On treatment selection in the face of uncertainty in expert systems

Artificial Intelligence

in Medicine I (I 989) I59-166

On treatment selection in the face of uncertainty in expert systems* Ronald R. Yager

Abstract. We look into the issue of selecting the appropriate treatment given a diagnosis in an environment in which there is some uncertainty in the effectiveness of a treatment for a diagnosis. We look at the ‘easiest first rule’ as a method of implementing possible treatments. Key words: treatment selection, uncertainty,

Symptom Set

expert systems.

1. Introduction In [l] we developed a diagnosis system which takes as its input a set of symptoms and gives as its output a set of possible diagnoses. The system in [l] was motivated by the work of Reggia et al. [2,3]. In this paper we extend the work started in [l] by providing for an expert component that takes the diagnosis set and indicates an appropriate action. This new additional component is made up of two units. The first unit, the remedy unit, takes the set of possible diagnoses and provides a set of potential remedies. The second unit selects from the set of potential remedies a prescribed treatment. These additional components discussed here provide the novelty and generality of allowing a mixture of probabilistic and possibilistic uncertainty as well as nonmonotonicity. In particular, we provide the capability of allowing for uncertainty between a diagnosis and a set of possible effective remedies. We also include in the treatment selection unit a rule based upon the principle ‘Do the easiest thing first!‘. This simplicity principle requires the implementation of a hierarchical default rule which in turn induces nonmonotonicity. We feel that such a principle, of implementing the easiest of any potential remedies first, is one which many experts follow. In order to develop expert systems that realistically follow the way experts do things we must provide a structure for realizing this highly non-monotonic rule. Figure 1 is a schematic of our overall system. As we have just indicated we shall, in this paper,

Figure 1.

concentrate on the portion of the system enclosed in the dashed field, units B and C. We note the complete system as envisioned would by dynamic in the sense that if a prescribed treatment did not provide a cure to the symptoms it would at least provide additional information to all the units which can be used in the selection of a second treatment. The remedy unit consists of a knowledge base which is constructed of rules which relate disorders to possible effective remedies. In our system we allow for uncertainty in the association between disorders and the success of remedies. The output of this unit is information of potential

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Ronald R. Yager

remedies based upon the diagnosis input. The representation of the knowledge base in B as well as the inference mechanisms used in B draws heavily upon our work on a general theory of reasoning with uncertainty [4-71. In particular, we use as a basic knowledge structure a possibility-probability granule which allows for the representation of possibilistic, probabilistic, and usuality type uncertainty. The treatment selection unit takes as its input the set of potential remedies, any information about treatments already tried, information about patient particulars such as allergies or other special problems and other relevant information. Its output is information on a choice of selected remedy called the treatment. Acentral feature of the unit is a knowledge base describing our default simplicity principle of ‘doing the easiest thing first’. The concept allows for appropriate definition. The representation and inference mechanism in this selection draws heavily upon our work on the representation of default rules [8-91.

at least one y E Y such that Aj(y) = 1. In addition, a basic probability assignment function m must satisfy

c,

m(Aj) = 1.

J

A possibility-probability statement of the form

granule (poss-prob granule) is a

U is m where m is a bpa on X. If Al, .... A, are the focal elements of m with weights m(Ai) = aj then the intent of a poss-prob granule ‘U is m’ is to represent the information that aj is the probability that ‘U is Ai’ is the appropriate canonical statement representing our knowledge about U. Poss-prob granules allow for a unified representation of various types of knowledge. For example, if our knowledge about U is in terms of a pure probability distribution on Y with Prob(U = yi) = pj then

2. Remedy selection unit Aj = ( yj} and m(Ai) = pj. The central feature of the remedy selection unit is the representation of information relating the association between diagnosis and potential remedies. It is assumed that there exists some uncertainty in these associations. In order to represent the uncertain information which may appear in the remedy selection unit we shall use the representation scheme called the general theory of uncertainty developed by Yager [4-71. We shall briefly describe the necessary ideas from this theory which we shall find useful. Assume U is a variable which takes a value in the set Y. A canonical statement is of the form U is A where A is a fuzzy subset of Y. The intent of a canonical statement is to indicate that for each y E Y, A(y) indicates the possibility that the value for U is y. In a general sense, a canonical statement is a reflection of the fact that the value for U must satisfy some property which is denoted by A. For example, if U is a person’s temperature then a canonical statement would be U is high. A basic probability assignment (bps) on Y is a mapping m: 1’ + [O,l] in which the fuzzy sets At, .... 4 for which m(Ai) > 0 are called the focal elements of m. The focal elements must be normal, in that if Aj is a focal element then there must exist

If our knowledge is a simple canonical statement U is A then m(A) = 1. If our knowledge is that ‘usually U is A’ then m(A) = a m(Y)=l-cc where a is close to one. The value of a reflects how usual the occurrence of A is. Assume V is another variable taking its value in the set X. A typical statement for representing our knowledge about the association between V and U is ifVisBthenUism where B is a fuzzy subset of X and m is a p-p granule on Y with focal elements Al;..., Aq whose weights are m(Aj) = aj. The effect of such a statement is to induce a p-p granule with respect to the joint variable (V, U) such that (V, U) is m* where m* is a bpa on X x Y with focal elements HI, .... H, and Hj=BUAi Hi(x, y) = (1 - B(x)) V Aj(y) m(Hj) = m(Aj) = aj.

(v = max)

Actually there are other possible implementations of Hi, one often used one is Hi(x, y) = 1 A (1 - B(x)) + Ai( However,

On treatment selection

when B(x) E (0,l 1,as will be the case we will be dealing with, all these different implementations become the same. Assume

we can infer that U is m+ where m+ is a bpa on Y with focal elements Bi, i=l, ... . k, and m+(Bi) = ai and Bi = Projy [Hi n D]

(V, U) is mT (V, U) is mT

that is.

are two independent propositions about the relationship of V and U where Hi and Cij are the focal elements of rni and rn: respectively and mi(Hi) = ai and m$(Gj) = bj. The effect of both these pieces of information, their conjunction, is a proposition (V, U) is m* where m*=minm;

Bi (y) = Max [Hi (x~)

A D(x)].

XEX

With this background we now can see the workings of the remedy selection unit. Let X = (xl, .. .. xq) be a set of possible disorders and let V be the variable ‘the diagnosed disorder’, the output of the diagnosis unit. Let Y = {ye. . .. . yp) be a set of treatments and let U be the variable ‘effective or potential treatments’. Our knowledge base in the remedy section consists of a set of q rules of the form Rt: “if V is Di then U is mi”

m* is obtained via Dempster’s each normal F c X x Y

rule [lO-111 as follows. For

m*(F) = (l/( 1 - K)) * c (ai * bj). and V is D over i, j such that HinGj=F

and where

K=C

161

ai * bj

with Hi n Gj not normal

i=l, . ... q

where Di = {Xi} and mi is a poss-prob granule on Y with focal elements Aii, j = 1, 2, .. . n(i) and mi(Aii) = aij. Thus for each disorder we have an associated rule-which indicates the potential treatments. Because of the structure of the bpa we can represent very complicated knowledge about treatments which includes uncertainty in the effectiveness of a treatment. By applying Dempster’s rule to the set of rules RI, .. .. q we obtain an overall representation of our knowledge base in this section as (V, U) is m*

i.j

Effectively, the focal elements of m* are all the normal sets &j = Hi n Gj. Parenthetically, we note that if Hi and Gj are fuzzy subsets of X x Y then

E&x, Y) = Hdx, Y) A Gj(x, y)

(A min).

Repeated application of this rule allows us to calculate the effect of more than two propositions about the association of U and V. Assume Pi, . ... Pq are a set of propositions of the type just described relating V and U. Let (V, U) is m* be the effective p-p granule representing all these associations. Assume HI, . . .. Hk are the focal elements of m* with weights m*(Hi) = ai. Let our knowledge about V be contained in the canonical statement V is D where D is a fuzzy subset of X. From these two pieces of knowledge (V, U) is m* and V is D

where m* =minm2n... the knowledge

n mg. Our input to this section is

V is D where D is a fuzzy subset of X indicating the set of possible disorders as obtained from the diagnosis unit, i.e., V is D is the output of the diagnosis unit. Applying the described inference mechanism we obtain as our output from this section Uism+ where U indicates the effective treatmenu for the diagnosis. We note that our knowledge of U, m+ may include various types of uncertainty. The following discussion provides an understanding of the meaning of the assignment of a pass-prob granule as the value of the effective treatment. Let ‘U is m’ be a poss-prob where m is a bpa defined on the set Y with focal elements Ai, i=l, .. .. n and weights m(Ai) = ai. Assume B is any subset of Y and Prob(B) indicates the probability that the effective treatment lies in the set B. Then

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Ronald R. Yager

Bel(B) 5 Prob(B) < PI(B)

m3(Al) = 0.7 rnJ(A.2)= 0.3

where PI(B), called the plausibility of B, is defined as We can represent each of these rules as a bpa on X x Y of the form

” PI(B) = CPOSS (B/Aj) * m(Aj). j=l

Ri: (VU) is rnr.

In the definition of PI(B) Poss[B/Aj] = Max [B(y) A Al(y)] YCY

and is called the possibility of B given Aj. In addition Bel(B), called the belief of B, is defined as n

Bel(B) = c Cert(B/Aj) * m(Aj)

In particular, if ml has focal element Aii, then rn; has focal element Bil on X x Y such that B&x, Y) = (1 - Q(x))

V &j(y).

Since Di = (Xi] then it follows that if X # Xi if X = Xi.

&(x7 Y) = 1 &(X7

Y)=

&j(y)

j=l

Thus for rule 1, rn: has one focal element, where al a2 a3

Cert[B/Al] = 1 - Poss[B/Aj]. Thus, if we have a diagnosis V is D and a knowledge base (V, U) is m*, and if this results in a remedy U is m+ then for each y E Y

dl Bll=&

d3

100

[

1 1 1 1 1 11

and m?(Bll) = 1. For rule 2, rnz has three focal elements, Bel+( y } I Prob+(y) I Pl+( y 1 ai a2 a3

indicates the probability that y is the remedy. The following example will illustrate the operations involved in manipulating the remedy selection unit. Example. Parr I: Assume X = { dt, dz, ds) is a set of potential diagnoses and Y = {al, a2, a3) is a set of remedies. Let V be the variable standing for the current diagnosis and let U be the potential remedies. Assume that our remedy selection unit consists of the following rules:

RI: if V is dl then U is ml R2: if V is d2 then U is m2 R3: if V is d3 then U is m3.

B2,;%i[

i;;]

ai a2 a3 dl 1 1 1 1 1 1 B23=d2 d3 [ 1 1 1

al a2 a3 B22=;;[

1

where rn@zr) = 0.5, m$(Bz) = 0.3, mi(B22) = 0.2. For rule 3, rn; has two focal elements al a2 a3

Let ml be a bpa on Y with one focal element A1 = (al ) and with ml(A1) = 1. Let m2 be a bpa on Y with three focal elementsAl=(a;!}, A2=(al,a3), A3=[at,az,a3}andwith weights mz(Al) = 0.5 mz(A2) = 0.3 mz(A3) = 0.2 Finally let m3 be a bpa on Y with two focal elements AI = (a3 1 and A2 = {al, a2, a3J where

4

B31=;;[

4

al a2 a3

B32=;;[

j

i 4

where rn;(Bjt) = 0.7 and m$J332) = 0.3. Using Dempster’s rule we can combine RI, R2 and R3 to get an overall bpa (V, U) is m where m=mlnm2nm3.InthiscasemisalsoabpaonX x Y with six focal elements

On treatment selection

ai a2 a3 4

d3 [ 010 1 1 11 Ht=d7, al a2 100

dl

d3 Hj=Q [

dl

a3

1 1 1 1 1 1 at a2 100

a3

d3 001 Hs=d2 1 IO 1

3. Treatment selection unit

al a2 a3

dl

100

100

d3 1 1 1 HZ=d’ [ 10 1 1

1 1

dt Hq=dz d3

dl Hg=d2 d3

[

al a2 a3 loo 0 1 0 001 ai

1

a2 a3

100

[

1 I 1 001

1

where the weights associated with these elements are m(Ht) = 0.15 m(H2) = 0.09

m(H3) = 0.06 m(H4) = 0.35

163

m(Hs) = 0.2 1 m(H6) = 0.14.

This above structure represents the knowledge base of the remedy selection unit. If we input to this structure our knowledge of the diagnosis, V is D, where D is a fuzzy subset of X then we get a potential treatment U is m* where m* is a bpa on Y with focal elements Gi, i = 1, .... 6, and m*(Gi) = m(Hi) Gi(y) = Max [D(x) A Hi(x

XEX

We shall look at the remedy selection in a number of cases of diagnosis. Case I. D = (dt ), the diagnosis is simply dt. In this case Gi = (ai} for all i and hence m* has just one focal element G = [at ) with weight 1. That is, the treatment is simply at. Case II. The diagnosis is d2 or d3, D = (dz, d3). In this case G1=G2=G3=G6=[a~,a2,a3},G4=(a2,a3l,andGs=(ai, as}. Thus in this case m* has just three distinct focal elements Ga= {at,a2,a3J,Gp= {a2,a3} andGy= [at,a3) withweights m*(Go) = 0.44, m*(Gp) = 0.35, and m*(G$ = 0.21. Case III. In this case we allow D to be a fuzzy subset, thus let D = { 0.7/dt, l/d2,0.3/d3 ), then the distinct focal elements of m* are Cl = (0.7/dt, l/a2,0.3/a3] G2 = 1l/at, 0.3/a2, l/a31 G3 = (l/at, l/az, l/a31 G4 = I l/at, O/az, l/a3 I with weights m*(Gl) = 0.5, m*(G2) = 0.09, m*(G$ = 0.2, and m*(Gq) = 0.2 1.

The treatment selection unit takes as its primary input the effectiveness of the various remedies in the face of the diagnosis. That is, the input to this unit is the output of the remedy selection unit. It also can use additional knowledge as input, such as patient particular restrictions to certain treatments and treatments already tried. The output of this unit is a prescribed treatment. The heart of this selection unit is a knowledge base and inference mechanism which contains our knowledge about how we shall select the appropriate treatment from the set of possible treatments. The advantage of separating the process of treatment selection from seIectin of potential treatments shows us to store the more medically-based association between diseases and remedies in the remedy unit while keeping the more ‘local’ information in the selection unit. By local information we mean things like availability of facilities, competence of personnel to administer treatment, local expertise, etc., as well as patient particular knowledge. In this section we shall consider one type of selection rule which we call the easiestfirst rule. Essentially this rule says ‘first try the potential treatment that is easiest to implement’. By the easiest we mean to convey considerations such as expense, time, pain, suffering, etc. From a formal point of view such an easiest first rule provides an example of a hierarchical default rule. Let Y be the set of treatments that are within the framework of the problem. Thus yi E Y is a treatment. Let U be a variable corresponding to the potential effective treatments based on the symptoms. As we indicated, the output of remedy unit is a datum U is m* where m* is a poss-prob granule on Y having focal elements GI, G2, .... G, with weights m*(Gi) = bi. At this point we can include any special knowledge about the special capabilities of the medical facility and relevant information about the patient history. For example, if we have already tried treatment yk and it didn’t work we can add this. Thus we would get Uism*andUisnot(yk]. After including all these special data we are now in the position of having to select a treatment to implement. Let W be a variable indicating the selected treatment. For purposes of clarification in discourse we shall let the set Z be the set from which we select the treatment. Thus in this case zi = yi and thus Z and Y are effectively the same set. Thus W will be an element in Z, actually in Y. The knowledge base in the treatment selection unit consists of our procedure for selecting the value of W based upon our knowledge of U. As we indicated we shall be using the ‘easiest first rule’ to select the treatment to implement. The most straightforward manifestation of this rule occurs when there exists a simple linear ordering on the elements of Z indicating

their ease of implementation.

Thus without loss

164

Ronald R. Yager

of generality we shall assume the elements of 2 are indexed in the order of increasing difftculty in implementation. Thus 21 > 23 > 23 ... > zp where zi > zj indicates zi is more easily implemented than zj. A typical manifestation of the rule can be seen in the following matrix where we consider only sets with three elements:

(Ylv Y? (Yl, Y3 (Y27 Y3 (Yl, Y29Y3

Zl

z2

z3

1 0 0 1 1 0 1

0 1 0 0 0 1 0

0 0 1 0 0 0 0

In our particular case, we have Ei = ( yi 1, and therefore E:(H) = Maq [H(y) A Ei(y)l = Wyi).

ET(H) = 1 - H(yi).

treatment in the set of possible treatments. We notice the nonmonotonicity of the structure in that if we know U E {yt, y3) then it says do zl. However, if we get more information which allows us to infer U E ( y3 ) then we would do z3. The knowledge contained in such an easiest first strategy can be captured by a set of production rules, one rule for each yi. The hierarchical aspect of the easiest first rule is explicitly seen in this representation. In particular, the knowledge base consists of a set of rules, one for each yi, of the form: Zfyi is

a possible treatment suggested by U and no easier treatments are suggested by U then implement Zi. Formally, we can capture such a rule within the theory of approximate reasoning [8] by Rl: If U is not Gi and U is Et is possible then W is Fi.

Before we proceed, a technical issue which arises in problems involving these types of default rules must be addressed. Assume Y is a set of objects. Let A be a subset of Y and let B be a subset of the power set of Y. In some situations we are faced with finding Poss[B/A]. In order to handle this problem, we can represent A, as a fuzzy subset of the power set of Y, A such that As = (Poss[A]/A) where the membership grade of A in As is Maxy[A(y)]. With this understanding

Poss[A] =

Poss[B/A] = Poss[B/A] = Poss[A] A B(A). We shall now continue our example by investigating the treatment selection unit.

Example. Part II: Let Y = (al, a2, a3) be the set of treatments and let Z = (bl, b2, b3) where bk = implement ak. We shall assume that al > a2 > a3 indicates the ease of implementation. The treatment selection unit can be represented by three rules: Rl: If U is (al) is possible then W is (bl). R2: If U is not (al ) and U is (a2) is possible then W is (b2]R3:IfUisnot (al,a2) thenWis (b3). .

Formally these rules are

In this formulation

Sl=E;uBl S~=LZUE;UBZ (note Gi = 0).

Using the translation rules of the theory of approximate reasoning [8] each rule gets translated into (U, W) is Si where Sl = Gi u --t (Et) u Fi. Furthermore, the overall rule

base can be represented as (U, W) is S where S=n

E;(H) = POSS[H/Ei] = Maxy [H(y) A Ei(y)].

Furthermore, since 7 (E:) which we shall denote as E: is the negation of ET then

The intent of this matrix is to indicate that if our knowledge of U is such that all we know is that U is an element of one of the sets on the left then implement the treatment with a one in its corresponding row. For example, if we know U E ( y 1, y3 ] implement zl. The above matrix is an example of a hierarchical default rule in that we default to the easiest

Fi = (zi] = implement treatment zi Ei = (yi] Gi = (~1, ... yi-1)) the set of treatments preferable to yi.

fuzzy subset of Y then E$ is a fuzzy subset of the power set of Y, such that for each fuzzy subset H of Y,

Si.

We note that in above definition Et is the translation of the statement Et is possible. As shown by Yager in [8], if Ei is a

S3 = L3u B3 whemBi=(b)andi=1,2,3: Lz=(al); L3=(at,a2]; and El is a fuzzy subset of the power set of Y standing for not (Ei is possible), thus E:(H) = 1 - H(ai). The conjunction of these three rules gives us the knowledge base of the treatment selection unit. Thus S=S1nSznS3

On treatment selection

165

hence T = B 1 and therefore W is B I. Since B 1 = (bl ) , the action is simply to implement treatment al.

hence S=

(L2nEy)u(L3nE?nEZ)u(BtnL2)

Case

u(B1nL3nE?)u(BznE?nL3) u (B3n ET n L2) u (B3 n Efn E:). Assume our output of the remedy selection unit is U is m* where m* is a bpa with focal elements Fi and weights m(Fi). In this case our selected treatment is

2: U is m* with three focal elements Fl = {at. a2, a3] F2=(a2,a3] F3= (at,a31

where m*(Ft) = 0.44, m*(F2) = 0.35, and m*(Fs) = 0.21. First consider F I :

Wismm where rn* is a bpa on Z with focal elements Ti where

Poss@$R2]

mo(Ti) = m*(Fi) and Ti = Projz(S n Fi). Thus if Fi is an arbitrary focal element of m* then T is an associated treatment selection where n FJ

Ti(z) = (Poss[Ei/L2 v Poss[EI

R2=L2nFt Rj=LJnFt

n ET/L3 n Fi]V B t(z)

A (Poss[E$/L~ n Fi] A Poss[E;A~

v

n Fi] v Poss[Ei/Fi]).

Let us now apply this formulation to the three cases discussed in the previous example. In the following we shall denote,

=0

PossfEI n ET/R31 = Poss[E; n E?/ (at, a2 )] = 0 POSS~z/Fl] = 1 Poss[Ei/R31 = Poss[EF/ 1ai. a2 13 = 1 A (1 - Poss[Et/{at, az}]) = 0 PossEt A ET/F11 = Poss[E; A E2/Y] = (1 - Poss[El/Y]) A (1 - Poss[Ez/Y])

A

1= O

hence in this case Tl(b) = Bl(b). i.e., TI = BI. For the second focal element F2 = (al, aj} Rz=L2nF2=0 F3=L3nF2=

(al}

POSS [ET/R21 Poss [ET/@] = 0 Poss[Es nEi/R3]=0 Poss [L$F2] = 0

L2 n Ft = R2 L3 n Ft = R3.

Poss [E2(a2 Case 1: U is m* where m* has just one focal element F = (al ] = Et. In this case R2=L2nF=(at} R3=bnF={al]=Et

= Poss[E;/Et]

Poss[L2/Fi]) v By

A Fi] v By

A (Poss[E$L2

= (at] = {al.az}.

Poss [ET n Es21

=0

Poss [ET/R31 = Poss [ET/a21 = 1 A (1 - Poss(Et/az))

= 1

thus Tz(b) = B2(b), i.e., T2 = B2. Similarly with F3 = (at, a3} we can show that T3 = Bl. Thus our proposed action is W is m* where me has two focal elements Bl = { bt ) and BZ = { b2) with weights m*(Bl) = 0.65 and m*(B2) = 0.35. In this case the appropriate action is to randomly select either bt or b2 with 0.65 probability that we select bl and 0.35 that we select b2.

=Et

hence Poss [ET/R21 = Poss [ET/E,] = 1 A (1 - POSS(El/El) = 0. Poss [ET n E$R3] = Poss [Ef n E&t] Poss [L2/Fj = Poss [E/El] = 1

Case3: =0

Poss [ET/R31 = Poss [ET/Et] = 0 Poss [ET n E2/FJ = Poss [Ei

)] =0

n ELI] = 0

therefore T(z) = 0 v 0 v Bt(z) A 1 v B2(z) A 0 v B3(z) A 0 = Bt(z)

H:ere the focal elements are

Ft = 0.1/a, l/al, 3la3 I F2 =

l/at, 0.3/a2, l/a3 I

F3 = ai, a2, a31 F4= Ial,a3] Let us look at Fl. In this case L2 n Ft = (0.7/at ) L3 n FI = (0.7/at, l/a2}

m*(Ft) = 0.5 m*(F2) = 0.09 m*(F3) = 0.2 m*(F4) = 0.2 1

166

Ronald R. Yager

In this case we select our treatment according to the performance of a random experiment in which P(bl) = 0.85 and P(b2 or b3) = 0.15.

PosslEi/L2 n F1] = 0.7 A 0.3 = 0.3 Poss[Ei n E%3 n F1] = 0 Possl@L3 A Ft] = 0 POSS[L2/-Fl]= 0.7 Poss[Ef/L3 n F1] = 0.3

Conclusion

PossIEi/Lz n FI] = 0.3 Poss[Ei n E/$F1] = 0 hence = 0.3 v (0.7 A B t(b)) v (0.3 A B2(b) v 0.3 A B3(b)) = { 0.7/b1, O.j/bz, 0.3/b3).

T1(b) Tl

L2 n F2 = 1 l/al F2 = { I/al,

References

I

0.3/a2}

[ 1] Yager, R.R.: Explanatory models in expert systems. Int. J. ofMon-Muchine Studies 23 (1985) 539-549. [2] Reggia. J., Nau, D., and Wang, P.: Diagnostic expert systems based on a set covering model. Int. J. of Man-Machine Studies 19 (1983) 437-

Poss[Ei/L2 n F2] = 0 Poss& n Eh

n F2] = 0

Poss[E%3 n F2] = 0.3 Poss[L2/F2] = 1 Poss[Ei/L3 n F2] = 0 Poss[Ei/L;! n F2] = 0 Poss[E; n Es21 = 0 hence Tz(b)

Note * This

research was in part supported by grants from the National Science Foundation and the Air Force Office of Scientific Research.

For F2

L3 n

While we have presented the methodology suggested using the flavor of a medical environment it should be carefully noted that the techniques presented here are applicable to any problem in which one must make some decision as to what action to take in the face of undesirable functioning of a system.

=

B 1(b).

It can easily be shown that T3 = B I and T4 = B 1.Therefore we get W is m where TI =( (0.7/B1,0.%2,0.3/b3 T2= Ib1]

1

m(T1) = 0.5 m(T2) = 0.5

460. [3] Reggia. J.. Nau. D., Wang, P., and Peng, Y.: A formal model of diagnostic inference. Informtim Sciences 37 (1985) 227-285. [4] Yager. R.R.: Toward a general theory of reasoning with uncertainty. Part I: Non-specificity and fuzziness. Int. J. of Intelligent Systems I (1986) 45-67. [5] Yager, R.R.: Towarda general theory of reasoning with uncertainty. Part fJ: Probability. Int. J. of Man-Machine Studies (to appear). [6] Yager, R.R.: Reasoning with uncertainty for expert systems. Proc. of Ninth IJCAI (1985). pp. 1295-1297. [7] Yager, R.R.: A general approach to decision making with evidential knowledge. In: L.H. Kanal and J.L. Hemmer (eds.), Uncertuinty in ArtificialIntelligence. North-Holland, Amsterdam. 1986, pp. 317-330. [8] Yager, R.R.: Using approximate reasoning to represent default knowledge. Artificial Intelligence 31 (1987) 99-l 12. [9] Yager, R.R.: Non-monotonic compatibility relations in the theory of evidence. Technical Report MJJ-615, Machine Intelligence Institute, Iona College, 1986. [lo] Dempster, AI?: Upper und lower probabilities induced by a multivalued mapping. Annals Mathematics Statistics 38 (1987) 325-339. [I 1] Shafer, G.: A Mathematical Theory of Evidence.Princeton University Press, Princeton, N.J., 1976.

Authors’ address Machine Intelligence

Institute, Iona College, New Rochelle, NY 10801, USA

0 1999 by Burgverlag.

Tecklentarg.

West Germany