; rl-l+pt+P2-p; r2-p; r2-rl] < y_+ _ Min[p2-p; {1; r2; rl-r2+pl-p]

d")

Max[0; ~I-P2+P; {wr2; {2-pl+p] _
These limits reproduce themselves if used in I)-IV). Therefore they define the solutions of the system if they are not contradictory i.e., if the upper limits are not smaller than the corresponding lower limits. From a") it can be concluded that (4.3) is sufficient for a non empty set of y++. The inequalities b")-d") do not make any further restraints upon p. [] Theorem (4.1) says that probabilities x++, x._, x_., x__ do not exist for all values of p. For instance, it is possible that p is too small. Let the probabilities of a breakdown in case of each positive signal Zj be comparatively large, while the probabilities of a breakdown in case of ~Zj are extremely small. If p is near to zero the two units almost never give alarm at the same time. Yet if one unit gives alarm, the probability of a failure is comparatively large; if the other does not give alarm, the probability of a failure is extremely small. This is a contradiction: A solution cannot exist. The second feature of Theorem (4.1) is that in most cases in which solutions exist, they are not unique. This means that the information used in this model is not sufficient to determine a single value for the probability of a failure in the case of two alarms sounding. This contrasts very sharply to those ideas which assume that a real number as a "combined probability" must exist even in cases where there is much less information than in the model used here. We shall see later, what type of information has to be added in order to secure that a real number exists which represents the probability of failure in case of two alarms being given.

72

One should interpret p with respect to Pl and P2 as a characteristic of the mutual dependence of the two units. If for instance p = Pl'P2, this means that the two signals are independent. But this type of independence is global independence: O-independence (see Chapter 3.5). It should not be confused with double independence for normal state and for breakdown. Only in cases which are of no importance to reality is it possible that global independence and double independence meet. We called this total independence. If one of these probabilities x++, x+_, x_+, x_ has t o b e estimated, the corresponding interval (4.4) gives the required information. Yet if for any purpose more than one of these probabilities has to be estimated for the same problem, then the functional dependence between x++ and x+_, for example, has to be taken into account. For each possible single value of x.+ the single value for every other probability is uniquely determined. We shall discuss these results ushlg the data of Example (3.11), which has already been used in discussing the Dempster-Shafer combination rule. Example (4.1): We use the same parameters as in Example (3.11): 0J1 = ~2 = 0.01

vl = 0J2 = 0.0001

For the probabilistic model used in this paragraph the information must be completed by the values of pj, e.g.: Pl = P2 = 0.02 For simplicity of calculation it is again assumed that the parameters for both units are the same. Due to this assumption Equation (4.1) is automatically true and we obtain: v = ply1 + (l-p1) 51 = p2~2 + (l-p2) v2 = 0.000298. For the parameter p Equation (4.3) leads to 0.000102 < p < 0.02. /

The maximal value of p, due to the symmetry between the two units, is 0.02. This value means that both units either give alarm at the same time or do not give alarm at all. In this case one could treat both units as if they were only one unit, because they are completely dependent upon each other. If p is smaller than 0.02, the possibility arises that one unit gives alarm and the other does not. In the case of p = 0.0004 the two units are globally independent, and if p is still smaller, there is a negative association between them: If one gives alarm, the probability that the other does likewise is reduced. In the following table the admissible values of x++ are given for some selected values of p: p = 0.02

:

x++ = 0.01

p = 0.01

:

0.0102 < x++ _ 0.02

p = 0.001

:

0.1020_< x++ _< 0.20

p = 0.0004

:

0.2550 < x++ < 0.50

p = 0.0002

:

0.5100 < x++ < 1.00

p = 0.000102

:

x++ = 1

73 It is seen that the intervals for x++ are comparatively large if p takes an intermediate value and that at the extreme points of p there is a unique solution for x++. Also it is seen that in the case of p = 0.02, if the two units operate as practically one, the probability model cannot be cheated: it reveals the same probability as in the case of only one unit. The smaller the value of p, the more valuable the set of the two units: if both give alarm the probability of a failure is now comparatively large; in the case of global independence (p = 0.0004) this probability is between 25.5% and 50%. It becomes still larger when there is negative association between the two units, and is 1.0 in the case of the smallest possible value of p (p = 0.000102). It should be noted that the probabilities as derived from the Dempster-Shafer combination rule are never found in the intervals of possible probabilities as determined by the model. The differences between the results of the two procedures applied to this problem are extreme.

D

We now alter the value of the probability for an alarm in the second unit, while all parameters for the first unit remain unchanged.

Example (4.2) Pl = 0.02

P2 = 0.01

o~i = 0 . 0 1

~2 = 0 . 0 1

~1 = 0.0001

~2 = 0.0002 ~o= 2.98.10 -4

We find by means of (4.3): 2.10 -6 < p < 0.01 and by means of(4.4a): p = 0.000002

:

x,+ = 1.0

p = 0.000010

:

0.200000 < x+÷ < 1.0

p = 0.000100

:

0.020000 < x++ < 1.0

p = 0.000200

:

0.010000 < x++ < 0.5000

p = 0.001000

:

0.002000 < x++ < 0.1000

p = 0.009902

:

0.000202 < x++ < 0.0101

p = 0.010000

:

x++ = 0.0100

There are two important differences between the results we obtain using these parameters and those obtained in the foregoing example: For some small values of p (around p = 0.0001) the intervals for x++ become very wide so that in such a case the parameters contain very little information about the probability x++. Secondly the lower limits of the interval for x÷+ are not a monotonous function of p. Instead they reach a minimum at a value of p which is determined by p = p 2 - ( 1 - p l ) ~ = 0.009902

74

and show a very sharp increase from there to the terminal point p = 0.01. If no exact knowledge about p exists, the only reliable treatment is to use the union of the intervals for x** with respect to all possible values of p. We shall later proceed to carry out a thorough investigation of situations in which parameters are defined only by intervals. Furthermore it should be noted, that our description of the methods in this paragraph, using the example of a power station, does not confine the application of this method. Evidently, in a medical diagnostic system for instance, symptoms may be used instead of our alarm units and types of illness instead of our "state of the power station". Finally, it must be said that in principle all the methods used here could be transferred to more complicated models where there are more than two states of nature or more than two different signals, which can be given by a unit or even more than two units. The formal treatment of such models is technically more complicated and is not shown here, because we shall now move onto another model, which uses assumptions about independence and therefore is not only easier to handle but also provides stronger results.

75

4.2 Solutions with Double Independence and Related Models The results of the analysis in Chapter (4.1) are comparatively weak. This is due to the fact that the models used there do not contain any statement about dependence or independence of the signals with respect to the two states of nature. These models use the global probabilities Pl, P2 and p, - w h i c h refer to O - while any measure of dependence in EF and EN must use conditional probabilities. When describing the formalism, we employ the notation already used in our example of the power station and find P(ZjAEF) = pj~aj

P(Zj^EN) : Pi(1-wj)

P(Zj IEF) = Pi~i

P(Z~IEN) :

P(Z1AZ2AEF) = p.x++

P(ZI^Z2AEN) = p(1-x++)

p(Z1AZ2IEF ) = p'X+÷ ~d

P(Z,^Z~IEN) :

1-0)

Now we define two measures of dependence between Zl and Z2. The one is the measure of dependence of ZI and Z2 with respect to EF and is called ~F. The other is the measure of dependence of Z1 and Z2 under EN and is called aN. To avoid indefiniteness we have to suppose 0
land0<

xj,~j
j=1,2.

These restrictions are well founded if the units are to work sensibly and therefore we shall also use them in the following chapters. Definition (4.1): P(Z1AZ2IEF)

P(Z1AZaIEN)

~F = P(ZIIEF)'P(Z2IEF)

~N = P{Z1]EN)'P(Z2JEN)

It is easy to show that Max [ 0 ;

1 t r ~

1 + P(Z2I EF)

1 1 - P(ZIIEF)'P(Z2IEF) ] -< ~F -<Max [P(ZIIEF),P(Za[EF)]

and the analogous relation for aN. Obviously nF = 1 has to be interpreted as EF-independence of Z1 and Z2. ,%F > 1 describes positive association between Z1 and Z2 with respect to EF: The probability that both events occur simultaneously is greater than it would be in the case of independence. Using Definition (4.1) we are able to derive the following theorem.

76 Theorem (4.2): If t~F and ~N describe the dependence of Z1 and Z2 with respect to EF and EN, then: EFtOI~2

(4.5)

X++ =

[]

Proof of Theorem (4.2): With the notation used above we obtain: px,, = ~Fplxl'p2°)2 and

pl(1-Xl)

p2(1-~d2)

Therefore: px++ = ~--~Fp,p2~l~2

and

p(1-x,,) : aN_i~p,p2(1-w,)(1-w2 )

(1-w2)lj P = PlP2 [ aFXI-L~w~+ aN (1-Wlll_w

By summation:

Dividing (*) by (**) we get Formula (4.5).

(*) (**) []

Due to the assumed knowledge of the kind of dependence between Z~ and Z2 under EF and EN we derive x++ as a number, not as an interval. Therefore we may conclude that the interval estimation of x,, in the preceding chapter is only due to the fact that no assumption was made about the conditional dependence or independence of Zl and Z2 with relation to the two states of nature. It should be noted, that (4.5) is a simple result of probability theory and does not contain any concepts which are not part of classical probability theory. Of course it does not use any ad hoc proposals for combining evidence. In fact results like (4.5) can be found in text books of probability theory. Any combination rule which is not in agreement with Formula (4.5) therefore contradicts elementary probability theory and cannot be justified by any argument about experimental verification. The most frequently used assumption about the conditional dependencies is the assumption about double independence of Z1 and Z2 in EF as well as in EN. This means nF = 1 and nN = 1. In this case Formula (4.5) is simplified to ~dl a/2

~0 0/

(4.6a)

+

This result is even more popular than (4.5), because the assumption of double independence is widely used. (4.6a) occurs when probabilities of a hypothesis are updated, provided that a total probability of the hypothesis is considered available - a situation which is often described as "Bayesian updating". The methods used in the expert system PROSPECTOR are of this kind and therefore may be regarded as almost identical [DUDA, HART, NILSSON, 1986]. The only relevant

77 difference between the two treatments is that P R O S P E C T O R uses likelihoods of the type P(Zj

IEF)

in its formula, while in (4.6a) probabilities of the type P(E~IZj) = wj are used. This changes the outfit of the results to some extent. The use of odds instead of probabilities by P R O S P E C T O R is purely superficial. It is possible to obtain, analogous to Theorem (4.2), the other probabilities of

EF,

for instance x÷_,

the probability of a failure in case of Z1 and -,Z2. In this case we have to use corresponding values of gF and gN. For double independence all those values for ~ take the value 1. Therefore formulas for x+_, x_+ and x__ can be derived in the same way as Formula (4.6a) for x++: 011~2

x+_ :

w aJl~2 + (1-~Ol)(1-~2) oJ 1-o~

(4.6b)

~la/2

x_+ =

~o + (1-~)(1-~2)

~

(4.6c)

~11~12

x__ =

~ ~a

(4.6d) i-~

Example (4.3): With the parameters used in Example (4.1) we achieve under the assumption of double inde~ pendence the following four probabilities of a failure of the power plant: 0.0001 gT0-O-f2-gN = 0.255 x,+= 0.0001 0.9801 gT0-0-02~ + (y79997N2 0.000001 x,_ = x_+ = 0.000001 0.989901 = 0.003377 g-_Ny0-2~ + 0TgggT~ 1.10-s ~:g0-O2-gg X__ = 1-10 -s 0.999800 = 0.000034 ~J:NY0-2~ + g:V99T0~ These are results in accordance with the purpose of an alarm system: If there are two units and both go off, the probability of a failure is much greater than in the case that there is only one unit and this goes off. One should remember the misleading result gained by the Dempster-Shafer combination rule in Example (3.11)! The corresponding p-value derived from equation (**)in Theorem (4.2) is p = 0.000526 and is remarkably different from the value p = 0.0004, which is a consequence of global independence.

78 It may be asked, how these results correspond to those of the preceding chapter, where for each p-value intervals for x+÷, x÷_, x_+ and x__ were obtained. If we use the p-value of 0.000526 to calculate those intervals we find, due to Equations (4.4a-d): 0.1939 _<x,+ < 0.3802 0 < x+_, x_+ __0.00503 0 _<x__ < 0.000102. Therefore all probabilities derived under the assumption of double independence lie in the corresponding intervals. [] The question, whether this statement is generally true, can be determined by the following theorem. Theorem (4.3): The probabilities deduced from (4.6) (under the assumption of double independence) lie in the corresponding intervals (4.4). [] Proof of Theorem (4.3): It can be checked that for [ ~1~02+ (1-~0t) (1-~o2) ] p = piP2 ~ 1-~a the values x++, x+_, x.+, x__ according to (4.6) are solutions of the System (4.2). For instance, p,,++ + (p,-p)x+_

= plp

+ p,(1-p2)

=

(***)

-

+

= p,+,,,+,,.

due to: p + pl(1-p2)

~1 2 +.

1-~0

= Pl P2 ---7-+ P2

l-w

{

=

-

(1-aq) (1-~2) }

i-~

= Pl { 7x, [p2w2 + (1-p2)~2] + l-x, [p2(i-~;2)+ (l-p2) (i-~2)]} = p,

~- x + i 7 7 (1-0o [ -

= Pl (l-w) (1-52)]

As the inequalities (4.4) are valid for all solutions of the System (4.2), they must be valid for the solutions x++, x+_, x_+, x_, provided the p-value for double independence is used. m It becomes obvious that p according to (***) is necessary for double independence, as shown in the proof of Theorem (4.2), but is not sufficient. It should be noted that double independence of Zl and Z2 in the case of E F and EN comes nearest to what might be understood by "independent sources of information" as used by Dempster. But of

79 course the two concepts are not comparable directly, in that double independence is defined exactly, whereas the meaning of "independent sources of information" is not totally clear. With respect to what we call similarity of the two concepts it is justifiable to compare the results of Dempster-Shafer's combination rule and (4.6). This is first done by means of Examples (3.11) and (4.3). The application of the Dempster-Shafer combination rule, in this case, gives results which are very different from those stemming from probability theory. If we compare the two formulas which are applied in both rules, the origin of the differences becomes obvious: while in (4.6) the total probability of EF is used as a denominator, it is not used at all in Dempster-Shafer's formula. Therefore the difference in the information used is described exactly by this "prior probability". It is easy to convert the Formulas (4.6) into Dempster-Shafer's formula, if all total probabilities are equal to one another, so that the denominators may be reduced. Therefore the application of the Dempster-Shafer combination rule leads to the same results as the application of (4.6), if the total probabilities are: P(EF) : P(EN) :

1

Such an assumption is of course very far from any estimation of the probability that a power station has a breakdown. This is the reason why Example (3.11) led to such obviously dangerous results, which are not seen in examples used by Sharer himself. As long as the regarded events Ei are more or less equally probable a-priori, the application of Dempster-Shafer' s rule comes near to the results of (4.6) and therefore leads to sensible results. It should be noted that the Dempster-Shafer rule applied to x+÷ does not use values of ~j, which influence the prior probability ~ by (4.1). Therefore it is possible to imagine values of ~j which lead - in combination with any pj - to w = 1/2. In this case the result of the Dempster-Shafer rule would be correct. For the example of the power station these ~j would have to be near to 1, in contrast to the true values of ~j which are much smaller than wj. Now the results of the application of the Dempster-Shafer rule in Example (3.11) become understandable. They are the same as those of the application of (4.6), provided ~j is near to 1, indicating that the signals Z1 and Z2 designate the state of tow danger while ~Zi designates high danger. In such a case it would be quite reasonable, that ZI^Z2 would signalize an extremely low danger. Therefore the probability of EF in case of ZI^Z2 would be in accordance with the results of the Dempster-Shafer rule. Of course this explication is good only for x,+. If one applies the Dempster-Shafer rule to x__, it amounts to the application of probability theory with the assumption that wj is very near to 1. In the cases of x+_ and x_+ the assumptions would be, that those wj or ~j which are not used in the formula itself, are near to 1. It is easily seen that using the knowledge of wI and Wl for the power station a value of w = 1/2 can be excluded, even if the values of Pl and p2 are not known. Therefore in such cases the application of Dempster-Shafer's rule amounting to such an assumption is not tolerable at all. The same is true in most medical diagnostic systems, when it is an important aim to detect rare diseases: If the Dempster-Shafer rule is applied, such a detection is impossible.

80 Concerning the strategy in cases of low information availability, generally it can be said: a)

It is hard to understand that the probabilities of a state E i in the case of Z1 and Z2 could be known, while the total probability of Ei is not known at all. Therefore in most cases it must be possible to give at least a good estimation of ~o, even if pl and P2 are not known (which is not necessary for the application of (4.6)).

b)

If it is absolutely impossible to estimate w, at least we know that, due to (4.1), w is a weighted average of wl and ~ ; therefore w lies between these two limits. And the same is true for w2 and ~2: these are also limits for oz Using the narrowest of these limits, interval estimates for x+, etc. can be derived from (4.6), even if "no information whatsoever about the prior exists".

Such interval estimates use the following facts: 1) 2)

Ma~ Min (0Jj ,~j) < 01< Min Ma~ (0Jj,~j) i J x** as well as x÷-, x_, and x_- are monotonously decreasing functions of w.

(4.7)

In the following example we demonstrate this kind of interval estimation: Example (4.4): Using the same values for wj and ~] as in Example (4.3), but assuming that w remains unknown (and p is not known either) we have: ~j = 0.0001 < 0J < 0.01 = wj Due to the monotony of the probabilities x++ etc. as functions of w, the upper limits are found by using a) = 0.0001 and the lower limits by using w = 0.01: 0.01 < x++ < 0.505 0.0001 _<x+_, x_+ < 0.01 0.99.10 -6 _<x__ < 0.0001 Although

these limits are far ranging, it is evident that the results of the application of

Dempster-Shafer's combination r u l e - as in Example (3.11) - l i e

far outside these limits in all

cases. This demonstrates: even if no information at all exists about the total probability, there is no excuse for the use of Dempster-Shafer's rule at least in this example, because the information about o3j and ~j is sufficient to show that the results of this combination rule are by far unreasonable.

[]

The derivation of x,, in (4.6) resulting from the Equation (4.5) was carried out under the assumption that ~;F ----gN = 1. Of course formulas which are analogous to (4.5) may be derived for x,_, x_+ and x__. Formulas (4.6) are special cases of these generalized formulas if the corresponding values of ~F and ~N are equal to one. In order to describe the necessary conditions for the validity of (4.6) we designate coefficients which must be used in formulas analogous to (4.5):

81 a ~ * = nF

n~-= K~+

- -

a~÷ = NN

P(Z1A-Z2IEF)

(4.8)

P(Z, IEF)'P(~Z2IEF)

P('ZI^Z2JEF) P(-ZaIEF)'P(Z2IEF) P(-Z1A-Z2JEF) P(-ZllEF)'P("Z2[EF)

a~-, a~*, ~fi- analogous. Now the question arises, whether the conditions that all aF and all aN are equal to 1 is necessary for the validity of Equations (4.6). It is obvious that (4.6a) is also derivable from (4.5) if ai~+ : aft +

(4.9)

Therefore the probability x++ may be described by (4.6a) not only in the case of double independence but also in those cases, in which both symptoms are dependent upon each other, but the kind of dependence under EF and EN is equal in the sense of (4.9). One should be reluctant to use this kind of "dependence of equal strength". In the same way as a ++ may be used as a measure of dependence, the values of a +-, ~-+ and a-- may also be used. Therefore it would only be justifiable to talk about dependence of equal strength, if all four measures were equal in the case of EF and EN: at + = ~+

a t = a~-

(4.10)

al~+ = ai~+

All these equations are true, if each of the eight values a is equal to one. This means double independence. The possibility, that (4.10) is true in other cases is investigated in Theorem (4.4). Theorem (4.4): (4.10) holds, iff either all values a are equal to 1 or P(ZjIEF) = P(Zj[EN) , j = 1 , 2 .

(4.11) []

According to this theorem the equality of all corresponding values of aF and aN is only possible in the case of double independence or in the trivial case that the signals have the same probability under EF and EN and therefore the units are not suitable for distinguishing between the two states (or for recognizing disease).

82 Proof of Theorem(4.4): Using thefollowing table for EF:

~}+P(ZllEF)P(Z21EF )

~}-P(ZI[EF)(1-P(Z2[EF))

P(Z1]EF)

~+(1-P(Z,]EF))P(Z2]EF)

~p- (1-P(Z1]EF))(1-P(Z2[EF))

1-P(ZllEF)

(4.12) P(Z2IEF)

1-P(Z2[EF)

and an analogous table for EN, we obtain

~--

1-~*P(Z2]EF)

~fi-=

1-P(Z2JEN)

1-P(Za]EF)

g~+ =

1-~*P(ZIIEF)

1-~+P(Z2IEN)

~+ =

1-P(ZllEF)

1-~+P(Z, IEN) 1-P(ZllEN)

1-P(Z, IEF)-P(Z21EF)+~ff+P(Z, IEF)P(Z2[EF) (t-P(ZIIEF))(1-P(Z2IEF)) and an analogous result for ~ - . By means of these results it can be shown that ~}+ = a~ + and a}~- = ~ - is together only possible if ~i~÷ = 1 or (4.11) holds. The analogous statements can be derived with respect to ~+, a~+, t;~- and KI~-.

[]

We demonstrate these considerations by use of the data to be found in Example (4.1): Example (4.5): Recalling that for the power station ~i = 0.01, ~j = 0.0001, Pi = 0.02 we find for the conditional probabilities P(Zj IEF) :

j = 1,2

P(EFIZi)'P(Zi) = wiPi : 0.6711. P(EF)

In an analogous way: P(Z i [EN) = 0.0198. These results characterize the two units: If there is a breakdown, each unit gives alarm with a probability of about 2/3, if the power plant is in normal state, the probability of a blind alarm is about 2 % for each unit. The probability of both units giving alarm can be determined, if n}~+and n~+ are chosen. For example we take n~+ = n~+ = 1.3 and obtain

83

P (ZlAZ2]EF) = O. 5856 P (Zl^Z2 ]EN) = O. 0005. Table (4.12) can now be completed by subtraction:

EF : 0.5856

0.0856

0.6711

0.0856

0.2433

0.3289

0.6711

0.3289

1

0.0005

0.0193

0.0198

0.0193

0.9609

0.9802

0.0198

0.9802

EN:

From this we obtain: ~ - = a~ + = 0.3878,

a~- = 2.2495

nil+ = ~ + = 0.9939,

nil- = 1.0001

If we now apply (4.5) to x**, we arrive at our previous result: x++ = 0.255

(because of ~ + = ~ ' )

while

x+_ = x_+ : 0.0013

(in contrast to 0.0034 in the case of double independence)

and

x__ = 0.000075

(in contrast to 0.000034).

[]

Up to now in this chapter we have used a description of the stochastic behaviour of the two units Z~ and Z2 with respect to the two states of nature: EF and EN. In previous chapters we used the concept of global independence which is related to O, the union of EF and EN. Global independence is assured by p = P(z,^z2)

= P(Z,).P(z~)

: pl.p2.

In Chapter (4.1) it was shown that for this p there results an interval for x++. If a further assumption is then made, either the independence in the case of EF or of EN, we describe situations which are called (O,EF)-independence or (O,EN)-independence. Because of the symmetry between them it is sufficient to discuss (O,EF)-independence. In this case x++ is derived in the following way: P(ZIAZ21EF)'P(EF)

P(ZI[EF)'P(Z21EF)'P(EF)

P (EF I ZIAZ2) P(ZIAZ2)

P(Z~).P(Z2)

P(EFIZ1)'P(EFIZ2) (4.13)

~(EF)

84 Of course the probability of EN in the case of (O,EF)-independence cannot be derived in an analogous way but must be calculated by P(EFIZ1)'P(EFIZ2) P(ENIZ1AZ2)

= 1-P(EFIZI^Z2) = 1

P(EF)

-

(4.14)

P(E~IZl)+P(~IZ2)-P(EN)-P(E~IZ,).P(ENIZ~) 1 - P(EN) The formulas for x,_, x_÷ and x__ correspond to (4.13) and (4.14). In case of (O,EN)-independence the Formulas (4.13) and (4.14) (or their corresponding ones) must be interchanged. The effect of the assumption "(O,EF)-independence" instead of "double independence" is shown in the following example. Example (4.6): Using the previous parameters and (O,EF)-independence it follows, that: P (EF I ZI^Z2) = 0. 3356 P (EN IZl^Z2) = 0.6644 If we use the respective formulas for (O,EN)-independence: P (EF I ZIAZ2) = 0. 0196

P (EN I ZlhZ2) = 0. 9804

In the case of double independence we had: P(EFIZI^Z2) = 0. 255

P(ENIZI^Z2) = 0. 745.

If these results are compared with previous results, we find that the value for x** lies in the interval which is derived frorn Theorem (4.1), if the value attributed to p describes global independence, i.e. p = 0.0004. This is a consequence of the fact that (O,EF)-independence is a special case of global independence. The analogous value of x++ in the case of (O,EN)-independence is not in concordance with the corresponding interval from Theorem (4.1). If the reason for this deviation is sought, it can be shown that (O,EN)-independence is not possible in this case. This is an example of the limitations for the construction of (O,EF)-independence and (O,EN)-independence, which we have already mentioned in Chapter 3.

[]

85 If we investigate the concept of these types of independence using the model of this chapter we find: Theorem (4.5): With the notations used in this chapter, the assumption of (O,Es)-independence is possible, iff a)

(1-~o,) (1-w2) < 1 1-to

h~j

P2 _< 1 - .(1-101).(i-102)

c)

Pl _<1

101 1-10 102

(1-101) (1-102)

-

1-10

d~_,

P132" ([1- (1-101) (1-102)11_10) - > pl" 101 10+p2" 10~_ 1

The assumption of (O,EF)-independence is possible, iff a)

101"102< 1 tO

-

1-0h

b)

p2 _<

c)

pl _<

d)

~

i - 10t'102 ~d

1

1-10~ -

~)1-~2 O2

(1 _ ~ } > _

pl(l-1001_+10p2(1-102)_ 1

The proof of this theorem relies upon the fact that for (O,E~)-independence, not only O and E N but also EF must be a set for which the conditional probabilities of Zl^Z2, gl^~Z2, ~Za^Z2, ~Zl^~Z2 are non-negative numbers. If this is controlled, the proof is then straightforward. It should be noted that in contrast to this result it is possible to conceive double independence for all parameters which define sensible subsets EF and EN. This remarkable difference between the two concepts shows the superiority of the concept of double independence. In fact it is not easy to describe a situation, in which the concept of (O,EF)-independence is obvious. On the other hand, double independence is a concept which is identical to those concepts most often used in practical statistical analysis. Finally we can conclude from the considerations described in this chapter: In all cases, in which the assumption of double independence is possible, Formulas (4.6) will provide a reliable and justifiable combination rule. No other "combination rule" should be applied in such a case. If there are serious doubts with respect to double independence, it might be easiest to collect reasonable information about p, Pl and P2 and to apply (4.4), even if it results in interval estimates for the probabilities x÷+ etc. While reliable information about the values of ~ would provide stronger estimates (point estimates) for x÷÷ and the other probabilities of this type, it will often be impossible to obtain such information.

CHAPTER

5

Generalizations 5.1 T h e F o r m a l i s m While in the preceding chapter we used the simplest type of a model to describe the states of nature, the observations and the conclusions drawn from them, in this chapter we shall discuss the general case of combining information provided that it consists of point-estimates of probabilities. At first we shall generalize the model with respect to the number of states of nature: k

O=oEi i=l

and the number of distinguishable signals or signs in both alarm units: A1 = (Zll, . . . , Zlal);

h2= (Z21, . . . ,

Z2n2).

At a later stage we shall also discuss situations in which there are more than two alarm units. We shall use a model, which is a direct generalization of the model of double independence used in the preceding chapter. For this model we require the following probabilities: V(Zll),

...,

P(Zlnl)

with

n1 rS=lP(Zlrl) = 1 1

P(Z21),

Definition

...,

P(Z2n2) w i t h

n2 r~=lP(Z2r2 ) = 1 2

(5.1):

As a generalization of double independence as used in the preceding chapter we describe k-independence of the two units by: P (Zlr ^Z2r2tEi) = P (Zlrl 1]~i) "]) (Z2r21]~i)

i : 1,...,

k

1,...,

nl

F1 =

r2=l,...,

(5.1)

n2

[]

In this chapter we shall always suppose that k-independence is given and additionally that the probabilities P(EilZlrl) , e(EilZ2r2) , rl = 1 , . . . , nt; r2 = 1 , . . . , n2 are known or that at least reliable point estimations of these probabilities are available. Just as for the case k = 2 Equation (4.1) must be fulfilled, there are also requirements for these probabilities: n1

n2

P(E~) = r~l:IP(E~LZ'r')"P(Z,r ) = r:~ P(EiIZur^).P(Z2r) ~ , ;

i--1

' " "

,k

(5.2)

88 This is a system of (k-l) equations which limits our freedom in estimating the k. (nl+n2)-2 probabilities appearing in (5.2). Of course any analysis depends on the validity of Equation (5.2). Due to the assumption of k-independence we can immediately derive a generalization of (4.6):

P(Ei* 1ZlrlhZ2r2) =

r(Zlr 1hZ2r2 IEi*)" P(Ei*) _ P ( z l r 1 ^z2r2 ] E i * ) ' P ( ~ i * ) k P(Zlrl^Z2r2) i~lP(Zl rl^Z2r21Ei)"P(Ei)

_P(ZlrlIEi*)'P(Z2r2IEi*)'P(Ei*)

_P(Ei*IZlrl)'P(Ei*IZ2r2)/P(Ei*)

k i=~lP(Z,r IEi)'P(Z2r2IEi)'P(Ei)

k i~,P(EilZ'r,)'P(Ei]Z2r2)/P(Ei)

i*= I , . . . , k ;

1"1= l , . . . , n i ;

r2 = 1 , . . . , n 2

(5.3)

As for (4.6), (5.3) is merely a consequence of probability theory and not a new result; this system of equations provides the combination rule which is supplied by classical probability theory for all cases in which the assumption of k-independence holds. It should be noted, that it does not use the values of P(Zjrj) directly, but these values are important for the calculation of the values P(Ei). Before discussing these results in detail we shall make a further generalization, this time assuming that there are not two but 1 alarm units: AI = ( Z 1 1 , . . . , A2 = ( Z 2 1 , . . . ,

Zl. 1) Z2.2)

: A1 : ( Z H , . . . ,

Zlnl)

In this case we must assume that the probabilities of the outcomes - or measurements - of each of the 1 alarm units are known:

P(Z11),..., e(Z~. 1) P(Z21),...

'

P(Z2n2)

P(Zll),- • •, P(Zlnl)

n. with •J P(Zjr ) : 1, r.=l j J

j = 1 , . . . ,1

89 Definition (5.2): The assumption of k-independence as made in the case of 1 = 2 must now be converted into the assumption of mutual k-independence which states 1

P (Zlrl ^ . . . A Zlrl IEi ) =i~lP(Zjrj IEi) rj = l , . . . , n i j=l,...,1

(5.4) []

i=l,...,k

This is a very strong assumption indeed, but it is justifiable if it may be taken that, whatever the state of nature, the alarm units are not influenced by each other. In addition we assume that the conditional probabilities P(EilZirj) , i=l,...,k; j=l,...,l; rj=l,...,nj are either known or that they are estimated by real numbers. We shall later discuss situations, in which only weaker estimates of these probabilities are available. Again a system of restrictions exists, which controls the credibility of the estimates used: nI

n1

P(Ei) = ~ P(EilZlrl)'P(Zxrl) . . . . . rl=l

~ P(EiIZI~.).P(Z~r.)

rl=l

1

l

i= 1 ...,k

(5.5)

'

Through these restrictions "prior probabilities" P(Ei) are once again defined which bear information contained in both the probabilities P(Zjr.) and the conditional probabilities P(EiIZjr.). J J It will be seen once again, that the probabilities of the outcomes are used in the final result only in this condensed form. The construction of the combination rule now means the calculation of conditional probability for the state of nature El* provided that Zlr, ..., Zlr I are given. The result is given in the following Theorem (5.1): Under the conditions of mutual k-independence the following combination rule can be stated: 1 P(Ei*]ZIr1 A. .. ^ Zlrl) : P ( E i * ) l - I k

1

II P(Ei*lZir.) i=l

J i

1 II P(Ei[Zjr.) i=1 p(Ei)l-1 j=l j

k;

(5.6)

ri = l , . . . , n j ; j =1,..., 1

[]

1' * = 1 , . . . ,

The proof of Theorem (5.1) follows exactly the procedure used in deriving Equation (5.3).

[]

90 Formula (5.6) may be converted into the following form 1 P(Ei*lZj r.) P(Ei*lZlr 1 h.-.h

Zlrl) =

P(Ei*) ,i~l PiEi,)

!

(5.7)

k P(Ei) ~ P(EilZJri) j~l

i~l

P(Ei)

which can be useful in computing. Formula (5.6), like more specialized formulas, does not contain the probabilities of the outcomes P(Zjr ) directly but only through their influence on the total probabilities P(Ei). Therefore the question arises about what knowledge is necessary in order to put (5.6) into practice: Obviously the conditional probabilities P(EilZiri) must be known; these are the main elements of the description of uncertainty in a diagnostic system. But this is not true in the same way for the probabilities P(Zjrj) of the outcomes. Is it sufficient that P(Ei) are known? For the mere use of Formula (5.6) the knowledge of the total probabilities P(Ei) -together with the conditional probabilities - is obviously sufficient. However, it is by no means obvious, that the given total probabilities and the conditional probabilities fit together in the model which constitutes the theoretical background of a well founded application of the formula. A set of conditional probabilities and a set of total probabilities are only in accordance with respect to the model, if probabilities of the outcomes exist which fulfill (5.5). Therefore the existence of suitable P(Ziri) should always be controlled if the Formula (5.6) is to be applied. It is easy to see that (5.5) consists of 1 subsystems, for instance when j = 1 n1

P(Ei) = rE 1P(Ei[Zarl ) "P(Zlrl)

i = 1,..., k

(5.5a)

1

Each subsystem represents k equations for nj unknown variables P(Zjrj) (J =- 1,..., 1). As the sum of these k equations must reveal the identity 1 = 1, the number of linear independent equations does not exceed k-1. On the other hand R.

~J P(Z~r) = 1

r.=l J

j

j = 1,...,

]

and therefore the number of linear independent unknowns is ni-1. Obviously such a subsystem cannot be soluble for every possible prior distribution P(E1),...,P(Ek), if k > nj. Although in special constellations solutions may exist which even lie in [0;1], one would generally expect (5.5) to be insoluble, ifk > nI holds for any j, j c { 1 , . . . , ]}. A necessary - but not sufficient - condition for the existence of solutions in [0;1] is obviously the following:

91 Max Min P(EilZjr.) < P(Ei) < llin Max P(EilZjrj) j

r.

j

-

-

j

J

r.

'

i = l,...,k

(5.8)

J

If (5.8) does not hold, at least one alarm unit exists for which all the conditional probabilities of Ei are greater than P(Ei) or at least one for which they are all smaller than P(Ei). As (5.8) is by no means a guaranty that all systems of the type (5.5a) are soluble and provide solutions which may be taken as probabilities of the outcomes, the general rule must be: Before applying (5.6) it has to be proven, that the estimates of the prior probabilities and those of the conditional probabilities are in accordance with the assumption of the existence of nj P(ZJri), 0_
which obey (5.5). This may be done by means of a program which solves linear equations. Should this control fail, (5.6) must not be employed! To illustrate this formula let us turn to our Example (4.3) about the power plant. Example (5.1): For convenience we shall use the same parameters as in Example (4.3), but assume that there are more than two alarm units which are all mutually double independent; this means that they are independent in the case of the normal state of the power station as well as in the case of a breakdown. One should remember that in the case of only one signal the probability of a breakdown if the signal went off was 0.01. If there were two signals and both went off the probability of a breakdown rose to 0.255. Now let us assume that there are three signals and that each of them goes off at the same time: OJ1 tO2 ~ 3

x+++ = mtm2e3 (1-0~t)(1-~2) (1-~03) = 0.921 +

(l_m)2

In case of four signals going off at tile same time, the result of the corresponding formula is 0.9975. On the contrary, the Dempster-Shafer rule would ill these cases have resulted in practical absolute security that the power station is in a quite normal state; this was shown in Chapter 3.

[]

It is useful to compare Formula (5.3) with the general form of the Dempster-Shafer formula. It is evident, that (5.3) can be converted into the Dempster-ShMer formula, if all denominators may be reduced due to the fact that they are all the same. This means that the application of the Dempster-Shafer formula is compatible with probability theory if the "independence of the sources of information" is interpreted as k-independence of Zlj and Z2j, and if the prior probabilities of the different states of nature are all the sane. In this case: P(Ei) = 1/k

i=l,...,

k.

92 The results we have discussed for 1 = 2 may be easily generalized to cover situations with more than two units. Again we must assume that the two concepts of independence are taken as compatible. Then a comparison of the formulas reveals, that both show commutativity and associativity: neither the order of the observations, nor the order in which these observations are combined is important for the result. And the results are the same for both methods, if all prior probabilities are taken to be equal. Therefore it may be summarized, that not only in the case of I = 2, but also in the general case the application of the Dempster-Shafer formula is equivalent to the use of equal a-priori probabilities. All considerations of the preceding chapter concerning interval estimation of the prior probabilities through the conditional probabilities - if this is necessary - can be generalized without difficulty. We shall demonstrate the behaviour of Formula (5.3) using the following example. Example (5.2): We assume that there are three possible states of nature, which we conceive as three states of a power station. The state E1 denotes the normal state of the station, the state E2 means that there are some disturbances and the state E3 that there is a severe breakdown. In addition, we assume that there are two alarm units which work independently from each other. For convenience we shall assume that each of the two units has a first sign: "green" which is to be interpreted as "no danger" (Zit), a second sign is "yellow" and is interpreted as "be careful" (Zi2) and a third sign: "red" signifying "high danger" (Zja); j = 1, 2. We assume that the prior probabilities for the three states of the power station are known as P(E1) =0.994050

,

P(E2) = 0.004455 ,

P(E3) = 0.001495.

In addition, we know the conditional probabilities of the three states of the power station given the signs of each of the units. They are formulated as three matrices:

P(EllZir.) j

:

[0.9990 0.9950

0.95 0.95

0.70] 0.75 ,I

P(E21Zjrj )

:

0.0009 0.0038

0.04 0.03

0.20} 0.20

P(E31Zirj )

:

0.0001 0.0012

0.01 0.02

0.10} 0.05

Now we could apply Formula (5.3) as all necessary values are given, but we should not do so before having controlled, whether the prior probabilities and the conditional probabilities fit together in the sense that values exist for the probabilities P(Zjrj) so that Equations (5.5) are fulfilled.

93

0.994050 = 0.999 P(Zn) + 0.95 P(ZI2) + 0.70 P(ZI3) = 0.995 P(Z21) + 0.95 P(Z22) + 0.75 P(Z23) 0.004455 = 0.0009 P(Zn) + 0.04 P(Z,2) + 0.20 P(Z,3)

= 0.0038 P(Z2,) + 0.03 P(Z22) + 0.20 P(Z23) 0.001495 = 0.0001 P(Zn) + 0.01 P(Z~2) + 0.10 P(Zla) = 0.0012 P(Z~I) + 0.02 P(Z22) + 0.05 P(Z23) We arrive at the following values: P(Zn) = 0.95 P(Z12) : 0.04 P(Z13) = 0.01 P(Z2~) = 0.98735 P(Z22) = 0.01075 P(Z23) = 0.00190. All solutions of these systems are in the interval [0;1]. Therefore all the solutions are admissible and a credible probability model exists for which the used values are parameters. Now we can apply Formula (5.6) and obtain the following results:

rl~

1

2

3

i

0.999153 0.957559 0.736359

0.992310 0.692504 0.199483

0.945142 0.251766 0.041096

r'~

1

2

3

0.000767 0.034358 0.179285

0.006299 0.205455 0.401602

0.050664 0.630758 0.698659

r~

1

2

3

i

0.000080 0.008083 0.084357

0.001391 0.102041 0.398915

0.004194 0.117476 0.260245

P(E11Zlrl^Z2r2) :

P(E21Zlrl^Z2r2) :

P(E31Zlrl^Z2r2) :

The behaviour of the combination rule for 3-independent units in this example is easily recognized: The conditional probability of E~ is at least 0.7 if we observe only one unit, even if this shows "red". In the case of two units this probability falls to 4% (0.041096) if both units show "red". On the other hand, the probability for E1 is increased from 0.999 at greatest if one unit shows "green", to 0.999156 if both units show "green". If we want to investigate this behaviour more thoroughly, we have to compare the two units and we see: unit one is more capable of distinguishing the states

94 of nature, because its conditional probabilities with respect to "green" and "red" show larger differences than the respective ones of the second. Therefore generally a red sign in unit one increases the probability of state Ea more than a red sign given by the second unit. Finally, it should be noted that, due to large differences between the probabilities for E1 and for the other states, the requirements for the conditional probabilities in order to secure admissible solutions of System (5.5) are very strict. If some of the conditional probabilities were changed only slightly, a situation is possible in which the solutions of the Systems (5.5) cannot be interpreted as probabilities.

[]

Of course it may be argued that the assumptions used in this chapter are often unrealistic: In many cases one has to deal with interval estimates for probabilities. The methods involved in solving such problems are developed in Chapter 6.

95

5.2 Some Aspects of Practical Application Let us start the consideration of the practical aspects with a short story.

Example (5.3): Mr. K. lives in a country in which 40% of the population suffer from a new type of disease, which can be cured if detected early enough. So he goes to a doctor who tells him that there are two tests for this disease, each working independently from the other. In addition, the doctor knows the probability of a person suffering from the disease, if one of the two tests produces a positive or negative result. They are: Test 1:

Test 2:

Positive result:

wl = 0.8

Negative result:

~1 = 0.1

Positive result:

~o2 = 0.9

Negative result:

~ -- 0.2

Both tests are applied to Mr. K. and both give a negative result. The doctor has a computer which informs him: "The probability that this person has the disease is 0.04." When the doctor tells Mr. K. this result, the latter asks him: "Does this computer use an appropriate program? I hope it doesn't use MYCIN or Dempster-Shafer[" The doctor asks: "What do you know about programs appropriate for diagnostic systems?" And Mr.K. replies: "Well, I am a statistician myself and therefore I don~t trust the programs used in some of these systems." The doctor says: "No, my system uses classical probability theory; but I'm glad you told me that you are a statistician, because I recently received information, which states that only 25% of the statisticians suffer from this new type of disease. So I am sure we shall find out, that together with the two negative results of the tests, the probability that you have the disease is smaller than 4%." He changes the input of the computer from w = 0.4 to o0 = 0.25 and reads: "The probability that this person has the disease is 0.077." The doctor now believes that his computer has a breakdown, as he does not find this result credible at all. But Mr. K. proposes: "Why don't you use the information about my profession as if it were a third test? Employ the formula for three independent tests!" The doctor then asks him: "Are we supposed to regard your profession and the two tests as mutually independent?" Mr. K. answers: "Why not? After all we need double independence. Take the case, for example, that somebody has the disease. I am quite sure that for each of the tests the probability of a positive result is the same, whether this person is a statistician or not. And the same is true for a person who is healthy: His profession will not influence the outcome of the test."

96 The doctor seems to be convinced and feeds the computer the supplementary input: w3 = 0.25 and makes it use the appropriate formula for three independent tests. The computer produces: "The probability that this person has the disease is 0.0204." While Mr. K. is quite satisfied, the doctor does not know which treatment he should prescribe. His rules order him to apply a certain kind of treatment to all people with a probability greater than 0.05 and another one to people for whom the probability is 0.05 or smaller. But Mr. K. comforts him: "Trust the result which says that the probability is about 2%, because this result is correct. I shall explain why your first result was wrong if you wish." Indeed Mr. K. is right: The computer program must never be used in the way the doctor used it the first time. We shall demonstrate this fact by considering the behaviour of Test 1. We know that for a person, for whom originally the probability of having the disease was 0.4, after a negative result of the test this probability is now 0.1. But is this also true for a person, for whom the original or prior probability was 0.25? Obviously it cannot be true for people whose prior probability was smaller than 0.1, because such total probability could never be produced as a weighted average of 0.1 and 0.8. Furthermore: If originally, for Person A, the probability of having the disease was smaller than for Person B, why should this difference vanish after the test has produced the same result for both people? If we accept that there should be a difference in the probabilities after the test, how is this difference determined? In the general case no answer can be found to this question, because the regarded difference must depend upon the behaviour of the test. However, it is sensible to assume that, if the person has the disease, the conditional probability of a positive result does not depend upon the facts which determine the prior probability w, and that the same is true if the person does not have the disease. This assumption produces a probabilistic formalism equal to that found in the case of double independence. Therefore Mr. K.'s explanation to the doctor is in the first place nothing more than a report on how we arrived at Formula (4.6). He shows that for a person with prior probability co' (instead of w), in the case of the single test j, we have to use a2 i • a2 T

(5.9)

~j =

~i.~' + (1-~j)(1-~') ~]

1-~

instead of ~0j and apply an analogous formula for calculating ~ .

For Mr. K. himsdf w ~ = 0.25 and for that reason the probabilities that he has the disease are as follows:

97 Test 1:

Test 2:

Positive result:

w[ = 0.6667

Negative result:

~ = 0.0526

Positive result:

w~ = 0.8182

Negative result:

~ = 0.1111

Now it is possible to derive, in two different ways, the probability that he has the disease, if both tests show a negative result. The first way is to combine ~l' and ~2' with regard to his prior probability w' 0.25 using (4.6): 0.0526 - 0.1111 =

;l';i

0.25 0.0526.0.1111 0.9474.0.8889 0.25 + 0.75

(1-;I)(1-;~)

0.0204

The other method uses w' as if it was produced by a third test. This is the method Mr. K. recommended to the doctor: ~1"~2"~'

0.1-0.2.0.25 0.4 '~ - 0.0204 0.1.0.2.0.25 0.9.0.8.0.75 0.42 + 0.62

X___ -

~1"~2"~' (1-~1) (1-;2) ( 1 - a ' ) ~-~-- + (l_a)2

The fact that both computations must produce the same result, can easily be shown if we transform the probabilities into: [ -'=

(1-~i) (1-w')~ } t 1+ - - - -

5j .~' (l-r)

~J

1-; I (1-;j)(1-~')~ -

j

~;

=

1,2

~i "~' (i-~) l-x__ (1-;I)(1-~)~'

and analogously:

= x__

~ I - ~ (1-~')

Therefore: l-x__

(1-~l)(1-w')w

(1-~2)(1-w')w

w'

(1-~,)(1-~2)(1-~')w 2

x__

~1"~o' (1-~)

~2"~'(1-w)

(1-~')

~1"~2"~'(1-~)~

Exactly the same expression is derived, if l-x--- is calculated through the formula which was used X---

for

the second method.

In conclusion, we may state the following rules for the use of diagnostic systems:

98

1)

A diagnostic system conceived for a situation with a prior probability distribution P(E2),...,P(Ek) must not be transferred to a situation with another prior probability distribution P'(E1),...,P'(Ek), without being adapted.

2) Adaptation supposes that P(Zjrj [Ei) remains unchanged if P(Ei) is altered. 3) Adaptation can be achieved by using P(Ei*IZj r . ) ' P ' (El*) 1

p,(Ei, i ZJrl) =

P(Ei*) k

Z i:1

(5.10)

P(Ei [Zj r . ) "P'(Ei)

i P(Ei)

- according to (5.3) -together with the prior distribution P'(E1),...,P'(Ek).

4) Adaptation can also be achieved by applying Formula (5.6) of Theorem (5.1). We define P(Ei[ZD2d):=P'(Ei), i = l , . . . , k , designating Z1+2,2 as the observation that the prior distribution has changed from P(Ei) to P'(Ei). Then we have for r1~1 = 1 : 1 P' (El* [Z2r1 ^ . . - ^ Zlrl) = P(Ei*[Ztr 1 ^ . . . ^ Zlr 1 ^ Zl÷2,1) =

k

X

1+2

• i IJ, P(E *IZjrj) 1

1÷2

(5.11)

• i--1J, P(E~IZj,.)

5)

Both methods of adaptation produce the same result.

6)

It must be admitted that often it will not be possible to gain reliable point estimates W(Ei), since one does not possess enough knowledge concerning the situation to which the diagnostic system is transferred. In those cases both methods of adaptation can be combined with procedures described in the next chapter.

99

CHAPTER 6 Interval Estimation of Probabilities in Diagnostic Systems The argument is widely used, that classical probability theory cannot be applied in diagnostic systems, because in many cases the available information is not sufficient to establish reliable point estimates for the probabilities. The latter statement has to be accepted, especially in those cases where the estimation of probabilities stems from sampling. In such a case only interval estimation may claim a certain amount of reliability. Shafer's theory of belief functions is merited with the ability to handle this problem. In this chapter we aim to derive solutions - using probability theory - for the problem of combining information which contains interval estimation of probabilities. We shall confine ourselves to models which assume k-independence and shall restrict ourselves to the simplest of situations: There are only two states of nature, two alarm units and each unit has two different signals. In this case the assumption of k-independence means double independence or (EF,Es)-independence. Now we suppose that the probability x** of the failure EF has to be estimated, if both alarm units go off; this event is denoted by Z1~^Z21 (instead of Z~AZ2 as in Chapter 4). For a problem of this type we suggested - if the necessary requirements concerning the underlying model are given - the application of Formula (4.6a), which uses the probabilities

P(EFlZ11) :

~

P(EFIZ21) = w2

P(~) : ~. While in Formula (4.6a) it was supposed that exact values for these probabilities are given, we will now allow three interval estimations to be given. We shall call this the basic information for x,,: Lj < wj < Uj L <~
j = 1,2

(6.1)

Due to the restrictions described in Chapter 4 we shall always suppose that L,Lj < 1 and U,Uj > 0. It should be noted that the symbols L] and Uj in this chapter are not used for the same variables as the symbols Li and Ui in Chapter 2. Normally, the index i distinguishes states of nature, the index j different alarm units or sources of information. However, in Chapter 2 the index j is sometimes used to specify a certain i. As long as we take a look at situations in which only two states of nature are distinguished, the limits for the probabilities of EN are determined by the basic information, if the estimates are feasible.

100

In this chapter we shall discuss different ways of treating this problem. We aim to show that the most adequate method depends upon the kind of additional information available. By additional information we mean all information beyond the basic information on the probability which is to be calculated. Therefore additional information for x÷, is basic information for x__. It is obvious that in diagnostic systems additional information will be automatically available in nearly all cases, because not only x÷÷ is of interest, but also the other probabilities. Nevertheless, we shall use the expression "basic information" in the following chapters with respect to x+,. For the purpose of comparison we shall start investigating a situation in which indeed only basic information is available.

101

6.1 An Approach without Additional Information A first method, which is rather simple, will lead to weak results. In applying this method we combine Equation (4.6a) with (6.1) in a simple mathematical way. The result is given by

Li .L2 U L1.L2 + (1-LI)-(1-L2) U

U1 .Us < x.

<

L U~.U2 + (1-Ut). (1-U2)

(l-U)

L

,

(6.2)

(l-L)

if U < 1 and L > O. In the case U = 1 the lower limit for x** is equal to zero; the upper limit can be calculated by (6.2). In the case L = 0 the upper timit for x , is equal to one; the lower limit is given by (6.2). It is easy to justify this result: At first, it has to be shown that the used constellations are not impossible, i.e. that simultaneously w~ may take the value L~, ~ the value L2 and w the value U to produce the m i n i m u m of x,., or that wj may take the value Uj (j = 1,2) and co the value L in order to produce the m a x i m u m of x,,. It must be taken into account, that the application of (4.6a) presupposes the model which was employed in Chapter 4 and therefore presupposes Equation (4.1). It should be noted that (6.1) does not contain any information about pl, p2, ~1 and 7_~. Above all no information is given which prevents us from taking Pl and P2 to be very near to zero. In this case the values of co, col and a~ are practically independent from each other and co can take its maxinmm, while col and ~ take their minima or vice versa. Secondly, it must be shown that - by using the smallest possible values for Wl and ~ and the largest possible value for w - the minimum of x++ is indeed achieved. This fact and the symmetric result for the m a x i m u m are easily seen, if (4.6a) is converted to ~01 " (-02

Xl'a~2 + (1-~t)" (l-x2) Lo 1-0o

= {1+ (1-a/l)" (1-a/2) el

X2

OJ } -1

" ]~d

(6.3)

which shows that x+, is a monotonously increasing function in oal and ~ , but a monotonously decreasing function in w. On the one hand, (6.2) produces the weakest estimation of x,+: Under no circumstances whatsoever can these limits be exceeded, if (6.1) holds. On the other hand, it provides the most accurate estimate, if only (6.1) and no other information about the parameters concerned may be taken to be known. Therefore additional information is necessary, if the result (6.2) is to be improved. The fact that these intervals are really very large will be shown in the following examples.

102 Example (6.1): Returning to the Example (4.3) we assume: 0.01 _<~j _< 0.02 j=1,2 0.0003_<~ < 0.0006 and obtain 0.1453 _<x÷, _< 0. 5812 Example (6.2): 0.1 _<xi _<0.2 0.055 <x <0.128

j =1,2

0.0776 _<x+÷ < 0.5178

In the following the estimates derived by (6.2) will be called the trivial estimates of x++.

103

6.2 A d d i t i o n a l

Information

a b o u t ~j

It is obvious, that an improvement of (6.2) is only possible, if more information about the parameters used in the model is taken into account than was done in the derivation of (6.1). We should consider that there is a symmetry between wi and wj, which corresponds to the symmetry between the signs Zil ("the alarm unit j goes off") and Zj2 ("the alarm unit j does not go off"). Therefore an obvious assumption is, especially for diagnostic systems, that the same kind of information is given for wl = P(EFIZJ2) as was given for wj = P(EF[Zjt), even if ~j is not apparent in Formula (4.6a). Accordingly we assume: I~i <-~i <-Ui j : 1,2 (6.4) with the restrictions Li < 1 and Uj > O. In the same way as in Chapter 4 we exclude the case: [Lj,Uj] = [Li,Vi]. If the estimates (6.1) and (6.4) are to be used simultaneously, obviously the fundamental Equation (4.1) must be taken into consideration. The prior probability must, for each of the two alarm units, result in a linear combination of ~vj and ~i with weights pj and l-p1 which must lie in the interval [0;1]. Therefore interval estimates which do not allow for the existence of probabilities pj should not be admitted. The control of admissibility, in this sense (the exact definition will be given later), is only possible through a calculation of these probabilities, taking into account that each pj can result not only in a single value but also in an interval. As each pj is determined by only co, a:j and ~j (not by coj,, ~j,, j'¢j), in the first part of our considerations we will confine ourselves to a situation in which pi is determined through a:, col and ~l.

Therefore the following problem arises, if possibilities to improve the trivial estimate (6.2) are studied: If

L~ _ wl _
I:1 _<~1 _
(6.5)

L <w
which of the triples (wl,~,w) are possible in the sense, that the three values co1,~1 and a: can occur simultaneously? This question is very closely related to the problem of the existence of solutions pl for Equations (4.1). It is easily seen, that under the condition 0:l ~ vi:

p~ :

01

-

~1

(6.6)

is a mathematical solution of this equation, which, because of Pl = P(Zn), must, in the present context, lie in the interval [0;1]. This is obviously granted, if and only if w lies between wl and ~l. Therefore the first answer to this problem is: A triple above, if w lies between wl and ~ .

(wl,~,w) is possible in

the sense described

104

Such an answer would not take into account that Pl is a quantity of the same kind as w, ws and 51. As for this type of quantities interval estimates have to be used in diagnostic systems, we are entitled to assume that the estimate for pl is also an interval. If (6.6) was used, the set of solutions would depend upon Wl and ~ and so would yield another interval estimate of Pl for every (ws,~), but would not produce one single interval. We therefore are in search of an interval of ps-values so that, for every element Pl of this interval and for every (wl,~) between their respective limits, a w-value between L and U is obtained through (4.1). Additionally, for each of the two limits L and U there must exist at least one triple (wl,wl,pl) so that (4.1) produces this value. Therefore our concern is to find: pL, p~E [0,1] ; p L < p ~ SO that [L,IJ] : {o~: w = psws + (1-ps)~s ; Ll_<Ws_(U1, ]~1_(~1_(]J1, 0 < p L _ < p l < p U < l }

(6.7)

This leads to the following definition Definition

(6.1):

A System (6.5) is interval-admissible, if pL, pU exist, so that (6.7) holds. A necessary condition for the admissibility of such a triple of intervals can easily be derived from (6.6): L must lie between L1 and Ls

and

U must lie between Ux and U1.

If these conditions are not fulfilled, System (6.5) can never have solutions pl which lie in the interval [0;1]. In this situation there is a striking discrepancy between the interval estimations used. In order to discuss further conditions of interval-admissibility, we have to distinguish four possible cases concerning the location of the limits of ws and

Case a)

Case b)

Case e)

~s:

Us _>Us Ls > 15s

(The limits of w~ are greater than those of ~1)

Vl _>Us Ls < ]:s

(The limits of 0h are outside those of 51)

Vl < Us La _>]~s

(The limits of wl are between those of 5s)

105

Ut < UI

Case d)

Li < 17t

(The limits of ~1 are smaller than those of ~1)

As equality between two limits does not raise special problems, we are able to treat Case a) and Case d) in the same way: The limits of oa are either both greater than the respective limits of ~1 or they are both smaller. In Case b) and Case c) the interval for one probability - Wl respective ~1 contains the interval for the other one. Therefore we may treat these two situations in an analogous way. Let us at first consider Case a). Taking into account that w is a monotonous increasing function in Wl as well as in ~ , we can conclude that the following inequality must hold for every admitted Pl: plLa + (1-pl)I71 < w(pl) = plwl + (1-pl)51 _
(6.8)

~1 the

respective limits are used. Furthermore, if U1 > U~, we have to take into account that the upper limit is a monotonously increasing function of Pl. If therefore pU is chosen so that U : pUU1 +

(i-p~)131

pU _ U-U1

(6.9)

Ut -UI then for all p < pU the inequality

o)(pl) _
is valid.

The same derivation can be carried out for the lower limit of p~ according to (6.8). If L1 > L1, this linear function in Pl is monotonously increasing and if we solve L : pL, L, +

by

pL_ L-];, , L 1-I~1 there will be w(p,) >_L for all p >_pL.

(6.10)

In the case that U1 = U1, the inequality v(pl) _
I~ < L < L1 U, < U _
(6.11)

As long as pL < pU, then for all Pl between these two limits the respective w(pl) lies between L and U and these two limits are indeed obtained. Therefore in this case System (6.5) is interval-admissible.

106

Although (6.11) holds, it is nevertheless possible that pV < pL. In this situation values of p~ do not exist, for which w(pl) lies between its limits L and U for all values of wl and wi, which are in accordance with (6.5). In this situation these three interval estimations for the prior probability w and for the conditional probabilities w~ and wi cannot be used simultaneously: they do not fit together. Due to pL > p~ and because of the characteristics of Case a) the two implications hold:

pLL, + (l-pL)l~i = L

L U Pi>Pl ~

p~LI + (l-p~)l:, < L

(6.12a)

p~U~ + (l-p~)U~ > u

(6.12b)

pbpV p~U~ + (l-p~)U, : ~

~

It is obvious that under these circumstances the estimates used in (6.5) have to be modified, if they are to be used in the diagnostic system. Roughly speaking, the limits for w are too narrow compared with the limits for wl and ~1, or the limits of w~ and ~ are too wide compared with the limits for w. As long as all estimates used are taken as valid - especially the estimates for w - the estimates for wl or ~ may be improved, but it is not determined in which way. One may, for instance, improve the estimation of w~ by increasing L~ until the inequality in (6.12a) becomes an equation. This means that, as long as p~ > 0, the new value of Li is given by

Ltnew-

L-(1-p~)~,

pO

( > El )

(6.13)

supposing that this new value of L1 is not larger than Uv Application of L1new in Formula (6.10) produces: pL = p~. We could apply the same procedure to L1 with the result of Z1new -

L-p~LI

( > ]~1 ) ,

(6.14)

1-p~ again with the restriction that the new value must not be larger than U1. (6.10) produces pL = p~ also in this case. If for any reason the use of (6.13) or (6.14) is too restrictive for wl respectively for ~t it is possible to increase L1 as well as ~t until the inequality on the right side of (6.12a) becomes u an equation. In all of these cases Pl remains unchanged and becomes the only admissible value of Pi so that the interval for p~ degenerates to a single point. All the upper limits of w~ and ~ remain of course unchanged by this procedure; so the equation on the left side of (6.12b) remains true. Alternatively an analogous procedure can be used to diminish the upper limits U1 or U1 in one of the two following ways: ul"e~ -

U-(1-p~)U1 L ( < Vl ) Pl

(6.15)

107

or as long as pL < 1:

~ llew

U-p~U1 - _

( < l~i ) 1-pL

(6.16)

or a combined treatment can be applied to U1 and 13"1 in an analogous way as to L1 and [71. In all these cases the interval for Pl degenerates to the single point pL. Furthermore, it is possible to apply a mixture of the two mentioned procedures so that upper limits as well as lower limits are modified - in the extreme case all four limits. Which of these many possible procedures is applied, depends only on material considerations, not on formal ones. The question is, which of the possible improvements can be justified best with respect to the matter under consideration. Nevertheless, the possibility that none of these improvements seems to be acceptable, must be taken into account too. If the respective source of information is to be used at all, the information included in System (6.5), which is not interval-admissible, has to be modified in the other direction, i.e. weakened. In this case the interval [L;U] for the prior probability co has to be expanded. By doing so it is admitted that the original estimate for co may be untrue. The expansion of the interval [L;U] again can be carried out in different ways. The first method defines the lower limit L by Lnew = pUL, + (1-pU)];1 (6.17) and uses pU as the degenerated interval for Pl; another possibility results in Unew = pLU1 + (1-pL)U1

(6.18)

and leads to p} as the only value of p,. It is evident that a mixture of these two procedures is also possible.

Case d) - apart from the treatment of the equality sign - is analogous to Case a), if wl and ~1 respectively Pl and (l-p1) are interchanged. Case b) is determined by the fact that the interval for o01 contains the interval for ~l. The requirements for w, which stem from the fundamental Equation (4.1), lead to the condition L1 _
(6.19)

which is necessary for interval-admissibility of System (6.5). It is obvious that (6.8) is also valid in this case, but while the upper limit is still a monotonously increasing function of Pl, the lower limit is now a monotonously decreasing function of Pl. Therefore, if L1 ¢ L1 and U1 ¢ Ui, one arrives at the following two limits for pi: lp~ = L-r~ , ~p~_ U-~I L 1-1;1 U,-U1

(6.20)

108

The case that either Lt = ]51 or U1 = U1 will be discussed under the section for case bT). Both of the limits (6.20) are upper limits and the only lower limit for Px is zero. If pl is very near to zero, the limits (6.8) are very near to the limits of ~

and therefore must be contained in the

interval [L;U]. If lp~ = 2pV = pV, a unique interval [0;p~] for Pl emerges as a solution to the problem under consideration. System (6.5) is therefore interval-admissible. If lpO1 ¢ 2p U, a solution cannot be found: Either the one limit for w is violated or the other one is not reached at all. We distinguish: Case ba)

XpU < 2pU:

Then the following implications hold ~p~L, + (1-'p~)]:1 = L ~ 2pULl + (1-2p~)151 < L 2p~U1 + (1-2p~)U1 = U ~

lpUU1 + (1-1p~)U1 < U

(6.21a) (6.21b)

As System (6.5) is not interval-admissible, modifications to the used estimates must once again be made. As long as all estimates of System (6.5) are trustworthy, some of them can be improved in order to obtain an interval-admissible system: Either L1 or ]:1 can be increased until the right side of (6.21a) becomes an equation. The two alternative solutions are either L1new and I:1 with L1 new -

L-(1-2p~)rl 2p~

( > L1 )

(6.22)

or L1 together with LlneW, where L-2p~L1

~|neW _ _ _

(>[1)

(6.23)

1-2p~ As for Case a), a mixture of both procedures which establishes the equation on the right side of (6.21a), is also possible. In all these cases the interval for Pl is: [0;2pV]. Another method which improves estimates used in (6.5) is to lower U so that the right side of (6.21b) becomes an equation: Unew = lpUU1 + (1-tpU)U1 ( < U )

(6.24)

If this modification is used the information concerning wl and ~1 remains unchanged. In contrast to Case a) this method allows us to improve the estimate of w in order to establish an interval-admissible system.

109

Again it is up to the experts to judge which of the possible moderations is most acceptable. If none of these improvements can be justified, the only solution is to accept expansions of the given intervals. It is possible to enlarge U1 or Ut in order to yield an equality for (6.21b). This leads to: U1new and U1 with

u-(I-'pV)N U1new

'pV

( > U1 )

(6.25)

or to U1 and U1new with U- lpUU1 Vlnew =- ( > UI ) 1-1p U

(6.26)

In this case the probability interval which is a solution of the modified System (6.5) is [0;~pU]. Otherwise L can be diminished until the right side of (6.21a) becomes an equation. This leads to L new : 2p~L1 + (l-2pU)]:1 ( < L ) (6.27) and in this case the interval for p~ is: [0;2pV]. Case b~): ~pU > 2pU: this case is treated analogously. The improvement of the estimates is achieved by decreasing U1 or U~ (or both) or by increasing L. If this is not possible, either L1 or L1 have to be decreased or U must be increased. Case b7): If Lt = g~ (=L) and U1 < U~ the relation L = p~L~ + (1-pl) E1 is true for all Pl E [0;1]. If we calculate 2pV through (6.20) the interval [0;2pU~] renders a solution for all 0 < Pt _<2p~. Inequality (6.8) holds and the limits L and U are reached. Therefore (6.5) is interval-admissible together with [0;2pU]. In the same way we can conclude that for U, = U~ and LI < L1 System (6.5) is interval-admissible together with [0;tp~] according to (6.20). Case c) is easily derived from Case b) if cat and ~ are interchanged. Of course this leads to an interchange between p~ and 1-pl, so that we obtain two lower limits of Pl in this case, while the maximum value is always equal to one. The expressions (6.22) - (6.27) remain valid, if it is taken into account that we have to use the lower limits p} instead of pU. From now on we shall always assume that modifications have been made if necessary and that System (6.5) is interval-admissible. This will be taken for granted in the following chapter for both sources of information; later for all sources of information.

110

Moreover, due to Equation (4.1) the same estimate of w has to be used, i.e. the same L and the same U, for all sources of information. Some practical problems may arise, if in the course of making a system admissible for one source of information, modification of L and U is taken into consideration. In such a case the computations concerning the admissibility of the systems stemming from the other sources of information have to be repeated. There are no formal methods concerning the problem of bringing different systems of the type (6.5) into agreement so that a common estimate [L;U] for w can be employed. This is only dependent upon the respective reliability of all the information used.

111

6.3 The Combination Rule for Two Units In this chapter we shall confine ourselves to a situation in which only two alarm units exist. Therefore the description of the problem is given by the following parameters: Lt < tot <_ Ut

L2<_to2<_U2

L1 _<~t _
L2 _<~2 _<02

(6.28)

L_
for j=l,2

A value for w can appear simultaneously with a pair ( ~ , ~ ) , if it can appear simultaneously with both wa and w2. If we denominate Max x(vl,to2) =Min {Max[pjtojL + (1-pL)Uj; p?vj + (1-p°)Uj]} j=l~2 and Min to(to,,to2) = Max {Min[pLwj + (1-pL)]:j; pjtoj U + (1-p~)[:j] } j=l~2 we find Min to(~l,to2) _
(6.31a)

(6.31b)

(6.31c)

as the range of all total probabilities ~ which can occur simultaneously with ~1 and ~2, provided that: Min to(tol,w2) _<Max ~d(tol,to2)

(6.31d)

112

It is, however, by no means evident, that (6.31d) must hold for all pairs (0d, ~2). If Min [pLoj1 + (1-pL)]:t; p~0:t + (1-p~)l~t] > Max [pLw2 + (1-pL)]:2; p2~z2 + (1-pV)I:2] then all total probabilities 0:, which can occur simultaneously with ~ , are greater than those, which can occur simultaneously with x2. Analogously L + (1-pL)]~2; p2w2 U + (1-pU)]]2] > Max [pL~, + (l_pL)i]l; P ~ I + (1-p~)I:t] Min [P2X2 states that all values of ~ occurring simultaneously with ~2 are greater than those occurring simultaneously with wl. In any of these cases a total probability x, which can appear simultaneously with both 0~t and 0J2, does not exist. This must be interpreted in the sense that ~1 and x2 can never occur simultaneously. Consequently all pairs (xl,x2), for which (6.31d) does not hold, should not taken into account in searching for possible results of the combination rule. It will, however, be seen that these exclusions do not have an effect on the final estimate of x++, due to the following lemma and Theorem (6.1). Lemma: (6.31d) holds for zJ = Uj, j = 1,2 and for ~i = Li, J = 1,2. Let ~j = Uj, j = 1,2: Because of interval-admissibility of both systems, L * (1-p )Uj; p Vj + (1-pC)Uj] : C Max [pjU] must hold for j = 1,2. Therefore: Max ~(U~,U2) = U,

while for

L < pjUj + (1-pi) q _
so that L ~ iinx

(h,U2) _
If ~j = Lj, j = 1,2, interval-admissibility leads to

Li+ Min [pjL

pUL1 +

=L

and therefore Min ~(L1,L2) = L , while for all pL _
j = 1,2

and therefore L _<Max w(L1,L2) _
113

T h e o r e m (6.1)

With respect to (6.28) and (6.29) the following inequality holds, if 0 < Max ~o(Lt,L2) < 1 and 0 < M i n w(UI,U2) < 1: L1L2 U1U2 Max w(LI~L2) < x+÷ Min w(U,,U2) (6.32) L,L~ + (I-L,)(I-L2) U1U2 + (1-U 0 (l-U2) Min z(UI,U2) 1-Min z(U1,U2) n Max ~(LI,L2) l-Max ~(LI,L2) The proof of this theorem usesthefollowing Lemina: ~1~2 Min w(wl,w~) xu(~l,~ ) = ~1~2 + (1-~)(1-e2) Min X(el,X2) 1-Min e ( ~ , e 2 )

(6.33a)

as well as ~l&2 xL(~l,~2 ) : Max ~ ( ~ , ~ ) ~ + (1-~,)(~-~) Max a(al,a2) 1-Max a(~1,~2)

(6.33b)

are monotonously increasing functions of oa and ~2. The proof of this lemma is only demonstrated with respect to xu(wl,~); for XL(aJ1,0J2) the proof is quite analogous. We have to distinguish four cases. Let us first assume that a) Mill a(WI,X2) = pLw1 + (1-pL1)I~I. Therefore Xu(tO,,~2)

= [ q + (l-t°1) (l-t°2) l [pLItjI+(1-pL)I~I] ] 0~1~2 { 1- [pL~0~+(1-p~)]~l] } [

I

L

=

L

(1-~02) " ~11 [Pla/I+(1-Pl)]~I] ]-1 = = 1 ~2.]~xl[pL(l_w~)+(l_pL)(l_i:l)]

[

+

L L ]~1 (1-0~2) • [p,+(1-pl) ~ ]]-' w2[pl+(1-pl)

l-w1 ]

whereby it is shown that in this case Xu(Wl,w2) is a monotonously increasing function of wl and of a.~.

114

b) Mi~ ~ ( ~ , ~ )

= p ~ , + (1-p~)~, :

p~ is always used instead of p}, but otherwise the proof and its result remain unchanged. The cases c) Min o~(el,e2) = P2LeO2 + (1-pL)I72 and d) Min e(e,,a2) = p~c02 + (1-p~)I72 are gained by interchanging the parameters for the first and the second source of information and they produce the same result.

[]

Proof of Theorem (6.1): Due to monotony the largest possible value of x++ is found if we take cat = U~, ~ = U2 and = Min w(Ut,U2). The smallest possible value of x++ is achieved if ~ol = L1, ~ = L2 and ~o = Max ~o(L1,L2). From this the inequality (6.32) follows immediately and the limits given there are seen to be the best possible ones. Because of the monotony of xu(el,~2) and XL(e~,e2) it iS easily seen that the lower limit of x++ can never be greater than the upper one.

[]

The application of Theorem (6.1) leads to the best available limits for x+÷. In most cases it will result in a remarkable gain in information compared with the trivial estimate. If on the other hand, the estimation which is derived from Theorem (6.1) is not sufficient with respect to a certain demanded accuracy, it must be understood that this cannot be helped by any formal method. In such a case the requirements of higher accuracy can only be met if either the accuracy of basic estimates is improved or any new material assumption is used. The use of Theorem (6.1) can be extended to the case in which Max 0J(LI,L2) = 1 or to the case in which Min L0(Ut,U2) = 0, while Max 0~(L1,L2) = 0, respectively Min e(UI,U2) = 1, can be excluded due to the restrictions described above. Since Lj < 1 and Uj > 0 (j = 1,2), it can be concluded that the lower limit for x+÷ is zero, if Max ~0(L1,L2) = 1 and that the upper limit for x++ is one, if Min ~0(UI,U~) = 0. We shall compare the results of Theorem (6.1) with the trivial estimate (TR) given in (6.2), using the parameters of Examples (6.1) and (6.2). Example (6.3): 0.01

< wj < 0.02

0.0001 < ~j _<0.0002

( j : 1,2)

O. 0003 < La < 0.0006 With this additional information about 5j we obtain: pL = pU = 0.0202. This is not a typical case, because the common estimate for pl and p2 results in a single number, not in an interval. Applying (6.32) - which becomes very simple in this case - we derive 0.2039 < x++ < 0.4533, a remarkable gain in accuracy compared with the result of Example (6.1): 0.1453 < x++ < 0.5812.

D

115

Example (6.4): 0.1 _<xj _<0.2 0.01 _<~j < 0.02

j = 1,2

0.055<~ <0.128 With this additional information about ~j the limits for pj (j = 1,2) are 0.5 and 0.6. These values are applied to (6.32) and yield: 0.1447 < x÷÷ _<0.3476 Therefore, also in this example, the accuracy of the estimate is improved considerably (compared with: 0.0776 < x÷÷ < 0.5178). [] In analogy to Formulas (4.6) we apply Theorem (6.1) to the remaining three probabilities of our model and obtain (by replacing wj with ~i and pj with 1-pj and using the analogous restrictions as for Theorem (6.1) and the conclusions analogous to those above): L1E2 U1~2 Max w(LI~2) < x+-< Min w(Ux:U~) L,~2 + (1-Lt)(1-~2) U,U2 + (1-U,)(1-U2) Max ~(L1,~2) 1-Max ~(L1,~2) Min ~(U1,U2) 1-Min ~(U,,U2)

(6.34)

where Max w(L,,I:2) : Min { Max[pLL, + (1-pL)V,; pUL, + (1-pU)Ux] ; Max[(1-pL)1:2 + LU Min w(U,,U2) : Max { Min[pLU1 + (1-pL)I:,; p~U, + (1-pU)]~,] ; Min[(1-pL)U2 + P2LL2; (1-pU)U2 + pUL2]}. ~,L2 UaU2 Max w(~,,L2) < x-+ < Min w(U~,U2) ~,L2 + (1-~x)(1-L2) U,U2 + (1-U,)(1-U2) Max w(~,L2) 1-Max w(Et,L2) Min w(U1,U2) 1-Min w(U1,U2) where Max w(]:,,L2) = Min { Max[(1-pL)]:, + pLu,; (1-pU)]~, + pVU,] ; Max[p~L L ~ + (I-pL)U~; p~C L 2 + (1-p~)U~]} Min w(U1,U2) = Max { Min[(1-pL)U1 + plLa;L (i_pU)U1 + pUL,] ; Min[pLU2 + (1-pL)I:2; p2u2U+ (l_p2U)i:2]}

(6.35)

116

~t~2 Max v(~1,~2)

r,r2 Max x(E1,E2)

+

(~-r,)(~-~2) 1-Max x(El,~2)

UtU2 Min ~(Ui,U2)

< x_-<

uiu~ Min

+

x(U1,U2)

(6.36)

(1-v,)(1-u~) 1-Min ~(U1,U2)

where U Max w(I:l,I~2) : Min { Max[(1-pL)ITj + pjLUj; (1-p~)l:j + pjUj]} j=l~2

Min ¢(Ua,U2) = Max { Min[(1-pL)Ui + piLLJ; (1-p~)% + p~Ljl } j=l~2 To avoid misinterpretations: Each of the symbols Max v(., .) and Min v(., .) has to be understood solely according to its respective definition, while x(., .) is not used as a symbol of its own and changes its role from one formula to the next. For practical calculations it should be noted that in the Formulas (6.32) - together with (6.31) and (6.34) to (6.36) each of the eight weighted averages is used four times: that is once in each formula. If this is taken into consideration the effort of calculation is reduced remarkably. We shall demonstrate these results by an example which is a slight variation of Example (6.4). Example (6.5): Lt = 0.1 ~1 = 0.01

Ul= 0.2 Ut = 0.02 L = 0.055

Like in Example (6.4) we have: In the same way we find:

L2 = 0.12 E2 = 0.012

U2= 0.18 U2= 0.018

U= 0.128

p~ = o.5 p~ = 0.39sl

p7 = 0.6 p~ = o.6790

Therefore the system of estimates is interval-admissible. It is possible to demonstrate by means of this example the existence of pairs (xl,0J2) which cannot occur simultaneously. If, for instance, vt = U1 and x2 = L2 are used, it is found according to (6.31a) and (6.31b) Max x(U1,L2) : Min {Max[0.1100; 0. 12801; Max[0.0586; 0.0873] } : 0.0873 Min ~(U1,L2) = Max {Min[0.1050; 0.12401; Min[0.0550; 0.08531 } = 0. 1050, a result, which indicates that ~a = UI and v2 = L2 can never occur simultaneously.

117

In order to prepare for the application of (6.32) and (6.34) to (6.36) we calculate:

Then we find

pLL, + (1-pL)U1 = 0.0600

p~L, + (1-p~)U, = 0.0680

L p~L2 + (1-pL)U2 = 0.0586

p2L2U + (1-p~)~2 = 0.0873

pLu, + ( 1 - p i ) i : , = 0.1050

p~U, + (1-pU)L, = 0.1240

p~u2L + (1-p~)~2 = 0.0789

U p2U2 + (1-p2U)I:2 = 0.1261

Max ~(L1,L2) = Min(0.0680; 0.0873) = 0.0680 Min ~(U1,U2) = Max(0.1050; 0.0789) = 0.1050 Max x(Ll,I~2) = Min(0.0680; 0.1261) = 0.0680 Min 0;(U1,U2) = Max(0.1050; 0.0586) = 0.1050

Max w(E1,L2) = Min(0.1240; 0.0873) = 0.0873 Min ~(U-1,U2) = Max(0.0600; 0.0789) = 0.0789

Max o~(L1,L2) = Min(O.1240; 0.1261) = 0.1240 Min x(UI,U2) = Max(O.0600; 0.0586) = 0.0600 This leads to 0.1720 _<x÷, < 0.3187 0.0182 < x+- < 0.0376 0.0142 _<x-, < 0.0497 0.0009 _<x__ < 0.0058

It should be noted, that Max 0;(El,L2) > Min 0;(UI,U2)

and

Max 0;(El,E2) > Min 0;(U1,U2) are results in accordance with the model used. They demonstrate that total probabilities may exist which can occur together with the pair of greatest values of the conditional probabilities as well as with the pair of smallest.

[]

118

6.4 The Combination Rule for More than Two Units In the following we shall confine ourselves to the discussion of the following problem: There are two states of nature. There are 1 alarm units each with two different signs. The random events Zjl (respectively Zj2) are mutually double independent, j = 1,...,1 (see Definition (5.2)). There exists an interval estimate for each of the probabilities wj and wj: [Li,Uj] respectively

[l:i,vi]. There exists an estimate for w: [L,U]. Under these circumstances we are able to generalize straightforwardly the results described previously in Chapter 6.3. The problem of admissibility has already been mentioned: It is necessary that all the interval estimates fit together: There must exist pL V L U J , Pi e[O;1] , Pi -< Pi , J = 1,...,l so that each of the systems of type (6.5) is interval-admissible. Without loss of generality we shall discuss the calculation of an interval estimate for P(EFIZu^ . . . ^Zll) =: Xl, Each other sequence of the signs (Zjl or Zj2) can be treated in the same way by interchanging the parameters ~oj and ~j, respectively pj and (!-pj). The formula which describes the calculation of this probability is a specialization of Formula (5.6): ~1

" • • • "~01

o)1-1 Xl+ = ~gl"..-"0)1 + ( l - X l ) ' . . . "

~11

(6.37)

(1-~1)

(1-o~)1-1

and at the same time it is a generalization of Formula (4.6a). Therefore the same considerations can be used as those applied in the case of 1 = 2: For the trivial solutions we arrive at the following result: L~.....L1 UI-i LI-...'LI

+ (1-LI).....(1-L1)

U1 1

(1-U) 1-1

UI.....U1 Ll-I

_<Xl÷ _<

,

(6.38)

U,'...-U1 + (1-Ua).....(1-Ut) L1 1

( l - L ) 1-

if U < 1 and L > 0. In the case that L = 0 the upper limit for x]÷ is equal to one; the lower limit is calculated by (6.38). In the case that U = 1 the lower limit for xl, is equal to zero; the upper limit is given by (6.38).

119

Analogously to the derivation of Formula (6.32) we take into account that for each 1-tuple (wi,...,wl) the greatest value of xl, emerges if w adopts the smallest possible value with respect to w~,...,oa. The analogous result holds for the lower limit and the largest w which is possible for wl,...,Wl. As for I = 2 we arrive at a fundamental theorem. Theorem (6.2): Ifpjg" _
j = 1,...,1,

and 0 < Max ~(L1,...,L1) < 1 as well as 0 < Min o~(U~,...,U1) < 1 , then

LI • . . • "LI [Max

~(Ll,...,]J1)]l-i

xl+ L,.....L1

+

[Max w(L1 . . . . ,L1)] 1-1

(6.39)

(1-LI).....(1-L1) [1-Max w ( L I , . . . , L 1 ) ] l-I U1 • • • • .U1

[Min x ( U 1 , . . . , U 1 ) ] 1-1 (

u1.....u~ [Min w ( U 1 , . . . , U l ) ] 1-1

+

( 1 - o l ) . . . . . (1-u~) [1-Min ~ ( U 1 , . . . , U 1 ) ] 1-,

where Max w(wl, • • ., wl) =

Min

L U {Max[pLwj + (1-pi)Uj; pjwj + (1-pV)Uj] }

(6.4oa)

{Min [pjwj L + (1-pj)]~j; L pUwj + (l_pV)~j]}

(6.40b)

j = l ~ • • • ,1

Mina(0~l , . . . ,

al) =

Max j=l~

• ..

,1

[]

The proof of this theorem uses a lemma which is a simple generalization of the one used in the case of 1 = 2: The upper as well as the lower limit are monotonously increasing functions of each of the wj, if (6.40) is used. The proof of this lemma is analogous to the proof of the lemma for the special case. Using this lemma, Theorem (6.2) follows in a straightforward manner: The extreme values of xl+ are produced by taking the extreme values of oa,...,~.

[]

The case that Max w(L1,...,L1) = 1, leads to the result that the lower limit in (6.39) is zero; the case that Min ~(U1,...,U1) --- 0, to the result that the upper limit is one. This is concluded in the same way as for Theorem (6.1). It should be noted that (6.40) results in, at least potentially, an additional limitation of w for each alarm unit. Therefore in the case of more than two sources of information, (6.39) has to be applied

120

instead of calculating xl. "step by step", a method for which an analogous derivation is not possible. This is different to situations in which all probabilities are given with precision. Up to now in this chapter we have not treated the problem which was introduced in Chapter 5.2 discussing the possibility that Mr.K. has a certain disease. If supplementary information about an individual case is available, the prior probability for this individual may be different from that used in the diagnostic system. In Chapter 5.2 two equivalent methods of adapting the diagnostic system to this situation were described, assuming that all probabilities are given as numbers.

If those probabilities are given as intervals, both methods of adaptation may still be used, but it seems to be much easier to employ the one which is mentioned under number 4. We shall briefly discuss this method and its execution. As long as the supplementary information has the same structure as information gained through questions, Formula (6.39) can be transformed by using Z].i,l in the same way as in Formula (5.11). This presupposes, however, that estimates are given for P1.1 and for ~1.1. In many cases such estimates will not be available or it will not be sensible to conceive probabilities like Pl~l and ~1+~ at all. This will be true, if the diagnostic system was calculated some time ago and the prior probability has changed in the meantime. In such a case it is impossible to extend (6.40a) and (6.40b) to (1+1). On the other hand, however, any extension of (6.40a) could at most lead to a diminished value of Max ~(~i,-..,~0, which would result in an increased lower limit of (6.39); in a symmetric way any extension of (6.40h) could at most lead to a decreased upper limit of (6.39). Therefore we recommend the use of Max ~(~1,...,~0 from (6.40a) and Min ~(wl,...,~1) from (6.40b) also in the extended version of (6.39). Further generalizations which increase the number of states of nature beyond 2 or the number of distinguishable signs for each unit to more than 2, are postponed for more specialized studies. Approximate solutions are possible if one state of nature, respectively one sign, is selected and the remaining states of nature or the remaining signs are considered collectively so that the distinction is only made between two states of nature, respectively two signs. The mode of operation of the methods described in Chapter 6 is demonstrated by means of a more detailed example in the next chapter.

CHAPTER

7

A Demonstration of the Use of Interval Estimation In this chapter we wish to demonstrate the performance of interval estimation, as described in the preceding chapter, by use of an example which consists of a tiny expert system. We assume that the aim of this expert system is to provide a statement concerning only two states of nature. Let us for simplicity call the one state of nature "failure" and the other one "normality". Then it is possible to use the same symbols as in the preceding chapter: EF and EN. The prior information contains an estimate of the probability of EF: This probability lies between 1% and 2.5%. If we employ the symbols L and U as in the last chapter, we have L = 0.010 and U = 0.025. The expert system poses four questions. Each of the questions allows only two answers. We shall designate them by "yes" and "no". We assume that intervals are given for both answers which describe the probability of failure if either answer is given. For the answer "yes" we designate these limits by Li and U] (j = 1,...,4) and for the answer "no" we use ~l and g I. Concerning Question 1 our information is given by: L1 = 0.050

U1 = 0.100

I~ = 0.000

U1 = 0.005.

The respective information for Question 2 is: L2 = 0.100

U2 = 0.150

1:2 = 0.010

C~ = 0.020,

for Question 3: L3 = 0.050

U3 = 0.150

I~3 = 0.005

U3 = 0.010

and for Question 4: L4 = 0.015

U4 = 0.020

I:4 : o. 005

v4 : o. 050.

It should be noted that for the first three questions the answer "yes" is obviously in favour of the state of failure while answer "no" favours normality. In the case of Question 4 the situation is different: answer "yes" contains much more information than answer "no", because answer "yes" gives narrower limits for the probability of a failure, while the limits in case of answer "no" are much wider and the interval [L4,U4] is contained in [L4,U4]. Using this information we shall calculate the probability of a failure in the case that a certain sequence of answers is given. At first we must prove the interval-admissibility of the four estimates

122

with respect to the prior information: [L,U]. This is done by calculating the probability intervals [ pL, p~ ] using the Equations (6.9) and (6.10). In the case of Question 1 this leads to: : 0.2

pV : o . 2 1 o 5 .

: o.o

: 0.0385.

In the case of Question 2 the result is:

If we use the same method for calculating the limits of P3 we get: pL=0.1111 pU:0.1071. This result informs us of the fact that the system is not interval-admissible. We have to decide whether we believe the estimates [L,U] or the limits [L3,U3] respectively [E3,U3]. We assume that there is no reason to doubt the prior information and also that we decide not to change the interval [Es,U3]. Therefore we can either derive a new lower limit L3 or a new upper limit Us. We decide to decrease the upper limit using Formula (6.15) and obtain U3ew = 0.145. With this solution we accept the degeneration of the interval for Ps to the single value P3 = 0.1111. Question 4 corresponds to what was referred to as Case c) in Chapter 6, where it was shown that this case is derived from Case b), if w4 is exchanged for 54 and p4 for l-p4. Therefore we have to derive an interval [pL4;1] as an estimate for P4, the probability of the answer "yes" to Question 4. The two solutions according to (6.20) are: lp L = 0 . 5 ,

2pL4=0.8333.

Again it was decided not to alter the prior probabilities but to use this information in order to improve the estimate for ~4, the probability of failure in case of the answer "no" to Question 4. Therefore we calculate ~ e w : o - ~p~ u4 : o.

03,

1 - lp L which corresponds to Equation (6.25). With the necessary corrections the interval-admissible system is as follows: L1 = 0.050

U, = 0.100

I;1 = 0.000

lJi = 0.005

0.2 < p~ < 0.2105 L2 = O. 100

U2 = O. 150

[:2 = 0.010

172 = 0.020

0.0 < P2 ~ 0.0385

123

L3 = 0.050 1;3= 0.005

II3= 0.145 U~= 0.010

P3 = 0.1111

L4 = 0.015

U4 = 0.020

1:4 = 0.oo5

u4 = o:o3o

0 . 5 _< P4 <- 1 . 0 .

Now we suppose that the answers to those four questions are double independent so that we can combine the information contained in the answers to different questions by means of the formulas derived in Chapter 6. First of all we calculate the probability of a failure if the answer to the first two questions is "yes". If we do not use any additional information we apply Formula (6.2), the trivial estimate, and arrive at the following interval for x,÷: 0.1857 < x** _<0.6600. If we apply the combination rule (6.32), we calculate, using (6.31): Max 0;(L1,L2) = 0.0145 (instead of U = 0.025 for TR) and Min x(U1,U2) = 0.02 (instead of L = 0.01 for TR) and obtain the following limits: 0.2848 < x** <_0.4900. Therefore the probability of a failure is at least 28%, although each individual probability does not exceed 15%. For the last time we report the result of the application of the Dempster-Shafer rule: 0.0144 < x** < 0.0173. Again this is an absurd one! Now let us assume that the answer to Question 3 was "yes" too. If we use the trivial solution we obtain: 0.3189 <_x+÷, _<0.9702. While this result shows the tendency to increase the probability of failure due to the third answer "yes", the interval is even larger than that in the case of only Question 1 and Question 2. If we employ the combination rule (6.39) to these three answers we obtain the following result: 0. 5890 _<x~++ <_O. 8830 (using Max v(Lt,L2,L3) = 0.0144 and Min 0~(U1,U2,U3) = 0.0206 apart from differences which are caused by rounding). Obviously a third answer "yes" has increased the probability of a failure considerably.

124

With respect to Question 4, we at first assume that once again the answer was "yes", remembering that this time the answer "yes" does not necessarily indicate an increase in the probability of failure, because the interval for 54 includes the interval for w4. The application of (6.38) leads to 0.2175 < x÷÷+÷ < 0.9850, which shows a loss of information even resulting from the answer "yes", because the lower limit decreased and the upper limit increased. Therefore, if the answer is "yes" and we employ the trivial solution, Question 4 is of no use. With the combination rule the result is 0.5982 < x÷÷.÷ < 0.8801 (with Max ~0(L1,L~,L3,L4) = 0.0144 and Min w(U1,U2,U3,U4) = 0.0206). Indeed this shows some progress compared with the result stemming from the first three questions. Therefore we may conclude that Question 4 may be useful to those who employ the combination rule (6.39) - as we in fact recommend. It should be taken into account that asking Question 4 contains the possibility that the answer is "no". The probability of this result, (1-p4), lies between 0 and 0.5 and therefore must be taken seriously into consideration. So it is useful to calculate additionally the interval estimate for the probability of failure in the case of the sequence of answers "yes, yes, yes, no". If TR is used the result is: 0.0840 < x÷÷÷_ < 0.9901. This is a very remarkable deterioration of the result for the first three questions. If TR is employed, any answer to Question 4 therefore results in a loss of information in the case that the first three questions are answered with "yes". Somebody who uses TR should avoid Question 4 in this case. If we employ the combination rule we obtain: 0.3296 < x+++- < 0.9175 which again is a much weaker statement than that we had gained from the first three answers. Since we are concerned with the application of the combination rule (6.39) we may conclude: If the answer to the first three questions is "yes", the answer to Question 4 results either in a small gain in information, if it is "yes", or in a serious loss of information, if it is "no". It is a good policy to exclude this question in this particular case. Furthermore, one should control whether the situation is at all different if other answers are given to the first three questions. Otherwise it would be better to avoid Question 4 from the beginning.

125

It is a remarkable feature of diagnostic systems using interval estimates for o4 ~j and ~j that questions like Question 4 may be a nuisance in all cases, because the intervals [Li,Uj] and [Lj,Ui] are so wide or because they belong to Case b) or Case c), as defined in Chapter (6.2). Therefore we should test to see which of the possible questions are of use and which of them would spoil the information gained from the others and should consequently be avoided. Concluding this argument we assume that Question 4 has been omitted and therefore x+÷÷, estimated through (6.39), may be taken as the final result for an individual case for which the estimation of the prior probability, as used in the diagnostic system, is true. Should, however, supplementary information consist of an increased prior probability, an adaptation of the result is necessary. Let us assume that for this individual case L'=0.04

and

U'=0.06

and that it is impossible to conceive probabilities P5 and ~5, which would be necessary if this information was to be used like information stemming from "Question 5". Due to the recommendation given in Chapter 6 we adapt the limits for x+++ gained so far by using L1.L2.L3.L' and U1.U2.U3.U' instead of L1.L2.L3 and U1.U2.U3, while Max ~o(LbL2,L3) and Min ~(U1,U2,U3) remain unchanged. This leads to a final result: 0.8029 < x+'++ < 0.9583 which demonstrates - compared with the corresponding limits for x+,+ (0.5890 and 0.8830) - the importance of the adaptation in this case. From further practical studies about diagnostic systems conceived in the way which is described in Chapter 6, valuable information concerning the application of these methods will be obtained.

APPENDIX Application of Formula (3.21) to Structures Defined by k-PPds.

P,(Ei*) We have

P(Ei*) = ~ P I ( E I )

i=l

and

" . . . " PI(Ei*)

(3.21)

"..." P I ( E i )

Sup Pj(Ei) = Uji PiES~. Inf Pj(Ei) = Lji PjES~.

We designate:

Ui*: =

i=l,...,k;

j=l,...,1

Sup~ P(Ei*) PjES~ j = l . . . . ,1

Li*:=

P(Ei*)

Inf P J.ES*J 3 = 1 , . . . ,1 and confine ourselves at first to the case 1 = 2. We start by calculating Ui*, for a fixed i*, if P j e S j, j = 1,2 : •

Ui*

PI(Ei*).P2(Ei*) = Sup [ 1 + i~i,Pl(Ei)'P2(Ei)]-1= P,(Ei*)'P2(Ei*) i=l 1-1= [ 1 + Inf~ = [ 1 + Infi~i,Pl(Ei).P2(Ei) i;~i*Pl (Ei).P2(Ei)] -1

= Sup ~ P , ( E i ) ' P 2 ( E i )

Sup PI(Ei*) "P2 (Ei*) Evidently the

sum

in the

numerator

U1 i* " U2 i* reaches its smallest

value, if P l ( E i * ) = Uli*

and

P2(Ei*) = U2i*. Therefore Ui* is calculated by solving a problem of Non-Linear Programming: the search for Infi~i,Pl(Ei ) "P2(Ei), under the restrictions defined by the two k - P R I s , combined with E Pt(Ei) = 1 - Ufi* and i~i*

i~¢i,P2(Ei) = 1 - U2i*

(*)

It is easy to see, that the infimum cannot be found in the interior of the two sub-structures defined by (*); it must appear in a situation which may be described as a combination of two corners belonging to both sub-structures defined by (*). Therefore the search for the infimum, which in this case is indeed a minimum, can be achieved by means of a computer program which constructs all pairs of corners belongingto the two sub-structures.

The values of Li*, if Pj E Sj , j = 1,2 are found in an analogous way:

128

PI(Ei*) "P2 (Ei*) Li* = I n f

+ S u P i ~ i , P l ( E i ) ' P 2 ( E i ) ]-1 =

1

L1 i*

E PI(Ei)'P2(Ei)

' L2 i*

if Lli* • L2i* ~ 0. If Lli* • L2i* = 0, then obviously Li* = 0. This time we search for Sup i~i,Pl(Ei)"P2(Ei) under the restriction defined by the k - P R I combined with i~i,P,(Ei)

: 1 - Lli*

~nd

i~i,r2(Ei)

: 1 - L21*

(**)

As this supremum must appear in a situation in which two corners of the sub-structures (**) are combined, it is indeed a maximum and may also be found by means of a computer program. Whilst the procedures described up to now are sufficient to apply (3.21) to a situation in which two k - P R I s are combined, the extension of these methods to situations in which more than two k-PRIs are involved, consists of a stepwise repetition of the same kind of analysis: Once the first two k - P R I s have been combined resulting in a single one, this can be combined with a third one, and so on. The final result does not depend upon the order of these operations because commutativity and associativity are granted. We shall demonstrate the application of Formula (3.21) to two k - P R I s with the following example. Example: Let k = 4 and Lll = 0.0

Ull = 0.2

L21 = 0.4

U~I = 0.5

L12 = 0.1

U12 = 0.4

L22 = 0.0

U22 = 0.2

L13 = 0.2

U13 = 0.5

L23 = 0. I

U23 = 0.3

L14 = 0.3

U14 = 0.7

L24 = 0.2

U24 = 0.4

4

The calculation of U1 requires the use of Inf Z Pi(Ei) .P2(Ei) i=2

when 4 i~2Pl(Ei) = 0 . 8

,

4 i~21)2(Ei) : 0 . 5

Controlling all pairs of corners, we find the minimum to be 0.12 (for instance if P,(E2) = 0.3 , PI(E3) = 0.2 , P,(E4) = 0 . 3 and P2(E2) = 0.0 , Pu(E3) = 0.3 , P2(E4) = 0.2 ). Therefore we can calculate: Ui=

[

1+0.2

• 0.5

]104 4

In the same way we derive U2 = 0.4706, U3 = 0.7143 and U4 = 0.9333. Concerning the lower limits it is immediately seen that LI = 0 and L2 = 0.

129

The calculation of L3 uses Sup [ PI(E1)"P2(E1) + Pt(E2)"P2(E2) + PI(E4)"Pa(E4) ] with

PI(E1) + PI(E2) + PI(E4) = 0.8 P2(E1) + P2(E2) + P2(E4) = 0.9

and results in: La = [ 1 + 0 . 2 0 : 3 0 . 1 ] - ' = i-~ = 0.0625. In the same manner we calculate L4 = 0.2143 so that our result of applying (3.21) to the combination of the two 4-PRIs is: Li = 0.0

U1 = 0.4545

L2 = 0.0

U2 = 0.4706

L3 = 0.0625

U3 = 0.7143

L 4 --

0.2143

U4 = 0 . 9 3 3 3 .

[]

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