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A comparative assessment of Dempster-Shafer and Bayesian belief in civil engineering applications Luo, Wuben 1988

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COMPARATIVE ASSESSMENT OF DEMPSTER-SHAFER AND BAYESIAN BELIEF IN CIVIL ENGINEERING APPLICATIONS by WUBEN LUO B.Sc. Tsinghua University, P.R. China, 1985 A THESIS SUBMITTED PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of C i v i l Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1988 © Wuben Luo, 1988 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of C/i/t't £nfj'^eehj/>^ The University of British Columbia Vancouver, Canada Date June s '7£& DE-6 (2/88) A b s t r a c t The Bayesian theory has long been the predominate method i n d e a l i n g w i t h u n c e r t a i n t i e s i n c i v i l eng ineer ing p r a c t i c e i n c l u d i n g water resources e n g i n e e r i n g . However, i t imposes unnecessary r e s t r i c t i v e requirements on i n f e r e n t i a l problems. Concerns thus a r i s e about the e f f e c t i v e n e s s of us ing Bayesian theory i n d e a l i n g wi th more genera l i n f e r e n t i a l problems. The r e c e n t l y developed Dempster-Shafer theory appears to be ab le to surmount the l i m i t a t i o n s of Bayesian theory . The new theory was o r i g i n a l l y proposed as a pure mathematical t heo ry . A reasonable amount of work has been done i n t r y i n g to adopt t h i s new theory i n p r a c t i c e , most of t h i s work being r e l a t e d to inexact inference i n expert systems and a l l of the work s t i l l remaining i n the fundamental s tage . The purpose of t h i s research i s f i r s t to compare the two t h e o r i e s and second to t r y to apply Dempster-Shafer theory i n s o l v i n g r e a l problems i n water resources e n g i n e e r i n g . In comparing Bayesian and Dempster-Shafer theory , the equ iva l en t s i t u a t i o n between these two t h e o r i e s under a s p e c i a l s i t u a t i o n i s d i scussed f i r s t . The divergence of r e s u l t s from Dempster-Shafer and Bayesian approaches under more genera l s i t u a t i o n s where Bayesian theory i s u n s a t i s f a c t o r y i s then examined. F o l l o w i n g t h i s , the conceptua l d i f f e r e n c e between the two t h e o r i e s i s argued. A l s o d i s cus sed i n the f i r s t par t of t h i s research i s the i ssue of d e a l i n g w i t h evidence i n c l u d i n g c l a s s i f y i n g sources i i of evidence and express ing them through b e l i e f f u n c t i o n s . In a t tempt ing to adopt Dempster-Shafer theory i n eng ineer ing p r a c t i c e , the Dempster-Shafer d e c i s i o n theory , i . e . the a p p l i c a t i o n of Dempster-Shafer theory w i t h i n the framework of conven t iona l d e c i s i o n theory , i s i n t roduced . The a p p l i c a t i o n of t h i s new d e c i s i o n theory i s demonstrated through a water resources eng ineer ing des ign example. Table of Contents Abs t rac t . . . . i i L i s t of Tables v i i L i s t of F igures v i i i Acknowledgements i x 1. In t roduc t ion 1 2. Review of Bayesian Theory and I t s A p p l i c a t i o n 7 2.1 Convent iona l Bayesian Theory 7 2.1.1 D i s c r e t e Form 8 2 .1 .2 Continuous Form 10 2.2 The A p p l i c a t i o n of Bayesian Theory to Water Resources Engineer ing 14 2.3 Comments on Bayesian Theory 20 2.4 Other Forms of Bayesian Theory 21 2.5 Summary 24 3. In t roduc t ion to Dempster-Shafer Theory 25 3.1 Representa t ion of Evidence v s . B e l i e f Func t ion . . 26 3.2 Dempster 's Rule of Combination . . . . . . 3 1 3.3 S p e c i a l Classes of B e l i e f Func t ions 36 3.3.1 Vacuous B e l i e f Func t ion 36 3.3.2 Simple B e l i e f Func t ion 38 3.3.3 Bayesian B e l i e f Func t ion 39 3.3.4 Consonant B e l i e f Func t ion 40 3.4 Summary 41 4. The Equivalence between Bayesian and Dempster-Shafer Theory 42 4.1 S t a t i s t i c a l S p e c i f i c a t i o n Model and Bayesian Theory 42 4.2 Dempster-Shafer Approach to S t a t i s t i c a l S p e c i f i c a t i o n Model 46 i v 4.3 A Further Example . 53 4 . 4 Summary . . . . 55 5. Divergence of Resu l t s from Dempster-Shafer and Bayesian Theories 57 5.1 In t roduc t ion 57 5.2 Discount ing A B e l i e f Func t ion . . . 6 0 5.3 Divergence of Resu l t s of Two Theor ies and S e n s i t i v i t y A n a l y s i s 63 5.3.1 Discoun t ing the P r i o r Bayesian B e l i e f Func t ion 64 5.3.2 Discoun t ing the Consonant B e l i e f Funct ion .67 5.4 Numerical Example 69 5.5 Summary 76 6. Conceptual Di f fe rence Between the Two Theor ies and Representat ion of Evidence 81 6.1 In t roduc t ion 81 6.2 Conceptual D i f f e rence Between the Two Theories . . 82 6.2.1 Two Types of U n c e r t a i n t i e s 82 6.2.2 Conceptual D i f f e r ence Between the Two Theories 84 6.3 General Cons ide ra t i on of Evidence 90 6.4 Express ing Evidence through a B e l i e f Funct ion . . . 9 3 6. 5 Summary 99 7. Demster-Shafer D e c i s i o n Making i n Water Resources Engineer ing 102 7.1 In t roduc t ion 102 7.2 Dempster-Shafer D e c i s i o n Theory 104 7.3 The A p p l i c a t i o n of Dempster-Shafer Dec i s ion Theory 108 7.3.1 D e s c r i p t i o n Of The O r i g i n a l Problem 108 7.3.2 Dempster-Shafer D e c i s i o n A n a l y s i s . . .111 v 7.4 Summary 8. Conclusions REFERENCES L i s t of Tables Table D e s c r i p t i o n Page 7.1 P r i o r p r o b a b i l i t i e s and u t i l i t i e s ( a f t e r R . J . 109 McAnif f et a l . ) 7.2 Sample l i k e l i h o o d of Zk=10.5% (a f te r R . J . 109 McAnif f et a l . ) 7.3 Expected cos t s based on Bayesian p o s t e r i o r s 110 7.4 Bas ic p r o b a b i l i t y assignment of a consonant 111 b e l i e f func t ion d e r i v e d from i n f e r e n t i a l evidence 7.5 Combined bas i c p r o b a b i l i t y assignments 112 7.6 B e l i e f and p l a u s i b i l i t y va lues of s i n g l e t o n s 113 7.7 The upper and lower expected cos t s 115 v i i L i s t of F igu res F igu re D e s c r i p t i o n Page 2.1 The p r i o r p r o b a b i l i t y dens i t y func t ion 11 2.2 Procedures for us ing Bayesian d e c i s i o n theory 19 3.1 An i l l u s t r a t i o n of set of p r o p o s i t i o n s of 0 27 5.1(a) m(A) v s . a when the two b e l i e f func t ions are 78 m i l d l y c o n f l i c t i n g and only the p r i o r b e l i e f func t ion i s d i scoun ted 5.1(b) The same as i n F i g . 5 .1 (a ) , but on ly the 78 consonant b e l i e f func t ion i s d i scounted 5 .1(c) The same as i n F i g . 5 .1 (a ) , but the two 79 b e l i e f func t ion are d i scounted s imul taneous ly 5.2(a) m(A) v s . a when the two b e l i e f func t ions are 79 h i g h l y c o n f l i c t i n g and only the p r i o r b e l i e f func t ion i s d i scoun ted 5.2(b) The same as i n F i g . 5 .2 (a ) , but on ly the 80 consonant b e l i e f f u n c t i o n i s d i scounted 5 .2(c) The same as i n F i g . 5 .2 (a ) , but the two 80 b e l i e f func t ion are d i scounted s imul taneous ly v i i i Acknowledgements The author wishes to express h i s s p e c i a l g r a t i t u d e to h i s s u p e r v i s o r , Dr . W.F. C a s e l t o n , for h i s i n v a l u a b l e encouragement, guidance and suggest ions throughout the process of t h i s research , and h i s e f f o r t s made i n rev iewing t h i s t h e s i s . The author a l s o thanks D r . A . D . R u s s e l l for h i s thorough review of t h i s t he s i s and h i s va luab le comments. The f i n a n c i a l support in the form of s c h o l a r s h i p provided by Chinese government i s g r a t e f u l l y acknowldged. The author a l so wishes to thank h i s parents and f r i ends for t h e i r he lp and encouragement which are important to the completion of t h i s research . i x 1 . INTRODUCTION U n c e r t a i n t i e s are always i n v o l v e d i n eng ineer ing des ign problems. D e a l i n g i n a p p r o p r i a t e l y w i t h u n c e r t a i n t i e s may r e s u l t i n the eng ine r ing p ro jec t being underdesigned or overdes igned, and t h i s i n turn may cause unnecessary e x t r a p ro jec t c o s t s , unexpected damage and other nega t ive impacts . There i s there fore a p r o f e s s i o n a l r e s p o n s i b i l i t y to dea l w i t h u n c e r t a i n t i e s i n the most r e spons ib l e and i n s i g h t f u l way p o s s i b l e . Eng ineer ing des ign under u n c e r t a i n t y c o n s i s t s of d e s c r i b i n g the c r i t i c a l random design v a r i a b l e s and then performing d e c i s i o n a n a l y s i s based on these d e s c r i p t i o n s us ing conven t iona l d e c i s i o n theory . The f i r s t par t of t h i s process i nc ludes d e a l i n g wi th the u n c e r t a i n t i e s of des ign v a r i a b l e s . Because the s p e c i f i c a t i o n of des ign v a r i a b l e s are very important for the whole p rocess , s i g n i f i c a n t e f f o r t s have been made over the l a s t two decades[26] i n t r y i n g to f i n d a c o n v i n c i n g method to dea l e x p l i c i t l y w i th the u n c e r t a i n t i e s i n v o l v e d . The most c o n v e n t i o n a l approach to t h i s problem u t i l i z e s po in t e s t ima t ion p rocedures [26] . Th i s approach r equ i r e s one to choose a p r o b a b i l i t y model which best de sc r ibes the random design v a r i a b l e and then es t imate the parameters of t h i s model us ing h i s t o r i c a l records or r e g i o n a l d a t a . The d e c i s i o n a n a l y s i s i s then undertaken as a seperate s tep based on the es t imated p r o b a b i l i t y model of the design v a r i a b l e . One weakness of t h i s approach i s tha t i t i s unable to subsequently incorpora te any newly obta ined i n f o r m a t i o n , such as from s i t e i n v e s t i g a t i o n , i n the 1 2 r e e s t i m a t i o n of the p r o b a b i l i t y d i s t r i b u t i o n of the des ign v a r i a b l e . A l s o , by c o n s i d e r i n g the d e c i s i o n a n a l y s i s as a separate s t ep , t h i s conven t iona l approach f a i l s to connect the e s t i m a t i o n of des ign v a r i a b l e s w i t h the d e c i s i o n a n a l y s i s i n a sys temat ic or l o g i c a l way. Bayesian d e c i s i o n theory has been s u c c e s s f u l l y adopted i n areas of c i v i l eng inee r ing des ign as w e l l as water resources eng ineer ing d e s i g n . Th i s i n v o l v e s us ing Bayes' equat ion i n the inexact inference about the random des ign v a r i a b l e and then making d e c i s i o n s based on t h i s in fe rence us ing conven t iona l d e c i s i o n theory . ' Bayesian theory makes i t p o s s i b l e to incorpora te a l l of the a v a i l a b l e p r i o r in fo rma t ion i n the inexac t in fe rence of the des ign v a r i a b l e and subsequently to update the i n f e r e n t i a l r e s u l t s a f t e r o b t a i n i n g new i n f o r m a t i o n . Furthermore, Bayesian d e c i s i o n theory p rov ides a framework which r e l a t e s the i n f e r e n t i a l r e s u l t s of the unce r t a in design v a r i a b l e w i t h the f i n a l d e c i s i o n a n a l y s i s , i . e . the subsequent updat ing of the i n f e r e n t i a l r e s u l t s for the des ign v a r i a b l e a f t e r o b t a i n i n g new in format ion i s a u t o m a t i c a l l y r e f l e c t e d i n the f i n a l op t ima l d e c i s i o n . A shor t review of Bayesian theory and i t s a p p l i c a t i o n i n water resources eng ineer ing design i s presented i n Chapter 2. In us ing Bayesian theory , the i n f e r e n t i a l problem has to be p l aced in to the s t a t i s t i c a l s p e c i f i c a t i o n model and both the p r i o r p r o b a b i l i t y assessments based on the p r i o r in fo rmat ion and the sample l i k e l i h o o d s based on the newly 3 obtained in format ion have to be expressed i n the form of conven t iona l p r o b a b i l i t y d i s t r i b u t i o n s . I f the a v a i l a b l e sources of in format ion are always s p e c i f i c enough so that they can e x p l i c i t l y spec i fy the r equ i r ed p r o b a b i l i t y d i s t r i b u t i o n s , Bayesian d e c i s i o n theory would be an i d e a l method for water resources eng inee r ing d e c i s i o n problems. U n f o r t u n a t e l y , s ince d e c i s i o n s of ten have to be determined under a wide v a r i e t y of c i rcumstances , i n many cases the a v a i l a b l e in format ion w i l l not be s u f f i c i e n t to s p e c i f y a conven t iona l p r o b a b i l i t y d i s t r i b u t i o n . Therefore a more general theory of d e c i s i o n i s needed. Dempster-Shafer theory , which was i n i t i a l l y proposed by Dempster and subsequently advanced s u b s t a n t i a l l y by Shafer , p rov ides an a l t e r n a t i v e to Bayesian theory . Compared w i t h Bayesian theo ry , Dempster-Shafer theory was fo rma l i zed from an e n t i r e l y d i f f e r e n t po in t of v i ew: i t uses a b e l i e f func t ion to represent a p iece of evidence and uses Dempster 's r u l e of combinat ion i n p o o l i n g a l l sources of i n f o r m a t i o n . Because of the g rea te r f l e x i b i l i t y of a b e l i e f func t ion i n r ep resen t ing evidence , one might a n t i c i p a t e that Dempster-Shafer theory o f f e r s at l e a s t the prospect of surmounting the l i m i t a t i o n s posed by Bayesian theo ry . The re levan t Dempster-Shafer theory i s in t roduced i n Chapter 3. An ex tens ive compar is ion of Bayesian theory and Dempster-Shafer theory at the most fundamental l e v e l i s undertaken i n t h i s r e sea rch . The equiva lence between the two t heo r i e s i s d i scussed in Chapter 4 . In those genera l 4 situations where Dempster-Shafer theory i s the appropriate scheme, Bayesian theory can only be used as an approximate approach and the results from the two theories d i f f e r . A general comparisions of results from the two approaches i s d i f f i c u l t because of the involvement of s u b j e c t i v i t i e s in choosing the approximation. However, the divergence of results from Dempster-Shafer and Bayesian methods can be analyzed t h e o r e t i c a l l y in a simple s i t u a t i o n , a situation in which the two methods are i n i t i a l l y i d e n t i c a l and the results from Dempster-Shafer and Bayesian schemes then diverge as doubts about the sources of evidence are introduced. A discussion of the divergence of the results from the two approaches i s presented in Chapter 5. In Chapter6, the conceptual difference between the two theories when expressing evidence is addressed. Since dealing with evidence i s of very important interest in engineering practice, the issues of c o l l e c t i n g and rearranging sources of evidence and expressing them in the forms of belief functions are also discussed in t h i s chapter. In recent years, a reasonable amount of work has been done by researchers in attempting to implement Dempster-Shafer theory in their own f i e l d s of interest, most of t h i s work having been related to inexact inference in expert systems. Because many i n f e r e n t i a l problems which arise in c i v i l engineering practice strongly resemble the i n f e r e n t i a l problems a r i s i n g in expert systems, 5 Dempster-Shafer theory may also be considered as a candidate for a general method for inexact inference in c i v i l engineering practice. Though considerable e f f o r t s have been made in tryin g to adopt the Dempster-Shafer theory in the real world, no s i g n i f i c a n t applications have so far been reported, most of the work being s t i l l at the fundamental stage. Furthermore, l i t t l e work has yet been done in applying t h i s new theory in c i v i l engineering practice, even at the fundamental l e v e l . The research reported in Chapter 7 therefore intended to make the f i r s t tentative step toward the application of the new theory into c i v i l engineering practice s p e c i f i c a l l y in connection with the representation of evidence. Quantitative decision making under uncertainty is an important by-product of inexact reasoning. The application of Dempster-Shafer theory in the decision analysis in water resources engineering, which i s referred to as the Dempster-Shafer decision theory, i s the last issue discussed. This, together with an example of application of Dempster-Shafer decision theory is presented in Chapter 7. F i n a l l y the research r e s u l t s , general conclusions and suggestions for future research are summarized in Chapter 8. It should be noted that t h i s thesis i s not intended to serve as an introduction to either Bayesian or Dempster-Shafer theory, i t s intent being to explain the common ground shared by the two theories and their points of departure. While the prospect is that of engineering application, the examples chosen are not intended to r e f l e c t 6 s t a t e o f t h e a r t a p p l i c a t i o n o f e i t h e r t h e o r y b u t a r e f o r t h e p u r p o s e o f c o m p a r i s i o n a n d c o n t r a s t . 2. REVIEW OF BAYESIAN THEORY AND ITS APPLICATION 2.1 CONVENTIONAL BAYESIAN THEORY The o r i g i n a l Bayesian theory was aimed at d e a l i n g wi th the exper imenta l problem. A t y p i c a l problem may be desc r ibed as f o l l o w s Suppose an experiment i s designed so that i t s outcome may be represented by one of a set of elements X={x^, i = 1 . . . m } , and the outcome w i l l be a s soc i a t ed wi th one of these elements w i th c e r t a i n p r o b a b i l i t y . The p r o b a b i l i t y d i s t r i b u t i o n which governs the e x p e r i m e n t i a l outcomes X i s c a l l e d a chance d e n s i t y func t ion and i s denoted by t9j=q^j(x). Because of the u n c e r t a i n t i e s a s s o c i a t e d w i t h the experiment , the t rue chance dens i t y func t ion i s unknown. Suppose there i s a set of chance dens i t y func t ions 0={q^j(x) j = 1 , . . . n } , one and only one of them being c o r r e c t . Let the p r i o r p r o b a b i l i t y judgements on t h i s set © be p ' ( 0 j ) . The problem of updat ing p ' ( 0 j ) a f t e r observ ing some exper imenta l outcome then a r i s e s . Th i s k ind of p r o b a b i l i t y model i s c a l l e d a s t a t i s t i c a l s p e c i f i c a t i o n m o d e l [ l 8 ] . In t h i s s e c t i o n , a b r i e f review of Bayesian theory d e a l i n g w i t h the problem desc r ibed above i s p resented . More comprehensive and d e t a i l e d d e s c r i p t i o n s of Bayesian theory can be found i n v a r i o u s textbooks d e a l i n g w i t h the a p p l i c a t i o n of p r o b a b i l i t y and d e c i s i o n theory i n c i v i l eng inee r ing p r a c t i c e [ 1 ] [ 3 ] . 7 8 2.1.1 DISCRETE FORM Consider the problem desc r ibed above, but assume that both sets X and 0 are d i s c r e t e . A f t e r an observa t ion of an experimental outcome x^ f the c o n d i t i o n a l p r o b a b i l i t y that x^ w i l l occur given each element 0j i n 0 being the t rue parameter (and therefore s p e c i f y i n g the t rue chance dens i ty funct ion) can then be obta ined from the set of chance dens i ty func t ions . This i s expressed as )=q 9 ] - (x i ) j = 1, 2 , . . . n (2.1) The updated p r o b a b i l i t y d i s t r i b u t i o n on 0 a f t e r the observat ion of x^ can then be obta ined from p r o b a b i l i t y theory, i . e . P(0 . j , x . ) The p r i o r p r o b a b i l i t y judgements on 0 are p'(0j) hence p(0j , x ^ ) can be expressed as p ( x j/0 j ) p ' (0 j ) and the t o t a l p r o b a b i l i t y of the occurrence of x^ can be replaced by p(x^)=I p ( x^/0 j ) p ' (0 j ) , the above equat ion can then be r ewr i t t en as p , ,(0j/x i)=kp(x i/0j)p' (6.) (2.2) where k= Z p ( x i/0 j ) p'(0 j) 9 The above equat ion i s known as Bayes ' equa t ion . On the r i g h t hand s ide of t h i s equa t ion , the term p'(0j) i s known and the c o n d i t i o n a l p r o b a b i l i t y p(x^/0j) i s g iven by E q . 2.1. F o l l o w i n g the obse rva t ion of x^ , the updated p r o b a b i l i t y d i s t r i b u t i o n can then be obta ined through the above Bayesian theory . Since the p r o b a b i l i t y p'(0j) i s obta ined p r i o r to the obse rva t ion of x^ , i t i s c a l l e d a p r i o r p r o b a b i l i t y . Th i s p r i o r p r o b a b i l i t y w i l l be obta ined from the p r i o r or o l d i n f o r m a t i o n . The p r o b a b i l i t y p"(0j / x ^ ) i s termed the p o s t e r i o r p r o b a b i l i t y , s ince i t i s ob ta ined a f t e r the obse rva t ion x ^ . The c o n d i t i o n a l p r o b a b i l i t y p ( x^/0j), which i s a func t ion of 0j, i s r e f e r r e d to as the sample l i k e l i h o o d f u n c t i o n . The constant k i s a n o r m a l i z i n g f ac to r which ensures tha t the c a l c u l a t i o n of p"(0j / x ^ ) y i e l d s a t rue p r o b a b i l i t y d i s t r i b u t i o n . Bayesian theory can be used to p r o g r e s s i v e l y update the p r o b a b i l i t y d i s t r i b u t i o n on 0 when a s e r i e s of independent experiments are performed and the exper imenta l outcomes are observed. One only needs to cons ide r the c a l c u l a t e d p o s t e r i o r p r o b a b i l i t y as the p r i o r p r o b a b i l i t y and use the sample l i k e l i h o o d func t ion obta ined from a newly observed exper imenta l outcome to get the new p o s t e r i o r p r o b a b i l i t y d i s t r i b u t i o n through E q . 2.2. An a l t e r n a t i v e approach to the same problem i s t ha t , f o l l o w i n g a s e r i e s of obse rva t ions x = { x , , x 2 , . . . x ^ } , the t o t a l sample l i k e l i h o o d func t ion i s f i r s t assessed and then the p o s t e r i o r p r o b a b i l i t y i s 10 obtained from a s imple implementat ion of Eq . 2 . 2 . The two approaches are mathemat ica l ly e q u i v a l e n t . The t o t a l sample l i k e l i h o o d func t ion L ( x / 0 j ) can be obta ined through the fo l lowing equation L(x/e.)= n p ( x i / e j ) (2.3) The pos t e r io r p r o b a b i l i t y i s then given as p"(69:J) = kL(x /c9 j )p ' (6.) ' In engineer ing p r a c t i c e , the exper imenta l outcomes xeX and the parameter value 6eQ may be best desc r ibed as continuous v a r i a b l e s . The cont inuous form of Bayesian theory i s descr ibed in Sec . 2 . 1 . 2 . 2.1.2 CONTINUOUS FORM Consider again the problem posed at the beginning of t h i s sec t ion and assume that the two sets are con t inuous . The p r i o r p r o b a b i l i t y d e n s i t y func t ion i s given in F i g . 2.1 The p r i o r p r o b a b i l i t y d e n s i t y func t ion i s g iven i n F i g . 2 . 1 . The p r o b a b i l i t y that a 6 value l i e s w i t h i n the i n t e r v a l (0j,0j+A0) i s g iven as p(0j<0<0j+A0)=f(0.)A0. After an obse rva t ion of an exper imenta l outcome x , the continuous sample l i k e l i h o o d func t ion l ( x / 0 . ) can then be 11 Figure 2.1 The prior probability density funct ion obtained from the cont inuous chance dens i ty func t ion q^^(x) as fo l lows l(x/d.)=qe.{x) (2.4) which i s a func t ion of 6y Consider the i n t e r v a l (9yd^+L\d), the use of Bayes' equation 2.2 y i e l d s l(x/d,)p'(8.<d<d .+Ad) p"[(d.<d<8.+L\d)/x]= 3 ^ D J 3 I l(x/0 k)p(0 k<e<0 k+A0)A0 12 using the p r o b a b i l i t y density function to express p r o b a b i l i t y , the above equation can be expressed as l (x /0 . )f ' (0 .)A0 t"(d./x)&d= — J 3 3 Z l (x /0 . )f '(0, )A0 Lett ing A0 tend to 0, the above equation becomes f"(0/x)=kl(x/0)f ' (0) (2.5) 1 where k= X o o l (x /0 ) f ' (0 )d0 + The above equation is known as the continuous Bayes' equation. Comparing Eq. 2.5 with Eq. 2.2, i t i s seen that those two equations are of e s s e n t i a l l y the same form. Analogous to the discrete form, f '(0) in Eq. 2.5 is named the pr ior p r o b a b i l i t y density function and f"(0/x) the posterior p r o b a b i l i t y density funct ion. The term l (x /0) i s a continuous sample l i k e l i h o o d funct ion. As for the discrete case, i f a set of observations x={x,, x 2 , . . .x^} are obtained from a series of independent experiments, Eq. 2.5 can be used to incorporate a l l of those observations and obtain a f i n a l poster ior probab i l i t y density funct ion . One way to deal with this problem is to use Eq. 2.5 sequential ly with the previous ca lculated posterior p r o b a b i l i t y density function as pr ior probab i l i t y density function u n t i l a l l of the observations have been 13 c o n s i d e r e d . A l t e r n a t i v e l y , the t o t a l sample l i k e l i h o o d of x={x, . . .x^} i s c a l c u l a t e d f i r s t and the E q . 2.5 i s then used to o b t a i n the f i n a l p o s t e r i o r p r o b a b i l i t y d e n s i t y f u n c t i o n . The t o t a l sample l i k e l i h o o d f u n c t i o n for x={x, . . .x^} i s g iven by the f o l l o w i n g equat ion L(x /0 )= II l ( x . / 0 ) (2.6) i which i s a cont inuous f u n c t i o n of 9. For cont inuous Bayes ian t h e o r y , the computat ion can be s i m p l i f e d by us ing a p a r t i c u l a r form of p r i o r d i s t r i b u t i o n . T h i s form of p r i o r d i s t r i b u t i o n i s c o m p a t i b l e wi th the sample l i k e l i h o o d f u n c t i o n , and i s c a l l e d a conjugate of t h i s f u n c t i o n . By us ing the conjugate p r i o r d i s t r i b u t i o n , the p o s t e r i o r p r o b a b i l i t y d i s t r i b u t i o n has the same form as the p r i o r p r o b a b i l i t y d i s t r i b u t i o n . I t shou l d be noted tha t a conjugate p r i o r d i s t r i b u t i o n i s adopted p r i m a r i l y to s i m p l i f y the mathemat ica l computa t ion . For a g i v e n form of l i k e l i h o o d f u n c t i o n , i t s conjugate f u n c t i o n can be used as p r i o r p r o b a b i l i t y d i s t r i b u t i o n p r o v i d e d i t does not c o n f l i c t wi th any idea about the r e a l p r i o r d i s t r i b u t i o n . Thus , i f the p r i o r i n f o r m a t i o n s t r o n g l y s u p p o r t s a p a r t i c u l a r form of p r i o r d i s t r i b u t i o n , then that form of p r i o r d i s t r i b u t i o n should be used i n s t e a d of the conjugate d i s t r i b u t i o n . 1 4 2.2 THE APPLICATION OF BAYESIAN THEORY TO WATER RESOURCES  ENGINEERING Var ious u n c e r t a i n t i e s are always i n v o l v e d i n water resources p r a c t i c e . These u n c e r t a i n t i e s can be d i v i d e d i n t o two types , n a t u r a l uncer ta in ty(NU) and i n f o r m a t i o n a l u n c e r t a i n t y ( I U ) [ 2 2 ] . NU i s the unce r t a in ty inherent i n the n a t u r a l random process i t s e l f , such as the annual f l o o d i n g on a r i v e r , and can be de sc r ibed by a s t a t i s t i c a l model. The IU i s due to the l ack of in format ion about the n a t u r a l random process . For example, one may r a r e l y have s u f f i c i e n t data to f i n d the exact s t a t i s t i c a l model and i t s parameters . I t i s seen that the IU can be d i v i d e d i n t o parameter u n c e r t a i n t y and model u n c e r t a i n t y . Because the NU i s about the p r o b a b i l i s t i c phenomenon i t s e l f , i t i s independent of one 's knowledge and i n f o r m a t i o n . The IU , on the other hand, can be e l i m i n a t e d i f s u f f i c i e n t in format ion i s ob t a ined . Water resources des ign cons ide r s both the u n c e r t a i n t i e s about h y d r o l o g i c a l v a r i a b l e s and the preferences of engineers towards some p a r t i c u l a r outcomes. For example, the c r e s t e l e v a t i o n of f l o o d p r o t e c t i o n d ike s on a r i v e r has to be s e l e c t e d based on the p r e d i c t e d f l o o d frequency curve and the s u b j e c t i v e preferences of engineers towards economic d e c i s i o n a l t e r n a t i v e s . The water resources des ign then i n v o l v e s two procedures , the inference or p r e d i c t i o n of a c r i t i c a l but random h y d r o l o g i c a l event , in t h i s example the peak annual f l o o d d i s cha rge , and the d e c i s i o n a n a l y s i s based on the inference and eng inee r ' s p re fe rences . 15 Because of the predominance of both short pe r iods and imperfect data concerning the h y d r o l o g i c a l events , water resources design under u n c e r t a i n t y has long been recognized as a p r a c t i c a l but c h a l l e n g i n g problem. The standard approach to t h i s problem us ing c l a s s i c a l s t a t i s t i c a l methods i n v o l v e s f i r s t choosing a p r o b a b i l i t y model which one b e l i e v e s best d i s c r i b e s the random charac te r of the c r i t i c a l des ign event and e v a l u a t i n g the parameter va lues of that model by i n f e r e n t i a l methods us ing e i t h e r h i s t o r i c a l records or r e g i o n a l da t a . The d e c i s i o n s are then made on the bas i s of some economic a n a l y s i s as a separate s t ep . I t i s seen that the standard approach f a i l s to cons ide r the u n c e r t a i n t i e s among the competing models. Furthermore, the u n c e r t a i n t i e s are not l o g i c a l l y r e l a t e d to the f i n a l d e c i s i o n p rocess . The water resources p ro j ec t based on such an approach may the re fore be i n a d v e r t e n t l y underdesigned or overdes igned. Bayesian d e c i s i o n theory , i . e . the Bayesian i n f e r e n t i a l framework togther w i th d e c i s i o n theory , p rov ides a more reasonable approach to the water resources des ign problem. I t begins by f i r s t performing the Bayesian in fe rence about the h y d r o l o g i c a l event . The d e c i s i o n a n a l y s i s i s then undertaken and a u t o m a t i c a l l y incorpora te s the u n c e r t a i n t i e s about the h y d r o l o g i c a l event , economic c o n s i d e r a t i o n s and the d e c i s i o n maker 's p references i n the f i n a l d e c i s i o n . The advantage of Bayesian d e c i s i o n theory i s that i t p rov ides a methodology to pool together a l l of the a v a i l a b l e 16 in format ion about the u n c e r t a i n t i e s of the h y d r o l o g i c a l event . T h i s would i nc lude r e g i o n a l and h i s t o r i c a l data as w e l l as s u b j e c t i v e judgements. Bayesian d e c i s i o n theory a l s o prov ides a t h e o r e t i c a l way to cons ide r both the parameter u n c e r t a i n t y and the model u n c e r t a i n t y i n the f i n a l d e c i s i o n . Jus t a few of the many a p p l i c a t i o n s of Bayesian d e c i s i o n theory i n water resources design problems are c i t e d here . R . M . Shane et al. [22] f i r s t suggested the use of Bayesian theory to incorpora te r e g i o n a l and h i s t o r i c a l data to reduce the u n c e r t a i n t i e s i n the e v a l u a t i o n of a h y d r o l o g i c a l v a r i a b l e . G. Tschanne r l [24 ] , D .R . Davis et al. [6] used Bayesian d e c i s i o n theory i n des ign ing h y d r o l o g i c a l s t r u c t u r e s i n s i t u a t i o n s where the h i s t o r i c a l records are short and a l s o examined the i n f l u e n c e of o b t a i n i n g new in format ion on the f i n a l d e c i s i o n . However, t h e i r s t ud i e s assumed only one model and cons idered only u n c e r t a i n t i e s about the parameters of that model. G . J . Vicens et al. [26] e x t e n s i v e l y d i scussed the in fe rence about the parameter u n c e r t a i n t i e s . E . F . Wood et al.[27] and B . Bodo et al. [14] a p p l i e d the more genera l Bayesian d e c i s i o n theory i n water resources des ign i n which the u n c e r t a i n t i e s about the parameter and model are cons idered together i n the d e c i s i o n p rocess . The procedures for us ing Bayesian theory i n water resources design i s summarized as f o l l o w s (and i s a l s o shown i n F i g . 2.2) 17 1) Assume that I r i s a r e g i o n a l data set which ac t s as p r i o r in fo rmat ion and X i s the h i s t o r i c a l r ecord which ac t s as new i n f o r m a t i o n . Accord ing to Bayesian theory , the p o s t e r i o r p r o b a b i l i t y dens i t y func t ion of a set of parameters 6 i s then given by f"(e / i r,x ) = k i(x/e ) f ' ( e / i r ) (2.7) where f ' ( © / I r ) i s the p r i o r p r o b a b i l i t y d e n s i t y func t ion on 0; l ( X / 0 ) i s the sample l i k e l i h o o d func t ion which may be c a l c u l a t e d by the method given i n Sec 2 . 1 . 2) A f t e r i n c o r p o r a t i n g the parameter u n c e r t a i n t y w i t h i n a p r o b a b i l i t y d e n s i t y func t ion for the h y d r o l o g i c a l event , the new p r o b a b i l i t y dens i t y func t i on which i s c a l l e d the Bayesian distri bution[3] i s g iven by f ( y ) = / e f ( y / e ) f " ( e ) d ( e ) (2.8) where f " (0 ) i s the same as f " ( 0 / l r , X ) which i s g iven by E q . 2 . 7 . 3) The model u n c e r t a i n t y i s then cons ide red i n b a s i c a l l y the same way as parameter u n c e r t a i n t y . Assuming p'(j3^) i s the p r i o r p r o b a b i l i t y tha t model i i s c o r r e c t , the p o s t e r i o r p r o b a b i l i t y i s then g iven by p" (0.)=(K./K (2 .9) 18 * where K i s a no rma l i z ing f a c t o r ; i s the marginal l i k e l i h o o d funct ion of the observed data determing from the i t h model. 4) The f i n a l p r o b a b i l i t y d e n s i t y funct ion for the h y d r o l o g i c a l event which c o n s i d e r s both types of u n c e r t a i n t i e s i s known as a composite Bayesian di st ri but i on[3] and i s expressed as f (y)= E p"(|3. )f • (y) (2.10) i where f^(y) i s the Bayesian d i s t r i b u t i o n of model i which i s given i n Eq 2 .8 . 5) The d e c i s i o n a n a l y s i s i s dependent upon the choice of d e c i s i o n s t ra tegy d^ from a set of a l t e r n a t i v e s D and the random outcome yj out of Y . A u t i l i t y func t ion u=U(d^,yj) must be d e f i n e d . Such a u t i l i t y func t i on expresses the d e c i s i o n maker 's preferences when measured n u m e r i c a l l y . 6) A f t e r the u t i l i t y func t ion i s ob ta ined , the d e c i s i o n can then be made based on the c r i t e r i o n of maximizing the expected u t i l i t y v a l u e . The expected u t i l i t y value cond i t i oned on d e c i s i o n d^ i s given by E ( u / d . ) = ; Y u ( d . f y j ) f ( y ) d y (2.11) where f (y ) i s g iven by Eq. 2 .10 . The Bayesian d e c i s i o n ru l e * i s then to choose d so that 1 9 Inference about parameter uncertainty: f " ( 0 / I r , X ) Bayesian d i s t r i b u t i o n of random design variable: f ( yW e f (y/e)f "(e)de Inference about model uncertainty: P " ^ ) Composite Bayesian di s t r i b u t i o n after considering both model and parameter uncertainties: f (y )=?p"( /3 i ) f i (y) U t i l i t y function: u = U(d Optimal decision making based on maximum expected u t i l i t y value E(u/d*)=Max E(u /d . ) d i • • 1 Figure 2.2 Procedures for us ing Bayesian d e c i s i o n theory 20 E(u/d*)=Max E(u /d - ) (2.12) d i 2.3 COMMENTS ON BAYESIAN THEORY Bayesian theory has been w e l l accepted i n the l i t e r a t u r e but to a much l e s s e r extent by p r a c t i o n e r s . I t s a b i l i t y to dea l w i th the u n c e r t a i n t i e s and to f a c i l i t a t e d e c i s i o n a n a l y s i s i n the face of these u n c e r t a i n t i e s i s appea l ing . I t p rovides a methodology which a l lows one to u t i l i z e a l l of the a v a i l a b l e i n f o r m a t i o n , both o b j e c t i v e data and sub j ec t i ve judgements, in the inference p rocess . Furthermore, the inference can be updated s e q u e n t i a l l y as new informat ion i s ob ta ined . Bayesian d e c i s i o n theory cons iders the u n c e r t a i n t i e s s imul taneous ly w i th the d e c i s i o n maker 's preferences , and thus mediates the d e c i s i o n maker 's two p r i n c i p l e areas of concern . However, there are some concerns about Bayesian theory . C l e a r l y the theory i s conf ined to the so c a l l e d s t a t i s t i c a l s p e c i f i c a t i o n model, which was o r i g i n a l l y designed to descr ibe an experiment. In p r a c t i c e , not a l l i n f e r e n t i a l problems can be f i t t e d i n to such a model. The a v a i l a b l e in format ion has to be d i v i d e d i n t o the " o l d " and "new" evidence and the p r i o r p r o b a b i l i t y and sample l i k e l i h o o d func t ion ( i . e . c o n d i t i o n a l p r o b a b i l i t y ) are then based on t h i s ass ignment[20] . The s i t u a t i o n may a r i s e where there i s not s u f f i c i e n t informat ion to ob ta in a l l of the needed p r o b a b i l i t i e s . Even though exper ts may be consu l t ed , one might s t i l l f e e l very uncomfortable about the p r o b a b i l i t y 21 judgements i f , for example, the number of these judgements becomes very l a rge or d i v e r g e n t . Furthermore, as w i l l be d i s cus sed i n Chapter 3, Bayesian theory may not be able to express the ignorance a p p r o p r i a t e l y . Another very important concern about t h i s theory i s tha t i t expresses both types of u n c e r t a i n t i e s wi th p r o b a b i l i t y d i s t r i b u t i o n s on s i n g l e elements . Th i s i s i n t u i t i v e l y unacceptable because i t uses the same approach to dea l w i th the two e n t i r e l y d i f f e r e n t types of u n c e r t a i n t y . In the p rev ious d i s c u s s i o n s about Bayesian theory , the new ev idence , from which the sample l i k e l i h o o d func t i on i s d e r i v e d , i s assumed to be observed w i t h c e r t a i n t y . In p r a c t i c e , however, the new evidence may not be obta ined without some e r r o r or other u n c e r t a i n t y . For example, an experiment which i s known to be accurate 70% of the t ime, g ives an outcome x ^ . I t i s s t i l l not known whether the experiment i s , i n t h i s p a r t i c u l a r i n s t ance , opera t ing a c c u r a t e l y or i n a c c u r a t e l y . Some of these concerns can be addressed by modifying the t r a d i t i o n a l Bayesian theory . This i s sue w i l l be d i scussed i n Sec. 2 . 4 . 2.4 OTHER FORMS OF BAYESIAN THEORY In t h i s s e c t i o n , two ways of modify ing the o r i g i n a l Bayesian theory to take i n t o account some of the concerns mentioned above w i l l be g i v e n . The f i r s t m o d i f i c a t i o n bases the Bayesian theory on the concept of l i k e l i h o o d r a t i o and 22 thereby reduces the task of a s ses s ing the r e q u i r e d p r o b a b i l i t i e s . The second one cons ide r s the s i t u a t i o n i n which the new evidence i s incomplete ev idence . 1) Bayesian theory based on l i k e l i h o o d r a t i o Assuming 0 = { 0 i , . . . 0 } i s a set of hypotheses and x = { x , . . . X | { } a set of obse rva t i ons , then the Bayesian theory can be r e w r i t t e n as fo l lows[20] p ( 0 i / x ) _ p ( 0 . ) p ( x / 0 . ) p ( 0 j / x ) p(0j ) p ( x / 0 j ) o r : p ( 0 . / x ) p (0 . ) - — 1 = - — i - - L ( x / 0 . : 0 . ) (2.14) p ( 0 j / x ) p ( 0 j ) 1 3 where L ( x / 0 ^ : 0 j ) i s c a l l e d the l i k e l i h o o d r a t i o f avo r ing 0^ over 0 j . Since the p r o b a b i l i t i e s p ( 0 k / x ) , k = 1 , 2 . . . n , must add to p ( 0 . / x ) 1.0, they are e n t i r e l y determined by t h e i r r a t i o s . p ( 0 , / x ) Therefore , one only needs to assess the l i k e l i h o o d r a t i o s of one hypothes i s over another to evalua te the p o s t e r i o r p r o b a b i l i t i e s , i n s t ead of e v a l u a t i n g the absolu te c o n d i t i o n a l p r o b a b i l i t i e s . The task of e v a l u a t i n g the p r o b a b i l i t i e s i s thus reduced. 2) Bayesian theory based on incomplete evidence 23 I f the new evidence on which the sample l i k e l i h o o d i s obta ined i s incomplete , i . e . there are some u n c e r t a i n t i e s about the evidence i t s e l f , the o r i g i n a l Bayesian theory cannot be used d i r e c t l y , and some m o d i f i c a t i o n i s necessary . The modi f ied f o r m u l a t i o n , f o l l o w i n g R . C . J e f f r e y [ l 3 ] and R .O . Duda[9] i s p ( 0 / x ' ) = p ( 0 / x ) p ( x / x ' ) + p ( 0 / x ) p ( x / x ' ) (2.15) where x ' represents incomplete ev idence . Here, x i s the evidence known to be t rue w i t h c e r t a i n t y and x i s the evidence known to be c e r t a i n l y f a l s e whi le p ( x / x ' ) and p ( x / x ' ) are the p r o b a b i l i t i e s of the evidence being t rue or f a l s e r e s p e c t i v e l y a f t e r the r e l evan t obse rva t ion x ' . Note that p ( 0 / x ) and p ( t V x ) are s imply the Bayesian p o s t e r i o r p r o b a b i l i t i e s of 6 c o n d i t i o n e d on x and x r e s p e c t i v e l y . The v a l i d i t y of E q . 2.15 can be demonstrated at the extremes by the f o l l o w i n g (1) i f p ( x / x ' ) = 1 . 0 , then p ( x / x ' ) = 0 . 0 . E q . 2.15 y i e l d s p ( 0 / x ' ) = p ( 0 / x ) (2) i f p ( x / x ' ) = 1 . 0 , then p ( x / x ' ) = 0 . 0 . Eq . 2.15 y i e l d s p ( 6 V x ' )=p(0/x) (3) i f p ( x / x ' ) = p ( x ) , then p ( x / x ' ) = p ( x ) . Eq . 2.15 y i e l d s p ( 0 / x ' ) = p ( 0 ) 24 The t h i r d c o n c l u s i o n means that i f the new evidence x ' i s no be t t e r than the p r i o r knowledge, the p r i o r p r o b a b i l i t i e s w i l l remain unchanged. 2.5 SUMMARY Even though there are some se r ious concerns about Bayesian theory , i t should s t i l l be cons ide red as an e f f e c t i v e method i n d e a l i n g w i th in fe rence about u n c e r t a i n t i e s and w i th d e c i s i o n a n a l y s i s i f the i n f e r e n t i a l problem can be f i t t e d i n t o the so c a l l e d s t a t i s t i c a l s p e c i f i c a t i o n model and the a v a i l a b l e in fo rma t ion ( i n c l u d i n g s u b j e c t i v e in format ion) i s s u f f i c i e n t to judge a l l of the r equ i r ed p r o b a b i l i t i e s . For the more genera l s i t u a t i o n s where the i n f e r e n t i a l problem i s hard to f i t i n t o the s t a t i s t i c a l s p e c i f i c a t i o n model, or the a v a i l a b l e in fo rma t ion i s not s u f f i c i e n t for one to judge the r equ i r ed p r o b a b i l i t i e s , or bo th , a more genera l theory i s necessary . The Dempster-Shafer theory seems to be a p o s s i b l e candidate to meet these requirements . Th i s theory w i l l be in t roduced i n Chapter 3 . 3. INTRODUCTION TO Dempster-SHAFER THEORY A b r i e f review of Bayesian theory and i t s a p p l i c a t i o n was g iven i n Chapter 2 . At the end of tha t chapter , a r e c e n t l y developed new theory c a l l e d Dempster-Shafer theory was mentioned. Th i s theory , o r i g i n a l l y developed by Dempster[7] and subsequently broadened s u b s t a n t i a l l y by S h a f e r [ l 8 ] , aims at f i n d i n g a more general and c o n v i n c i n g method for the inference of u n c e r t a i n t i e s . Even though Bayesian theory can be proved to be a s p e c i a l case of Dempster-Shafer theory (see Chapter 4 ) , i t was fo rma l i zed from an e n t i r e l y non Bayesian po in t of v iew. I t in t roduces the concept of us ing a b e l i e f func t ion to represent evidence ins tead of the c l a s s i c a l p r o b a b i l i t y d i s t r i b u t i o n used i n Bayesian theory . The p o o l i n g of d i f f e r e n t sources of in fo rmat ion i n Dempster-Shafer theory i s through Dempster 's r u l e of combinat ion which i s q u i t e d i f f e r e n t from the Bayesian method based on the Bayes' equa t ion . In recent years , cons ide rab l e i n t e r e s t has a r i s e n i n the use of Dempster-Shafer theory for inference i n expert systems. Because i n f e r e n t i a l problems i n c i v i l eng ineer ing p r a c t i c e s t r o n g l y resemble i n f e r e n t i a l problems i n expert systems, Dempster-Shafer theory appears to be a l s o worthy of s e r i o u s c o n s i d e r a t i o n as a genera l method of d e a l i n g w i t h u n c e r t a i n t i e s i n eng ineer ing p r a c t i c e . Dempster-Shafer theory p rov ides a sys temat ic framework for d e a l i n g wi th u n c e r t a i n t i e s . The e s s e n t i a l pa r t s of t h i s theory are the concept of a b e l i e f func t ion by which 25 26 evidence i s represented , and Dempster 's r u l e of combinat ion by which d i f f e r e n t bodies of evidence are poo led . In t h i s chap te r , a b r i e f i n t r o d u c t i o n of t h i s new theory i s p resen ted . More d e t a i l e d d e s c r i p t i o n s and p r o p e r t i e s of Dempster-Shafer theory can be found i n S h a f e r [ l 8 ] . 3.1 REPRESENTATION OF EVIDENCE VS. BELIEF FUNCTION Suppose there i s a set of p o s s i b l e answers 0 to some q u e s t i o n , one and on ly one of them being the c o r r e c t answer. The set 0 i s then exhaus t ive and mutua l ly e x c l u s i v e , and i t i s termed the frame of discernment by Shafe r . Any subset A i n 0 i s c a l l e d a proposition. The set of a l l p r o p o s i t i o n s of © corresponds to the set of subsets of 0 and i s denoted by 0 2 . Among the p r o p o s i t i o n s are the whole frame 0 i t s e l f , the s i n g l e elements and the empty set 0. Example 3.1 demonstrates the concept of a frame of d iscernment . Example 3.1. Finding the type of material I t i s necessary to f i n d out the type of m a t e r i a l on some s i t e where a dam w i l l be b u i l t . Assume the m a t e r i a l can on ly be one of three p o s s i b l e types : {sand}, {rock} and { s o i l } . The frame of discernment i s then 0={sand, rock , s o i l } The set of p r o p o s i t i o n s 2® can be expressed by a t r ee as 27 shown i n F i g . 3 . 1 . The p r o p o s i t i o n at each node i s so arranged that i t imp l i e s i t s ances to r s . {sand, rock, s o i l } F igure 3.1 An i l l u s t r a t i o n of set of p r o p o s i t i o n s of 0 The l o g i c a l r e l a t i o n s h i p between p r o p o s i t i o n s such as con junc t ion , d i s j u n c t i o n , i m p l i c a t i o n and negation can be t r a n s l a t e d i n t o the more g r a p h i c a l s e t - t h e o r e t i c r e l a t i o n s h i p between two subsets such as i n t e r s e c t i o n , un ion , i n c l u s i o n and complementation. Man ipu la t ion of l o g i c w i t h i n an expert system i s another advantage of the Dempster-Shafer theory which enr iches i t s c a p a b i l i t i e s . Th i s top ic i s not , however, a concern of t h i s t h e s i s . I f a body of evidence i s ob ta ined , i t a s s igns p r o b a b i l i t y mass over the p r o p o s i t i o n s . The p r o b a b i l i t y number ass igned to some p r o p o s i t i o n A i s c a l l e d a basic probability assignment and denoted by m(A). I t i s obvious 28 that the bas ic p r o b a b i l i t y assignment should s a t i s f y the f o l l o w i n g cond i t i ons m(0)=O. 0<m(A)<1.0 L m(A)=1.0 A where 0 represents the n u l l or empty s e t . The p r o p o s i t i o n s on which the basic p r o b a b i l i t y numbers are not zero are c a l l e d focal elements. The b e l i e f value m(A) which i s ass igned to p r o p o s i t i o n A i s a l s o committed to any p r o p o s i t i o n which A i m p l i e s . The t o t a l b e l i e f committed to A i s therefore the summation of a l l bas ic p r o b a b i l i t y numbers to the p r o p o s i t i o n s which imply A . In other words, a l l b e l i e f committed to subpropos i t i on which f a l l w i t h i n the j o i n t p r o p o s i t i o n A are cons ide red to c o n t r i b u t e to the b e l i e f in A . Thus, Bel(A)= I m(B) (3.1) BC A Bel (A) :2® [0 ,1] i s c a l l e d a b e l i e f func t ion over 0 . A b e l i e f func t ion i s sometimes a l s o c a l l e d a support function. The two concepts are d i f f e r e n t from the pure mathematical po in t of v i ew. Th i s d i f f e rence i s d i s cus sed i n S h a f e r [ l 8 ] . Here i t i s s u f f i c i e n t to use the term of b e l i e f func t ion o n l y . A 29 b e l i e f func t ion should s a t i s f y the f o l l o w i n g c o n d i t i o n s Bel (0)=O Bel(0)=1 .O 0.0<Bel(A)<1.0 for Ae0 The value Be l (A) i s only the support which i s p rov ided by the evidence on p r o p o s i t i o n A . To descr ibe the impact of the evidence on A, one a l so needs to acknowledge the b e l i e f va lue , imp l i ed by the evidence which i s not agains t A, i . e . not suppor t ing the negation A " . Th i s i s c a l l e d the degree of plausibility on A and i s denoted by P I ( A ) . Accord ing ly P1(A)=Z m(B) AriB*0 ( 3 . 2 ) or P l ( A ) = 1 - B e l ( A - ) The degree of p l a u s i b i l i t y i s analogous to the concept of upper probability. The d i s t i n c t i o n between the two concepts can be found in S h a f e r [ l 8 ] . The formulat ion g iven i n Eq . 3.2 i s c a l l e d the plausibility function. The relative plausibilities of s i ng l e tons Rpl(0) of b e l i e f f unc t i on Bel(0) i s expressed as R p l ( e ) = c P l ( 0 ) ( 3 . 3 ) where c i s a non-zero constant which i s independent of 9. 30 S i m i l a r to the b e l i e f f u n c t i o n , the p l a u s i b i l i t y va lue i s between 0 and 1.0; i t w i l l be 0.0 when the evidence i s wholly against A and 1.0 when there i s no evidence aga ins t A at a l l . The p l a u s i b i l i t y va lue for the empty set 0 i s obv ious ly ze ro . I t should be noted that the three concepts , basic p r o b a b i l i t y assignment, b e l i e f func t ion and p l a u s i b i l i t y f u n c t i o n , e s s e n t i a l l y represent the same t h i n g , i . e . once one of them i s determined, the other two w i l l be f i x e d a c c o r d i n g l y . For example, i f the b e l i e f func t ion i s known, the basic p r o b a b i l i t y assignment can be obtained from m(A)= Z ( - 1 ) l A _ B l B e l ( B ) BC A This interdependence should not in any way d i m i n i s h the importance of the three concepts as they cons ide rab ly e n r i c h the d e s c r i p t i v e c a p a b i l i t i e s of the Dempster-Shafer scheme and enhance i t s p r o p e r t i e s for use fu l a p p l i c a t i o n i n the r e a l wor ld . In e f f e c t , the b e l i e f value Bel(A) i s the minimum support a body of evidence can provide w i t h 100% conf idence , whi le the p l a u s i b i l i t y value P l ( A ) i s the maximum p o s s i b l e support the evidence can conce ivab ly provide for p r o p o s i t i o n A . The Dempster-Shafer theory therefore uses a band [ B e l ( A ) , P l ( A ) ] to express the f u l l impact of a p iece of evidence on A, in con t ras t to the Bayesian theory which cons ide r s on ly the s i ng l e tons and i s cons t ra ined to use p r o b a b i l i t i e s on s i ng l e tons to represent the impact of the ev idence . One may i n t u i t i v e l y conclude[28] that the p r o b a b i l i t y of a p r o p o s i t i o n A, namely P ( A ) , should 31 be somewhere between the b e l i e f va lue and the p l a u s i b i l i t y v a l u e , i . e . Bel(A)<P(A)<Pl(A) (3.4) though t h i s p r o b a b i l i t y value may never be known. I f P i ( A ) = B e l ( A ) , then Be l (A) i s the same as p r o b a b i l i t y va lue P ( A ) . The d i f f e r e n c e between P1(A) and Be l (A) can be i n t e r p r e t e d as the degree of u n c e r t a i n t y about p r o p o s i t i o n A, i . e . the ambigui ty i n the evidence (or whether some of the in fo rma t ion i s p r o v i d i n g the b e l i e f s ) concern ing the b e l i e f i n A . The b e l i e f va lue ass igned to the whole frame 0 i s c a l l e d the ignorance of the evidence as t h i s component of b e l i e f i s unable to r e so lve the d i f f e r e n c e i n b e l i e f between any p r o p o s i t i o n s . A b e l i e f func t ion can be ob ta ined , once a p iece of evidence i s ob ta ined , us ing s eve ra l t echniques , some of which w i l l be d i scussed i n Chapter 6. I f s e v e r a l sources of evidence are ob ta ined , Dempster 's r u l e of combinat ion can be used to combine a l l of the b e l i e f func t ions to y i e l d a r e s u l t a n t b e l i e f f u n c t i o n . Dempster 's r u l e of combinat ion i s in t roduced i n Sec. 3 . 2 . 3.2 DEMPSTER'S RULE OF COMBINATION As was d i scussed i n Sec. 3 . 1 , the bas i c p r o b a b i l i t y assignment expresses e x a c t l y the same in fo rmat ion as the b e l i e f f u n c t i o n . Since Dempster 's r u l e of combinat ion i s 32 simpler to desc r ibe i n terms of the bas i c p r o b a b i l i t y assignment, the d i s c u s s i o n s i n t h i s s e c t i o n w i l l be based mainly on t h i s assignment. Assume there are two d i s t i n c t (or independent) bodies of evidence bear ing on a frame 0. The corresponding bas i c p r o b a b i l i t y assignments are denoted as m,(A) and m 2 ( B ) . Dempster's ru le of combinat ion then g ives a new bas i c p r o b a b i l i t y assignment m(C) which r e s u l t s from the combination of the two bodies of ev idence . I t i s expressed as m(C)=m,(A)+m 2(B)=(1-k)" 1 Z m,(A)m 2 (B) ADB=C (3.5) m(0)=O.O where k= Z m,(A)m 2 (B) ADB=0 k i s the bas ic p r o b a b i l i t y number ass igned to empty p r o p o s i t i o n 0. Because m(0) should have zero v a l u e , the basic p r o b a b i l i t y numbers on the other non-empty p ropos i t i ons should therefore be i n f l a t e d by a f ac to r ( 1 - k ) " 1 . The f o c a l elements of the combined b e l i e f func t ion given by m(C) are the non-empty i n t e r s e c t i o n s of the f o c a l elements given by m,(A) and m 2 ( B ) . I f s eve ra l p ieces of evidence are ob ta ined , the corresponding b e l i e f func t ions can then be combined by us ing Eq. 3.5 r epea ted ly . The f i n a l 33 bas ic p r o b a b i l i t y assignment w i l l not depend on the order i n which the bas ic p r o b a b i l i t y assignments are combined. S i m i l a r to the case of combining two bodies of ev idence , the f o c a l elements of the combined bas ic p r o b a b i l i t y assignment are the i n t e r s e c t i o n s of the f o c a l elements coming from each bas ic p r o b a b i l i t y assignment. I t i s recognized that the value k i s the bas i c p r o b a b i l i t y number ass igned to the empty p r o p o s i t i o n . Th i s value there fore i n d i c a t e s the c o n f l i c t between the two b e l i e f f u n c t i o n s . In f a c t , Shafer suggests that the degree of c o n f l i c t between two b e l i e f func t ions be c a l c u l a t e d from c o n ( B e l , , B e l 2 ) = l o g K = - l o g ( 1 - k ) (3.6) which i s s imply a t r ans fo rmat ion of k. E q . 3.6 i n d i c a t e s that i f the two b e l i e f func t ions are comple te ly c o n s i s t e n t , then k=0.0 and the degree of c o n f l i c t i n g logK=0.0; i f the two b e l i e f func t ions f l a t l y c o n t r a d i c t each o ther , then k=1.0 and the degree of c o n f l i c t logK=°°. In t h i s l a t t e r extreme case , the combinat ion of the two b e l i e f func t ions does not y i e l d a v a l i d b e l i e f f u n c t i o n . The f o l l o w i n g example i l l u s t r a t e s the a p p l i c a t i o n of Dempster 's r u l e of combinat ion i n combining b e l i e f func t ions Ex amp I e 3.2 (Com i nuat i on of Exampl e 3.1) 34 Suppose there i s one p iece of evidence which suggests that there i s 40% chance that the m a t e r i a l i s rock , 20% chance that i t i s sand and 40% chance that i t may be any of the three p o s s i b i l i t i e s . The bas ic p r o b a b i l i t y assignment then i s m, ({rock})=0.4 m, ({sand})=0.2 m,(0)=O.4 and the corresponding b e l i e f func t ion i s B e l , ( { r o c k } ) = 0 . 4 Be l , ({sand})=0 .2 Bel , (0)=1 .O Assume that there i s another p iece of evidence which g ives r i s e to m 2 ({sand})=0.3 m 2 ({sand, rock})=0.5 m 2 ({sand, so i l } )=0 .1 m 2(0)=O.1 35 and the cor responding b e l i e f func t ion i s B e l 2 ( { s a n d } ) = 0 . 3 B e l 2 ( { s a n d , rock})=0.8 B e l 2 ( { s a n d , s o i l } ) = 0 . 4 B e l 2 ( 0 ) = 1 . O The combinat ion of m,(A) w i t h m 2 ( B ) , us ing Eq . 3 . 5 , y i e l d s : m 3 ({sand})=0.381 m 3 ({ rock})=0.286 m 3 ({sand, s o i l } ) = 0 . 0 4 8 m 3 ({sand, rock})=0.238 m 3 (0)=O.O47 and the corresponding b e l i e f func t ion i s Bel 3 ( { sand})=0.381 B e l 3 ( { r o c k } ) = 0 . 2 8 6 B e l 3 ( { s a n d , s o i l } ) = 0 . 4 2 9 B e l 3 ( { s a n d , rock})=0.524 B e l 3 ( 0 ) = 1 . 0 36 3.3 SPECIAL CLASSES OF BELIEF FUNCTIONS In Sec. 3 . 1 , the genera l concept of b e l i e f func t ion was in t roduced . In t h i s s e c t i o n , some s p e c i a l c l a s s e s of b e l i e f func t ions w i l l be presented . These s p e c i a l c l a s s e s of b e l i e f func t ions are not only s imple i n express ion but a l s o , as w i l l be d i scus sed i n d e t a i l i n Chapter 6, meaningful when represen t ing evidence i n p r a c t i c e . 3.3.1 VACUOUS BELIEF FUNCTION The vacuous belief function i s fea tured by a s s i g n i n g one ' s t o t a l b e l i e f to the whole frame 0 and none to any s i n g l e t o n s or j o i n t p r o p o s i t i o n s i n 0 . I t has the express ion Bel(0)=1.0 Bel(A)=0.0 Ae0 From t h i s , i t can be e a s i l y shown that the p l a u s i b i l i t y va lue for any p r o p o s i t i o n A i s 1.0, i . e . P1(A)=1.0. The vacuous b e l i e f func t ion can be used to desc r ibe complete ignorance . By c o n t r a s t , i n Bayesian theory the ignorance i s represented by a s s i g n i n g equal p r o b a b i l i t i e s to a l l s i n g l e t o n s . In t e rp re t ed from the b e l i e f func t ion po in t of v iew, t h i s treatment of ignorance suggests that one has s p e c i f i c b e l i e f on each s i n g l e t o n p r o p o s i t i o n . To say the l e a s t , t h i s i s d i f f i c u l t to r e c o n c i l e w i th the idea of complete ignorance . As a matter of f a c t , i t can be e a s i l y shown that the combinat ion of a b e l i e f func t ion w i th a 37 vacuous b e l i e f function w i l l not change the o r i g i n a l b e l i e f function. This is consistent with a conventional view of the influence of belief characterized as the "ignorance". By contrast, the combination of the same be l i e f function with the Bayesian representation of ignorance w i l l d e f i n i t e l y change the o r i g i n a l b e l i e f function. This point may better be i l l u s t r a t e d by a simple example. Exampl e 3. 3 Consider a bel i e f function B e l , ^ 0 , 0={0,, 62 , 0 3}. Its basic probability assignment i s given as m,({01})=O.1 n M C t f , , 02})=O.3 m,({0,, 03})=O.2 m1(0)=O.4 The combination of m^A) with a vacuous belief function on 0 m2(A)=0 for a l l Ae© m2(0)=1.0 w i l l keep m, (A) unchanged. The combination of m^A) with the Bayesian representation of ignorance m 3(0,)=m 3(0 2)=m 3(0 3)=1/3 w i l l y i e l d 38 m({0,})=O.348 m({0 2})=0.391 m( { 0 3 } ) =0 . 261 which i s obv ious ly d i f f e r e n t from the o r i g i n a l bas ic p r o b a b i l i t y assignment m ^ A ) . One might reasonably conclude from t h i s that Dempster-Shafer theory provides a more appropr ia te way to represent ignorance than does Bayesian theory . 3.3.2 SIMPLE BELIEF FUNCTION A b e l i e f func t ion B e l : 2® [0 ,1] i s c a l l e d a simple belief function i f there e x i s t s a non-empty subset A of 0 such that m(A)=s Ae© m(B)=0.0 Be0 B*A m(0)=1-s The b e l i e f func t ion i s then expressed as Bel(B)=4 0.0 i f B does not con ta in A s i f B con ta ins A and B * 0 1.0 i f B=0 When two simple b e l i e f funct ions have the same f o c a l element A, the combination of them w i l l g ive another simple b e l i e f func t ion again wi th the same foca l element A . I f they do not 3 9 have the same f o c a l element, t h i s combinat ion w i l l not lead to a s imple b e l i e f f u n c t i o n . 3 . 3 . 3 BAYESIAN BELIEF FUNCTION I f the f o c a l elements of a b e l i e f func t ion are a l l s i n g l e t o n s , the corresponding b e l i e f func t ion i s named a Bayesian belief function. I t i s so named s imply because i t i s i d e n t i c a l to the p r o b a b i l i t y d i s t r i b u t i o n on s i n g l e t o n s used i n Bayesian theory . This type of b e l i e f f unc t i on i s c o n s i s t e n t w i th i n t u i t i v e frequency i n t e r p r e t a t i o n of p r o b a b i l i t y . In f a c t , i f one has enough obse rva t ions or exper ience about some unknown parameter, he can g ive the frequency (or chance) of each p o s s i b i l i t y being the true parameter and by doing t h i s , he i s a c t u a l l y g i v i n g a Bayesian b e l i e f f u n c t i o n . A d e t a i l e d d i s c u s s i o n of the p r o p e r t i e s of t h i s b e l i e f func t ion and the evidence cor responding to such type of b e l i e f func t ion can be found i n Chapter 6 . I t i s i n t e r e s t i n g to note that for each s i n g l e t o n , the b e l i e f v a l u e , p l a u s i b i l i t y value and bas ic p r o b a b i l i t y number are the same for Bayesian b e l i e f f u n c t i o n . The combinat ion of Bayesian b e l i e f func t ion wi th any b e l i e f func t ion i s s t i l l a Bayesian b e l i e f f u n c t i o n . The same r e s u l t can be obta ined by combining t h i s Bayesian b e l i e f func t ion w i th the r e l a t i v e p l a u s i b i l i t i e s of the s i n g l e t o n s of the other general b e l i e f f u n c t i o n , i . e . . Bel ( e)=kBel o(0)RplU) ( 3 . 7 ) 40 This concept w i l l also be discussed again in Chapter 4. 3.3.4 CONSONANT BELIEF FUNCTION If the focal elements of a be l i e f function can be ordered so that they are nested, th i s type of b e l i e f function' i s refered to as a consonant belief function. Assuming the focal elements of such a be l i e f function are A i = {6»,}, A2={0,, 62] . . . A ={0 1 f... 0n} the belief function and the p l a u s i b i l i t i e s for singletons then must s a t i s f y the following conditions[5] Bel(A,)<Bel(A 2)< <Bel(A n) Pl(0,)>P1(0 2)...>Pl(0 n) and PHe, ) = 1 .0 which in turn y i e l d s Bel({0 l})=1-Pl(0 2) Bel({0 1 r 0 2})=1-Pl(0 3) • • • • BelUfl, , . . .0 n_ 1 }) = 1-Pl(0 n) Bel(0)=1.O 41 3.4 SUMMARY An e n t i r e l y new theory which i s known as Dempster-Shafer theory i s in t roduced i n t h i s chap te r . Th i s theory i s based on the concept of a b e l i e f func t ion and Dempster 's r u l e of combina t ion . T h i s i s e n t i r e l y d i f f e r e n t from the bas i s of Bayesian theory . With an understanding of t h i s new theory , one might reasonably expect that i t would be used as a more genera l approach to the problem of in fe rence about u n c e r t a i n t y . Th i s seems p a r t i c u l a r l y t rue when the l i m i t a t i o n s posed by Bayesian theory are c o n s i d e r e d . A l s o of some s i g n i f i c a n c e a f t e r almost three decades of a p p l i c a t i o n i n c i v i l e n g i n e e r i n g , Bayesian theory can be shown to be a s p e c i a l case of Dempster-Shafer t heo ry . While the genera l b e l i e f func t ion may be compl i ca t ed , some s p e c i a l c l a s s e s of b e l i e f f u n c t i o n s , which have s imple mathematical p r o p e r t i e s and are p o t e n t i a l l y meaningful i n r ep re sen t ing ev idence , have been presented . As w i l l be d i s cus sed i n d e t a i l i n Chapter 6, these c l a s s e s of b e l i e f func t ions can p lay a major r o l e i n r ep resen t ing ev idence . The consonant b e l i e f func t ion i s of s p e c i a l i n t e r e s t when i n v e s t i g a t i n g the r e l a t i o n s h i p between Bayes ' and Dempster-Shafer methods. In fac t a l l of the b e l i e f func t ions i n the c l a s s e s , except the Bayesian type of b e l i e f f u n c t i o n , are s p e c i a l cases of consonant b e l i e f f u n c t i o n . I t w i l l be seen i n Chapter 6 that there i s an even s t ronger case for us ing consonant b e l i e f func t ions to represent ev idence . 4. THE EQUIVALENCE BETWEEN BAYESIAN AND DEMPSTER-SHAFER THEORY 4.1 STATISTICAL SPECIFICATION MODEL AND BAYESIAN THEORY The concept of a s t a t i s t i c a l s p e c i f i c a t i o n model has been mentioned at the beginning of Chapter 2 . A more formal d e f i n i t i o n of t h i s model i s now g i v e n [ l 8 ] . Suppose the frame of discernment 0 c o n s i s t s of the p o s s i b l e va lues of a parameter 6 and suppose that an experiment i s governed by one of a c l a s s {q^ (x )} , de®, of chance d e n s i t i e s on a set of exper imenta l outcomes X , the c o r r e c t va lue 6 corresponding to the c o r r e c t chance d e n s i t y func t ion q^(x) over X . The s p e c i f i c a t i o n of se ts 0 and X , togther w i t h the c l a s s of chance dens i ty func t ions {q^ (x )} , OeQ, i s de f ined as a statistical specification model. In a s t a t i s t i c a l s p e c i f i c a t i o n model, an observed outcome x of the experiment i s a p iece of evidence about the dens i ty func t ion q ^ ( x ) , and hence the value of 0, being c o r r e c t . S ince the p o s s i b i l i t i e s conta ined i n 0 can be const rued as the causes and the exper imenta l outcomes X as the e f f e c t s , the evidence g iven by outcome x i s then re fe red to as inferential evidence, i . e . the evidence of cause that i s p rov ided by an e f f e c t . As was d i scussed i n Chapter 2, Bayesian theory i s based on the s t a t i s t i c a l s p e c i f i c a t i o n model. I f the p r i o r in fo rmat ion about the frame of discernment 0 can be expressed by a p r i o r p r o b a b i l i t y d i s t r i b u t i o n p ' ( 0 j ) on 0, 42 43 then the updated p r o b a b i l i t y d i s t r i b u t i o n on 0 a f t e r observ ing an outcome x. can be obta ined from Bayes' equat ion p " ( 0 j / x i ) = k p ' ( 0 j ) p ( x . / 0 j ) (2.2) where k i s a n o r m a l i z i n g f ac to r as def ined p r e v i o u s l y . Accord ing to Eq . 2.1 i n Chapter 2, the sample l i k e l i h o o d p ( x ^ / 0 j ) can be expressed as p ( x i / 0 j ) = q e j ( x i ) (2.1) There fore , the Bayes ' formula can be expressed as p " ( 0 j / x i ) = k p ' (6.)qe.Ui) (4.1) where Q g j ( x £ ) i s a func t ion of 0 j . The f o l l o w i n g s imple example i l l u s t r a t e s the procedures of us ing Bayesian theory for the i n f e r e n t i a l problem which can be put i n to the s t a t i s t i c a l s p e c i f i c a t i o n model. Example 4.1 (From Tang and Ang Vol. 1 pp.332 Example 8.1) P i l e s for a foundat ion were i n i t i a l l y designed for the c a p a c i t y of 250 tons each. However, on some rare occas ions , i t i s es t imated that some of the p i l e s may be subjected to load as h igh as 300 tons . The task i s to determine the p r o b a b i l i t y of the p i l e s f a i l i n g under 300 tons l o a d . 44 Suppose the p r o b a b i l i t y of f a i l u r e ranges from 0.2 to 1.0 w i t h i n t e r v a l 0 .2 . The frame of discernment 0 then i s 0={0, , 0 2 , 0 3 , 0 „ , 0 5 }={O.2, 0 .4 , 0 .6 , 0 . 8 , 1.0}. The p r i o r p r o b a b i l i t y d i s t r i b u t i o n on 0 i s s p e c i f i e d as p ' ( 0 1 ) = O . 3 p ' (0 2 )=O.4 p ' ( 0 3 ) = O . l 5 p ' (0 f l )=O.1 p ' ( 0 5 ) = 0 . 0 5 A t e s t of the p i l e under the load of 300 tons i s conducted and the p o s s i b l e r e s u l t s of the t e s t form another set X = { x , = f a i l , x 2 = s u r v i v e } . The chance d e n s i t y func t ion q^(x) for each 0 which governs the exper imenta l outcomes X={x, , x 2 } i s g iven as q e ( x ) X i x 2 q 0 i ( x ) 0.2 0.8 q 0 2 ( x ) 0.4 0.6 q ^ 3 ( x ) 0.6 0.4 q e « ( x ) 0.8 0.2 q e s ( x ) 1 .0 0.0 Accord ing to the d e f i n i t i o n , t h i s i n f e r e n t i a l problem can be f i t t e d i n t o the s t a t i s t i c a l s p e c i f i c a t i o n model. S ince the p r i o r p r o b a b i l i t y d i s t r i b u t i o n and the chance d e n s i t y func t ions are known, Eq . 4.1 can then be used to 45 update the p r o b a b i l i t y d i s t r i b u t i o n on 0 a f t e r the obse rva t ion of outcome x ^ . For example, i f one t e s t i n d i c a t e s that the p i l e f a i l s to s u r v i v e the 300 tons l o a d , i . e . x^=x 1 f the updated p o s t e r i o r p r o b a b i l i t i e s c o n d i t i o n e d on x , = f a i l then i s p " , ( 0 , ) = 0 . 0 6 k p " 1 ( 0 2 ) = O . 1 6 k p " , ( 0 3 ) = O . O 9 k p " , ( 0 » ) = 0 . 0 8 k p " , ( 0 5 ) = O . O 5 k 1 The n o r m a l i z i n g fac to r k should be k= —'•— =2.273, 0.44 t h e r e f o r e , the p o s t e r i o r p r o b a b i l i t i e s a r e : p " , ( 0 , ) = 0 . 1 3 6 p " , ( 0 2 ) = O . 3 6 4 p " , ( 0 3 ) = O . 2 O 5 p " , ( 0 f t ) = 0 . 1 8 2 p " , ( 0 5 ) = O . 113 Because the t e s t shows that the p i l e f a i l s under the 300 tons l o a d , the r e v i s e d p r o b a b i l i t y d i s t r i b u t i o n r e f l e c t s t h i s in fo rmat ion by s h i f t i n g i t s supports towards the elements which i n d i c a t e h igher f a i l u r e p r o b a b i l i t i e s . Another t e s t i s made and i t i n d i c a t e s that the p i l e s u r v i v e s the 300 tons t e s t l o a d . The p r e v i o u s l y ob ta ined p o s t e r i o r s c o n d i t i o n e d on x ^ f a i l can then be used as p r i o r s and the new p o s t e r i o r s ob ta ined the same way as before , y i e l d i n g 46 p" 2(0,)=0.244 p" 2(0 2)=O.49O p" 2(0 3)=O.184 p " 2 ( 0 « ) = 0 . 0 8 2 p" 2 (0 5 )=O.O Since th i s test indicates that the p i l e succeeded in holding the 300 tons load, the new poster iors re f l ec t th i s information by s h i f t i n g to provide more support to those elements with lower f a i l u r e p r o b a b i l i t i e s . Note that p" 2 (0 5 )=O.O. In th is s i t u a t i o n , th i s p r o b a b i l i t y value w i l l remain at zero even though the p i l e s f a i l to survive the 300 tons load in a set of succeeding t e s t s . Since Bayesian theory becomes ine f fec t ive in deal ing with the s i tuat ion l i k e t h i s , care should be taken to avoid the occurrence of zero p r o b a b i l i t y . The best way to do th i s i s to subst i tute a c lose - to -zero value for a zero p r o b a b i l i t y and a c l o s e - t o - u n i t value for unity whenever these a r i s e . 4.2 DEMPSTER-SHAFER APPROACH TO STATISTICAL SPECIFICATION  MODEL Reca l l ing the discuss ion in Chapter 3, applying Dempster-Shafer theory requires one to f i r s t construct b e l i e f functions from the ava i lab le pieces of evidence and then combine them by using Dempster's rule of combination. In the spec ia l case in which the i n f e r e n t i a l problem can be put into the s t a t i s t i c a l s p e c i f i c a t i o n model, the Dempster-Shafer approach involves construct ing b e l i e f functions from the p r i o r evidence and the i n f e r e n t i a l 47 evidence ( i . e . exper imenta l outcome) and combining them by the combinat ion r u l e . For the purpose of comparison w i th the d i s c u s s i o n s i n s e c t i o n 4 . 1 , the p r i o r evidence i s assumed to p rov ide a Bayesian b e l i e f func t ion which i s the same as the p r i o r p r o b a b i l i t y d i s t r i b u t i o n i n Bayesian theory . (The p r i o r ev idence , of course , does not n e c e s s a r i l y g ive r i s e to a Bayesian b e l i e f f u n c t i o n ) . For such a Bayesian b e l i e f f u n c t i o n , the b e l i e f v a l u e , bas i c p r o b a b i l i t y number and p r o b a b i l i t y for each s i n g l e t o n are the same, i . e . Be lo (0 j )=m o (0 . j )=p , (0 j ) The c a l c u l a t i o n of the b e l i e f func t ion from the i n f e r e n t i a l evidence i s not so d i r e c t and needs some assumptions and analyses[18 ] . I f a p iece of i n f e r e n t i a l evidence x^ i s obta ined about the s t a t i s t i c a l s p e c i f i c a t i o n model ( q ^ ( x ) } , 0e0, one may i n t u i t i v e l y f e e l tha t evidence x^ favors those elements of 0 tha t provide g rea te r chance about the occurrence of x ^ . A c c o r d i n g l y , i t may be assumed that the p l a u s i b i l i t y of s i n g l e t o n P l ( 0 j ) i s p r o p o r t i o n a l to the chance that q^j (x) a s s igns x^ , i . e . P l ( 0 j ) = c q 0 j ( x . ) 0j*0 (4.2) where c i s a constant which i s independent of 9y The above i n t u i t i v e idea a l s o leads to another reasonable assumption: tha t the i n f e r e n t i a l evidence should best be de sc r ibed by a 48 consonant b e l i e f f u n c t i o n . I f the elements fl/e© are ordered so that the element ^j+i has l e s s tendency to produce the evidence x than dy then 0 , , having the greates t tendency to produce x , w i l l ob ta in a degree of suppor t . The element 0 2 w i l l not deserve any p o s i t i v e support as t h i s would c o n f l i c t wi th support for 0 , . The set { d X r 6 2 } however should have more support than 0, alone s ince {d:,d2} has grea te r tendency to produce x than 0, a l one . A c c o r d i n g l y , a consonant b e l i e f funct ion de r ived from i n f e r e n t i a l evidence x w i l l have ranked b e l i e f s , i . e . B e l ( 0 , , . . , 0 n ) > B e l ( 0 1 , . . . 0 n - 1 ) > . . . > B e l ( 0 , , 0 2 ) > B e l ( 0 , ) The two assumptions, i . e . the b e l i e f func t ion being consonant and the p l a u s i b i l i t y of a s i ng l e ton being p r o p o r t i o n a l to the chance value that QLQ(X) ass igns x , completely determines the consonant b e l i e f funct ion!18] Max q g ( x ) B e l ( A ) = 1- ^ (4.3) Max q f l (x) 0c0 y The corresponding p l a u s i b i l i t y func t ion P l (A) and bas ic p r o b a b i l i t y assignment m(A) can be c a l c u l a t e d by the methods given i n Chapter 3. Once the b e l i e f funct ions based on the p r i o r informat ion and i n f e r e n t i a l evidence are c a l c u l a t e d , they can be combined by Dempster 's r u l e of combina t ion . In the case where the p r i o r evidence can be represented by a 49 Bayesian b e l i e f f u n c t i o n , the combinat ion of those b e l i e f func t ions w i l l y i e l d a new Bayesian b e l i e f f u n c t i o n . In f a c t , as was mentioned i n Sec. 3 . 3 , ( a l so see Ref. [18]) t h i s combinat ion can be s i m p l i f i e d by combining the p r i o r Bayesian b e l i e f func t ion w i th the r e l a t i v e p l a u s i b i l i t i e s of s i n g l e t o n s of the consonant b e l i e f func t ion d e r i v e d from the i n f e r e n t i a l ev idence , i . e . B e l ( 0 j ) = kBe l o ( e . j )Rp l (0 j ) (4.4) The r e l a t i v e p l a u s i b i l i t i e s of s i n g l e t o n s R p l ( 0 j ) have the express ion R p l ( 0 j ) = c P l ( 0 j ) (3.3) The p l a u s i b i l i t y func t ion P l ( 0 j ) t a k i n g the form given by E q . 4 . 2 , the combined Bayesian b e l i e f func t ion Be l ( t9 ) the re fore can be expressed i n the f i n a l form B e l ( 0 : J ) = k B e l o ( 0 j ) q & ; j ( x i ) (4.5) Comparing Eq . 4.5 w i t h the Bayesian formula Eq . 4 . 1 , i t can be seen that the two equat ions are e s s e n t i a l l y the same once one r e a l i z e s that the p r i o r Bayesian b e l i e f func t ion B e l o ( 0 j ) i n E q . 4.5 i s e x a c t l y the same as the p r i o r p r o b a b i l i t y d i s t r i b u t i o n p ' ( 0 j ) i n E q . 4 . 1 . I t can then be concluded t h a t , for t h i s s p e c i a l case of the s t a t i s t i c a l 50 s p e c i f i c a t i o n model w i t h the p r i o r evidence expressed by a Bayesian b e l i e f f u n c t i o n , Dempster-Shafer theory y i e l d s the same r e s u l t as Bayesian theory . Since Dempster-Shafer theory i s not n e c e s s a r i l y conf ined to the s t a t i s t i c a l s p e c i f i c a t i o n model, and s ince not a l l of the i n f e r e n t i a l problems encountered i n p r a c t i c e can be f i t t e d i n t o the s t a t i s t i c a l s p e c i f i c a t i o n model, and, furthermore, not a l l of the p r i o r evidence can be s a t i s f a c t o r i l y expressed by a Bayesian b e l i e f f u n c t i o n , Dempster-Shafer theory can the re fore be used to dea l w i t h more genera l i n f e r e n t i a l problems. In other words, Bayesian theory i s only a very s p e c i a l case of Dempster-Shafer theory . Example 4.2 i l l u s t r a t e s the Dempster-Shafer approach to the i n f e r e n t i a l problem g iven in Example 4 . 1 . I t demonstrates that the Dempster-Shafer approach w i l l y i e l d the same r e s u l t as the Bayesian approach to t h i s p a r t i c u l a r problem. Example 4.2 (Continuation of Example 4.1) The b e l i e f func t ion B e l o ( 0 ) , based on the p r i o r ev idence , i s a Bayesian b e l i e f func t ion which i s i d e n t i c a l to the p r i o r p r o b a b i l i t y d i s t r i b u t i o n . The bas i c p r o b a b i l i t y assignment for t h i s p r i o r Bayesian b e l i e f func t ion i s m o(e ,)=0.30 m o (0 2 )=O.4O m o (0 3 )=O.15 m o ( 6 , « ) = 0 . l 0 m o (0 5 )=O.O5 51 A test i s conducted and i t s result indicates x 1 f i . e . the p i l e f a i l s to survive the 300 tons load. This test result w i l l give r i s e to a consonant b e l i e f function which can be calculated by Eq.4.3. The resulting consonant b e l i e f function and i t s corresponding basic pr o b a b i l i t y assignment are Bel,(0 5)=0.2 Bel,(0 5,0„)=O.4 Bel,(0 5,0 a,0 3)=O.6 Bel,(0 5,0 4,0 3 r0 2)=O.8 Bel,(0)=1.0 m, (05)=O.2 m,(0 5,0„)=0.2 m,(05,0,,03)=O.2 m,(0 5,0„,0 3,0 2)=O.2 m,(6)=0.2 The combination of m0(A) with m,(A) through Dempster's rule of combination yields a new Bayesian b e l i e f function with i t s basic p r o b a b i l i t y assignment given as m2(0,)=O.136 m2(02)=O.364 m2(03)=O.2O4 m2(0,)=0.182 m2(05)=0.114 which is i d e n t i c a l to the Bayesian posterior p r o b a b i l i t i e s after the observation of x,. As was mentioned e a r l i e r in th i s section, the same result can be obtained by combining the prior Bayesian be l i e f function with the r e l a t i v e p l a u s i b i l i t i e s of singletons of the consonant b e l i e f 52 func t ion B e l , ( A ) . I f another t e s t y i e l d s the r e s u l t x 2 = s u r v i v e , the consonant b e l i e f func t ion B e l 3 ( A ) and the corresponding bas ic p r o b a b i l i t y assignment m 3 (A) d e r i v e d from x 2 are given as B e l 3 ( 0 , ) = 0 . 2 5 B e l 3 ( 0 , , 0 2 ) = O . 5 B e l 3 ( 0 i , 0 2 , e 3 ) = O . 7 5 B e l 3 ( 0 , , 6 2 , 6 3 , 6 * ) = ] .0 1113(0, ) = 0.25 m 3 ( 0 i , 0 2 ) = O . 2 5 m 3 ( 0 , , 0 2 , 0 3 ) = 0 . 2 5 m 3 ( 0 , , 0 2 , 0 3 , 6 u ) = 0 . 2 5 The combinat ion of m 3 (A) w i th m 2 (A) y i e l d s : m 4 (0,)=O.244 ' m,(0 2 )=O.49l m „ ( 0 3 ) = 0 . 1 8 3 m,, ( 0« ) =0 . 082 m „ ( 0 5 ) = 0 . 0 which i s the same as the Bayesian p o s t e r i o r p r o b a b i l i t i e s a f t e r the two observa t ions x , and x 2 . A g a i n , t h i s r e s u l t can a l s o be ob ta ined by combining the Bayesian b e l i e f func t ion g iven by m 2 (A) w i t h the r e l a t i v e p l a u s i b i l i t i e s of s i n g l e t o n s g iven by the consonant b e l i e f func t ion B e l 3 ( A ) . 53 4.3 A FURTHER EXAMPLE The f o l l o w i n g example shows once more the whole procedure for the Dempster-Shafer approach to the i n f e r e n t i a l problem, based on the s t a t i s t i c a l s p e c i f i c a t i o n model. Example 4.3 (From Shafer[)8] pp.243) Cons ide r ing a c o i n - t o s s i n g experiment. The r e s u l t i s e i t h e r Heads or T a i l s . The chance dens i ty func t ion which governs the experiment i s unknown and i s assumed to be q 0 ( H ) = 0 / l O and q^(T)=1-0/lO where deQ and 0 = { 0 , 1 , 2 , . . . 1 0 } . I f the toss r e s u l t i s head, i t w i l l produce a consonant b e l i e f func t ion which can be c a l c u l a t e d from the f o l l o w i n g equation d e r i v e d from Eq. 4.3 Max q f l ( H ) , , - 0cA u B e l u ( A ) = 1- = H Max q f l ( H ) 0c0 0 Note that q f l (H)=0/lO, therefore Maxq f l(H)=1.0, and the above equation becomes Be l u (A)=1-Max(0 / l0 ) H 0c A Furthermore, the f o c a l elements A of B e l H a re : {10}, {9 ,10} , { 8 , 9 , 1 0 } , . . . { 0 , 1 , . . . 1 0 } . The e v a l u a t i o n of Be l^(A) acco rd ing 54 to above equation and the corresponding basic p r o b a b i l i t y assignment m^(A) i s Bel H({lO})=l/lO mH({10})=1/10 BelH({9,10})=2/10 mH({9,10})=1/10 Bel H({8,9,10})=3/l0 mR({8,9,10})=1/10 BelH({1,2,..10})=1 m H ({l,2,..10})=l/lO Similarly, a toss resulting in t a i l w i l l give another consonant belief function which is Max qa(T) Bel T(A)-1- 6 C k 6  1 Max g«(T) The focal elements for Bel^,( A) are {0}, {0,1}, ... {0,1,..9}. The calculations of Bel^(A) and the corresponding basic probability assignment are given as Bel T({0})=l/l0 mT({0})=l/10 Bel T({0,1})=2/10 mT({0,1})=1/10 Bel T({0,1,2})=3/l0 mT({0,1,2})=1/10 BelT({0,1,..9})=1 mT({0,1,..9})=1/10 If n tosses are recorded, and k out of n are heads, then the f i n a l b e l i e f function Bel(A) can be obtained by combining k consonant b e l i e f functions Bel„(A) with n-k the 55 consonant belief functions Bel T(A) using Dempster's combination rule. It should be noted that, though each of these individual b e l i e f functions i s consonant, the combined belief function Bel(A) w i l l be disonant unless k=0 or k=n, i.e. unless a l l the records are either t a i l s or heads. If prior information i s available, the b e l i e f function derived from the prior information can be combined with Bel(A) obtained above to y i e l d a new b e l i e f function. In the case where the prior b e l i e f . function can be expressed as a Bayesian b e l i e f function, the result of the combination with Bel(A) i s s t i l l a Bayesian be l i e f function. 4 . 4 SUMMARY Bayesian theory and Dempster-Shafer theory can both be used to deal with the i n f e r e n t i a l problem. But these two theories are di f f e r e n t in the sense that Bayesian theory can only be used to deal with the i n f e r e n t i a l problem which can be put into the s t a t i s t i c a l s p e c i f i c a t i o n model and in which the prior evidence i s s p e c i f i c a l l y expressed by a Bayesian prior p r o b a b i l i t y d i s t r i b u t i o n . Dempster-Shafer theory, on the other hand, can be used to deal with more general i n f e r e n t i a l problems which do not n e c e s s a r i l l y f i t into the s t a t i s t i c a l s p e c i f i c a t i o n model. In the case where Bayesian theory i s appropriate, Dempster-Shafer theory can s t i l l be used and y i e l d s the same results as Bayesian theory. The conclusion therefore i s that Bayesian theory i s a very special scheme in dealing with i n f e r e n t i a l problem which i s 56 conta ined w i t h i n the more genera l Dempster-Shafer theory . The concept of a s t a t i s t i c a l s p e c i f i c a t i o n model i s very a t t r a c t i v e t h e o r e t i c a l l y and revea l s both s i m i l a r i t i e s and d i f f e r e n c e between Bayesian theory and Dempster-Shafer t heo ry . I t i s a l s o a t t r a c t i v e i n p r a c t i c e s ince a great number of problems can be f i t t e d or i n t e r p r e t e d i n t o the scheme of such model. T h i s chapter has concent ra ted l a r g e l y on the d i s c u s s i o n s about the equiva lence between Bayesian theory and Dempster-Shafer t heo ry . In the f o l l o w i n g two chap te r s , s i g n i f i c a n t d i f f e r e n c e s between the two t h e o r i e s w i l l be d i s c u s s e d . The d i s c u s s i o n s i n Chapter 5 w i l l concent ra te on the divergence of the r e s u l t s of Dempster-Shafer and Bayesian t h e o r i e s . In Chapter 6, the conceptua l d i f f e r e n c e between the two t h e o r i e s and the r ep re sen t a t i on of evidence through b e l i e f func t ions are d i s c u s s e d . The advantages of Dempster-Shafer theory over Bayesian theory can subsequently be viewed from these d i s c u s s i o n s . 5. DIVERGENCE OF RESULTS FROM DEMPSTER-SHAFER AND BAYESIAN THEORIES 5.1 INTRODUCTION The c o n d i t i o n for equiva lence between Bayesian theory and Dempster-Shafer theory was d i scussed i n Chapter 4. I t was concluded that for the s p e c i a l case of a s t a t i s t i c a l s p e c i f i c a t i o n model, i n which the p r i o r evidence can be expressed as a Bayesian b e l i e f f u n c t i o n , the two t h e o r i e s w i l l y i e l d the same r e s u l t s . In the case where the p r i o r evidence can only be s a t i s f a c t o r i l y expressed as a non-Bayesian b e l i e f f u n c t i o n , Dempster-Shafer theory i s c l e a r l y the more appropr i a t e approach to the i n f e r e n t i a l problem. In t h i s l a t t e r case , there i s no equ iva l en t Bayesian approach. Dempster-Shafer theory can a l s o be used i n the more genera l s i t u a t i o n i n which the i n f e r e n t i a l problem can not be f i t t e d i n t o the s t a t i s t i c a l s p e c i f i c a t i o n model. In tha t s i t u a t i o n a l s o , as one might expect , there i s no s u i t a b l e r i go rous Bayesian approach. For the s i t u a t i o n where Dempster-Shafer theory i s the only appropr i a t e approach to an i n f e r e n t i a l problem, Bayesian theory might s t i l l be adopted but recognized to be only an approximate approach. For example, i n a case where the i n f e r e n t i a l problem can be f i t t e d i n t o the s t a t i s t i c a l s p e c i f i c a t i o n model but the " t rue" p r i o r evidence can c l e a r l y be expressed only as a non-Bayesian b e l i e f f u n c t i o n , one may adopt a surrogate conven t iona l p r o b a b i l i t y 57 58 d i s t r i b u t i o n , based as c l e a r l y as p o s s i b l e on t h i s p r i o r ev idence , and then use the Bayesian theory for the i n f e r e n c e . In doing so, s u b j e c t i v e judgements are c l e a r l y i n v o l v e d . For the more genera l case where the i n f e r e n t i a l problem does not f i t i n t o the s t a t i s t i c a l s p e c i f i c a t i o n model, one may force the problem i n t o the context of the Bayesian p r o b a b i l i t y model. Doing t h i s i n v o l v e s a s s i g n i n g some of the evidence as " p r i o r " evidence and the remaining evidence as " i n f e r e n t i a l " ev idence . A p r i o r p r o b a b i l i t y d i s t r i b u t i o n has to be then cons t ruc ted from the " p r i o r " evidence and a sample l i k e l i h o o d func t ion cons t ruc ted from the " i n f e r e n t i a l " ev idence . Th i s process i n v o l v e s fur ther s u b j e c t i v e judgements. As one would expect , t h i s approximate Bayesian approach w i l l be q u i t e d i f f e r e n t from a more appropr i a t e Dempster-Shafer approach and would e v e n t u a l l y produce d i f f e r e n t r e s u l t s . A fur ther d i s c u s s i o n about the d i f f e r e n c e between the two t heo r i e s when d e a l i n g wi th i n f e r e n t i a l problems can be found i n Chapter 6 . A genera l compar is ion between the Dempster-Shafer and Bayesian approaches i n the s i t u a t i o n s where Dempster-Shafer theory i s the appropr i a t e approach and there i s no t rue equ iva l en t Bayesian approach, may be very d i f f i c u l t because of the s u b j e c t i v e elements i n v o l v e d i n the n e c e s s a r i l y approximate Bayesian approach. But i t i s p o s s i b l e to access the divergence of the two approaches i n a s i m p l i f i e d s i t u a t i o n . Assume that an i n f e r e n t i a l problem can be f i t t e d 59 i n t o the s t a t i s t i c a l s p e c i f i c a t i o n model and the p r i o r evidence can be expressed by a Bayesian b e l i e f f u n c t i o n , then the Bayesian theory can be p r o p e r l y used. At t h i s p o i n t , the r e s u l t s obta ined by e i t h e r Dempster-Shafer approach or Bayesian approach w i l l agree . Suppose now that some doubts a r i s e concerning the p r i o r evidence and the i n f e r e n t i a l evidence and should be taken i n t o account by modifying the p r i o r Bayesian b e l i e f func t ion and the consonant b e l i e f f u n c t i o n . Th i s m o d i f i c a t i o n can be made us ing S h a f e r ' s discounting met hod[18] which w i l l be in t roduced i n Sec. 5 .2 . The Dempster-Shafer r e s u l t s based on the d i scounted b e l i e f func t ions w i l l d iverge from the o r i g i n a l Bayesian r e s u l t s . As w i l l be seen i n l a t t e r s e c t i o n s , t h i s divergence w i l l depend on the degree of the doubt one cas t s on the two b e l i e f func t ions and the degree of c o n f l i c t between the two b e l i e f f u n c t i o n s . A t h e o r e t i c a l compar is ion between the two approaches de sc r ibed above i s presented i n Sec. 5 . 3 . The d i s c u s s i o n w i l l be conf ined to a two element frame of discernment to s i m p l i f y the a n a l y s i s . In Sec . 5 .4 , numer ica l examples are given to fur ther i l l u s t r a t e the c o n c l u s i o n s obta ined i n Sec. 5.3 as w e l l as to prov ide a s e n s i t i v i t y view of the divergence of r e s u l t s . 60 5.2 DISCOUNTING A BELIEF FUNCTION Assume a p iece of evidence can be represented by a b e l i e f func t ion B e l ( A ) , i t s cor responding bas i c p r o b a b i l i t y assignment being denoted as m(A). Assume fu r the r that one has some doubts about the t r u t h of the ev idence . Then he can modify the b e l i e f func t ion us ing S h a f e r ' s d i s c o u n t i n g method. S h a f e r ' s d i s c o u n t i n g method i n v o l v e s f i r s t determing a d i scount ra te a, where 0<a^1.0. The a value i s a measure of the degree of doubt one has on the ev idence : the greater the a va lue i s , the higher the degree of doubt about the ev idence . When a i s equal to ze ro , i t i n d i c a t e s tha t one has no doubt at a l l about the ev idence . In the case where a reachs 1.0, i t means that one has no conf idence at a l l about the ev idence . This i m p l i e s that the evidence conveys nothing but complete ignorance of the s i t u a t i o n . The value 1-a the re fore can be cons idered as the degree of trust one has on the ev idence . The d i s c o u n t i n g of a b e l i e f func t ion based on the d i scount ra te a i s done by m u l t i p l y i n g the b e l i e f func t ion Be l (A) by the degree of t r u s t 1-a and by a s s i g n i n g the d i s p l a c e d b e l i e f f r a c t i o n a (which i s reass igned to ignorance) to the whole frame 0. The d i scoun ted b e l i e f func t ion B e l ' ( A ) can then be expressed as 61 B e l ' ( A ) = ( 1 - a ) B e l ( A ) Ae© (5.1) B e l ' ( 0 ) = 1 . 0 and the corresponding b a s i c p r o b a b i l i t y assignment i s m'(A)=(1-a)m(A) Ae© (5.2) m ' ( © ) = ( 1 - a ) m ( 0 ) + a S h a f e r ' s d i s c o u n t i n g method can be used to d i scoun t any b e l i e f func t ion when one f ee l s l e s s than comple te ly conf ident about that b e l i e f f u n c t i o n . Th i s method can a l s o be used i n the s i t u a t i o n where one wants to reduce the in f luence of a b e l i e f f u n c t i o n . For example, assuming a s i t u a t i o n w i t h a g iven set of b e l i e f f u n c t i o n s , where one of them i s s t r o n g l y c o n f l i c t i n g wi th the others wh i l e the other b e l i e f func t ions only m i l d l y c o n f l i c t amongst themselves . Under some c o n d i t i o n s , such as where the source of c o n f l i c t i n g b e l i e f i s low i n c r e d i b i l i t y , one may wish to d iscount the odd b e l i e f func t ion to reduce i t s e f f e c t on the f i n a l combina t ion . Two simple examples are presented to i l l u s t r a t e Sha fe r ' s d i s c o u n t i n g method. Example 5.1 Suppose there i s a Bayesian b e l i e f func t ion B e l ( 0 ) : 2 ® , ©={0i , 02f 83), and i t s bas ic p r o b a b i l i t y assignment i s 62 given by m 1(t9 1) = 0.5 m 1 (0 2 )=O.3 m,(0 3 )=O.2 I f one i s 80% conf ident about the b e l i e f f u n c t i o n , then the b e l i e f func t ion can be d i scounted by a d i scount ra te a=0.2 through E q . 5 .2 . The bas ic p r o b a b i l i t y assignment of the d i scounted b e l i e f func t ion i s m ' 1 (0 1 )=O.4 m' , (0 2 )=O.24 m ' , (0 3 )=O.16 m' 1 (G)=0.2 Note that a d i scounted Bayesian b e l i e f func t ion becomes a non-Bayesian b e l i e f func t ion for a l l values of a grea ter than 0. Exampl e 5.2 Suppose there i s a consonant b e l i e f func t ion Bel (A) on 0 = { 0 , , 0 2 ,#3 } and i t s ba s i c p r o b a b i l i t y assignment i s g iven by m 2 (0 i )=O.5 m^e, , 0 2 ) = O.3 m 2 ( © ) = 0 . 2 I f one i s 80% conf ident about t h i s b e l i e f func t ion then the b e l i e f func t ion can be d i scounted by a d i scount r a te a=0.2. The d i scounted b e l i e f func t ion w i l l be m ' 2 (0 1 )=O.4 m ' 2 ( 0 , , 0 2 ) = O . 2 4 m' 2 (0)=O.36 63 Note that d i s c o u n t i n g a consonant b e l i e f func t ion s t i l l r e s u l t s i n a consonant b e l i e f f u n c t i o n . 5.3 DIVERGENCE OF RESULTS OF TWO THEORIES AND SENSITIVITY  ANALYSIS Assume that there i s an i n f e r e n t i a l problem, the two element frame of discernment being ®={di,62}, and the i n f e r e n t i a l problem being able to f i t i n t o the s t a t i s t i c a l s p e c i f i c a t i o n model. Suppose the p r i o r evidence can be expressed by a Bayesian b e l i e f func t ion w i t h i t s ba s i c p r o b a b i l i t y assignment m l (e , )=p 1 m, (0 2 )=p 2 (5.3) and the sample l i k e l i h o o d s of the i n f e r e n t i a l evidence x are g iven as l ( x / 0 , ) = l i l ( x / 0 2 ) = l 2 1!>12 S h a f e r ' s consonant b e l i e f func t ion based on the i n f e r e n t i a l evidence x can then be expressed as m 2 ( 0 , ) = 1 - l 2 / l 1 m 2 ( 0 , , 0 2 ) = l 2 / l i (5 .4) In t h i s case both Bayesian theory and Dempster-Shafer theory can be used p r o p e r l y to perform the i n f e r ence . The i n f e r e n t i a l r e s u l t w i l l be a p o s t e r i o r Bayesian b e l i e f 64 func t ion P i l i P l l l + P 2 l 2 (5.5) P2I2 P l l l + P 2 l 2 Assume fur ther that one has some doubts about the p r i o r evidence on which the Bayesian b e l i e f f unc t i on i s based and about the i n f e r e n t i a l evidence on which the consonant b e l i e f func t ion i s based. The two b e l i e f func t ions can be d i scounted us ing Sha fe r ' s d i s c o u n t i n g method as g iven i n Sec. 5 .2 . Dempster-Shafer b e l i e f combinat ion based on these d iscounted b e l i e f func t ions w i l l produce r e s u l t s which d iverge from the o r i g i n a l Bayesian r e s u l t s . The divergence of r e s u l t s from the Dempster-Shafer and Bayesian approaches under p r o g r e s s i v e l y i n c r e a s i n g d i s c o u n t i n g w i l l be assessed i n t h i s s e c t i o n . Two separate cases w i l l be c o n s i d e r e d . One where on ly the p r i o r Bayesian b e l i e f func t ion i s d i scounted and the other where only the consonant b e l i e f func t ion i s d i s c o u n t e d . 5.3.1'DISCOUNTING THE PRIOR BAYESIAN BELIEF FUNCTION As the f i r s t case , cons ider that the Bayesian b e l i e f func t ion m^(8) i s d i scounted by a d i scoun t ra te a. The d i scounted p r i o r Bayesian b e l i e f func t ion then i s m(0, ) = m(0 2)= m 1 i ( 0 i ) = P i ( 1 - a ) m ' , ( 0 2 ) = p 2 ( 1 - a ) m ' 1 ( 0 1 , 0 2 ) = a 65 The combinat ion of m ' , ( A ) w i th m 2 (A) y i e l d s p , (1 -a )+a( I - I2 / I1 ) m(e, ) = 1 - p 2 ( l - a ) d - l 2 / l 1 ) p 2 ( 1 - a ) 1 2 / 1 1 m(62)= - (5.6) 1 - p 2 ( 1 - a ) ( 1 - 1 2 / 1 , ) l 2 / l , a m{6,,92)= 1 - p 2 ( 1 - a ) ( 1 - l a / l , ) Note tha t when a=0.0, E q . 5.6 w i l l be i d e n t i c a l to Eq . 5 .5 ; when a>0.0, Eq . 5.6 w i l l d iverge from Eq . 5 .5 ; when a=1.0, Eq . 5.6 w i l l be i d e n t i c a l to E q . 5 .4 . Therefore the greater the a va lue i s , the higher the degree of divergence of Eq . 5.6 from E q . 5 .5 . The ra te of divergence i s g iven by the f i r s t d e r i v a t i v e s of Eq . 5 .6 , i . e . d[m(0, ) ] - p , + k - k 2 p 2 = (5 .7 .1 ) da [ 1 - p 2 ( 1 - a ) k ] 2 d[m(0 2 ) ] _ - p 2 ( l - k ) ' da ~ [ 1 - p 2 ( 1 - a ) k ] 2 d [ m ( 0 1 f 0 2 ) ] ( 1 - k ) ( 1 - p 2 k ) (5.7.2) da [ 1 - p 2 ( 1 - a ) k ] 2 (5 .7 .3 ) where k = 1 - 1 2 / l , . Note tha t once p , , p 2 and k are determined, E q . 5.7 w i l l be func t ions of the d i scount ra te a . Eq . 5.7 there fore g ives the s e n s i t i v i t y in fo rmat ion of the divergence of Dempster-Shafer and Bayesian r e s u l t s w i th change i n the d i scount r a te a . 6 6 An interesting situation to consider is the rate of divergence of the Dempster-Shafer and Bayesian results when a i s close to 0.0. Setting a equal to zero, Eq. 5.7 w i l l give s e n s i t i v i t y information about the rate of divergence with a s l i g h t increase of a from zero, i . e . d[m(0,)] _ -pi+k-k 2p 2 da" ' a = 0 ~ [ 1-p 2k] 2 d[m(0 2)] _ -p 2d-k) da" a = 0 " [ l - p 2 k ] 2 d[m(0 1,0 2)] (1-k)(l-p 2k) da" ' a = 0 [ 1-p 2k] 2 Note that the term p 2k in the denominator of the above equations i s the degree of c o n f l i c t between the o r i g i n a l two bel i e f functions m,(A) and m 2(A). Therefore, the rate of the divergence of the two theories close to a=0 w i l l depend on the degree of c o n f l i c t of the two o r i g i n a l b e l i e f functions. When the degree of c o n f l i c t i s not too high, i . e . the value p 2k substantially less than 1.0, Eq. 5.8.1 w i l l give a f i n i t e value, indicating that the Dempster-Shafer approach w i l l diverge from the Bayesian approach gradually as a increases. When the degree of c o n f l i c t i s very high, i . e . the value p 2k i s very close to 1.0, Eq. 5.8.1 w i l l give a very large value. This indicates that in a situation where the two b e l i e f functions are strongly c o n f l i c t i n g , the Dempster-Shafer approach w i l l diverge s i g n i f i c a n t l y from the Bayesian approach when only the sl i g h t e s t amount of (5.8.1) (5.8.2) (5.8.3) 67 d i s c o u n t i n g i s i n t roduced . In other words, when the two b e l i e f func t ions are s t r o n g l y c o n f l i c t i n g , Bayesian theory w i l l g ive an i n f e r e n t i a l r e s u l t which i s not i n d i c a t i v e of the s l i g h t e s t doubt concerning the p r i o r evidence or the exper imenta l r e s u l t . Whi le i t might be concluded that the Bayesian approach should be avoided i n t h i s s i t u a t i o n , and S h a f e r ' s d i s c o u n t i n g method w i t h Dempster 's r u l e of combinat ion should be cons idered as the appropr i a t e approach, the ques t ion of assess ing the c o r r e c t cho ice of a va lue for a i s s t i l l un re so lved . The d i s c u s s i o n s and conc lu s ions i n t h i s s e c t i o n have been based on Eq . 5 . 8 . 1 . S i m i l a r c o n c l u s i o n s can a l s o be drawn from E q . 5 .8 .2 and Eq . 5 . 8 . 3 . 5 .3 .2 DISCOUNTING THE CONSONANT BELIEF FUNCTION Assume one i s conf iden t of the Bayesian b e l i e f func t ion but has some doubts about the consonant b e l i e f f u n c t i o n . The consonant b e l i e f func t ion should then be .d iscounted, accord ing to the d i scount ra te a, a.s f o l l o w s m ' 2 ( 0 , ) = ( 1 - 1 2 / 1 , ) ( 1 - a ) m ' 2 ( 0 , , 0 2 ) = ( 1 - a ) l 2 / l , + a The combinat ion of m,(A) w i t h m ' 2 ( A ) y i e l d s m(0 1)= Pi (5 .9 .1 ) p,+p 2 [ ( 1 -a) l 2 / l - ,+a] m(0 2 )=1-Pi (5 .9 .2 ) p , + p 2 [ ( l - a ) l 2 / l , + a ] 68 The ra te of the divergence of the Dempster-Shafer approach from the Bayesian approach, as a r e s u l t of a , can then be obta ined by the d e r i v a t i v e s of Eq . 5.9 d[m(0,) ] " P i ( l - l 2 / l i ) (5 .10.1) da [ P i + p 2 ( 1 - a ) l 2 / l 1 + a ] 2 d[m(0 2 ) ] _ p , ( 1 - l 2 / l 1 ) da [ p , + p 2 ( 1-a ) l 2 / l 1+a] 2 (5 .10.2) S e t t i n g a=0.0, the s e n s i t i v i t y in fo rmat ion about the rate of divergence w i t h a s l i g h t increase of a from zero can be obta ined i . e . d [m(0 1 ) ] - p 1 d - l 2 / l 1 ) da ' a = 0 ( p t + p . l . / l , ) 2 d[m(0 2 ) ] _ p! (1 - 1 2 / l i ) da" ' a = 0 ~ ( p , + p 2 l 2 / l , ) 2 Note that the denominator t p i + p 2 l 2 / l 1 ] 2 i n the above equat ions can again be w r i t t e n as ( 1 - p 2 k ) 2 . The above equat ions can then be w r i t t e n as d[m(6,)] - p ^ l - l a / l , ) | n= (5 .11.1) da a " 0 ( l - p 2 k ) 2 d[m(0 2 ) ] p , ( 1 - l 2 / l 1 ) L - n = — 5.11.2) da a = 0 ( 1 - p 2 k ) 2 The comparison of E q . 5.11 wi th Eq . 5.8 shows that they have the same denominator. This i n d i c a t e s that the ra te of divergence of the Dempster-Shafer approach from the Bayesian 69 approach, i n t h i s s i t u a t i o n , a l s o depends on the degree of c o n f l i c t between the two b e l i e f f u n c t i o n s . Therefore the same c o n c l u s i o n s as tha t i n Sec 5 .3 .2 can be obta ined through the a n a l y s i s of Eq . 5 .11 . Th i s confirms the symmetr ical t reatments of the two sources of the b e l i e f i n the Dempster-Shafer scheme. The d i s c o u n t i n g of Bayesian b e l i e f f unc t i on and consonant b e l i e f func t ion has been cons idered separa te ly i n Sec. 5.3.1 and Sec. 5 . 3 . 2 . In a s i t u a t i o n where both b e l i e f func t ions are d i scounted s imu l t aneous ly , the a n a l y s i s becomes more d i f f i c u l t because of the two d i scoun t v a r i a b l e s i n v o l v e d . However, t h i s s i t u a t i o n can be e a s i l y i n v e s t i g a t e d through a numer ica l example. In Sec . 5 .4 , numerica l examples w i l l be presented to i l l u s t r a t e the conc lu s ions obta ined i n t h i s s e c t i o n and to demonstrate the s i t u a t i o n where the two b e l i e f func t ions are d i scounted at the same t ime . 5.4 NUMERICAL EXAMPLE In Sec. 5 .3 , i t was concluded that the divergence of the r e s u l t s from the Dempster-Shafer and Bayesian approach, when d i s c o u n t i n g b e l i e f f u n c t i o n s , w i l l depend on the d i scount ra te a and the degree of c o n f l i c t between the two o r i g i n a l b e l i e f f u n c t i o n s . Two numer ica l examples are presented i n t h i s s e c t i o n and address two d i f f e r e n t s i t u a t i o n s : where b e l i e f func t ions are on ly m i l d l y c o n f l i c t i n g and where b e l i e f func t ions are s t r o n g l y 70 c o n f l i c t i n g . In each -o f these two s i t u a t i o n s , three cases w i l l be d i s c u s s e d , i . e . the d i s c o u n t i n g of the p r i o r Bayesian b e l i e f func t ion o n l y , the d i s c o u n t i n g of the consonant b e l i e f func t ion only and the d i s c o u n t i n g of bo th . The eamples are conf ined to the two element frame of d iscernment . I t i s assumed that the i n f e r e n t i a l problem can be f i t t e d i n to the s t a t i s t i c a l s p e c i f i c a t i o n model c o n s i s t e n t w i th the d i s c u s s i o n s i n Sec. 5 . 3 . Example 5.3 The belief functions are mildly conflicting Assume a two element frame of discernment © = { 0 i , 0 2 } . The p r i o r evidence i s expressed by a Bayesian b e l i e f func t ion w i t h i t s bas ic p r o b a b i l i t y assignment m 1 (0 1 )=O.65 m 1 (0 2 )=O.35 The sample l i k e l i h o o d of a p iece of i n f e r e n t i a l evidence x i s known to be l ( x / 0 1 ) = O . 7 5 and 1 ( x / 0 2 ) = 0 . 4 5 . A consonant b e l i e f func t ion can then be obta ined based on the i n f e r e n t i a l evidence and i t s bas i c p r o b a b i l i t y assignment i s g iven as m 2 (0 1 )=O.4 m 2 ( 0 , , 0 2 ) = O . 6 The degree of c o n f l i c t between m,(A) and m 2 (A) i s 0 .14, which i s s u b s t a n t i a l l y l e s s than 1.0. Th i s s i t u a t i o n therefore represents a m i l d l y c o n f l i c t i n g s i t u a t i o n . 71 Now cons ider the three d i f f e r e n t cases of d i s c o u n t i n g of the b e l i e f f u n c t i o n s : Case 1. m,(A) i s d i scounted by d i scount ra te a wh i l e m 2 (A) remains unchanged The d i s c o u n t i n g of m,(A) by a d i scount ra te a w i l l y i e l d a new b e l i e f func t ion which i s g iven by m ' , (0 , )=0 .65(1-a ) m ' , (0 2 )=0 .35 (1 -a ) m * 1 ( 0 1 f 0 2 ) = a The combinat ion of m ' , ( A ) wi th m 2 (A) w i l l y i e l d 0 .65-0 .05a m(0,)= 0.86+0.14a 0.21(1-a) 111(0,)= (5.12) 0.86+0.14a 0. 6a m(0 , , 0 2 )= 0.86+0.14a Case 2. m 2 (A) i s d i scounted by d iscount ra te a wh i l e m,(A) remains unchanged The d i s c o u n t i n g of m 2 (A) by a d i scoun t ra te a w i l l y i e l d a new b e l i e f func t ion which i s m ' 2 (0 , )=0 .4 (1 -a ) m ' 2 ( 0 , , 0 2 ) = 0 . 6 + 0 . 4 a Combining m,(A) w i t h m ' 2 ( A ) by us ing Dempster 's r u l e of 72 combina t ion , a new b e l i e f func t ion w i l l be obta ined 0.65 m(0, ) = 0.86+0.14a (5.13) 0.35(0.6+0.4a) m(0 2)= 0.86+0.14a Case 3. both m,(A) and m 2 (A) are d i scounted s imul taneous ly Assume the two b e l i e f func t ions are d i scoun ted by the same d i scount ra te a. The combinat ion of the d i scounted b e l i e f func t ions m ' , ( A ) and m ' 2 ( A ) g iven i n case 1 and case 2 w i l l y i e l d 0 .65(1-a)+0.4a(1-a) m(e,)= 1-0 .14(1-a ) 2 0 . 3 5 ( 1 - a ) + 0 . 1 4 d - a ) 2 m(0 2)= (5.14) 1 -0 .14(1-a ) 2 a-0.4a(1-a) m(0 , , 6 2 )= 1-0 .14(1-a ) 2 Figu re s 5.1(a) to 5 .1(c) p l o t s the r e l a t i o n s h i p m(A) v s . a as s p e c i f i e d by Eq . 5.12 through Eq . 5.14 r e s p e c t i v e l y . From F i g . 5.1(a) to F i g . 5 . 1 ( c ) , i t i s seen that for the s i t u a t i o n i n which the degree of c o n f l i c t between the two b e l i e f func t ions i s not very h i g h , the divergence of the r e s u l t s of Dempster-Shafer approach from the Bayesian approach w i l l increase g r a d u a l l y as the d i scount ra te a i n c r e a s e s . In other words, the r e s u l t a n t b e l i e f s a f t e r the d i s c o u n t i n g of the o r i g i n a l b e l i e f 73 func t ions for a l l the three cases approximate ly l i n e a r l y i n t e r p o l a t e w i t h a between the p o s t e r i o r s and the b e l i e f s of the non-discounted b e l i e f func t ion (the vacuous b e l i e f func t ion i n case 3 ) . Th is obse rva t ion i s c o n s i s t e n t w i t h the c o n c l u s i o n s obta ined i n Sec. 5 . 3 . Example 5.4 The belief functions are strongly c o n f l i c t i n g Assume i n t h i s example tha t the p r i o r Bayesian b e l i e f func t ion can be expressed as m , ( 0 1 ) = O . O 1 m , ( 0 2 ) = O . 9 9 and the consonant b e l i e f func t ion based on the i n f e r e n t i a l evidence can be expressed as m 2 ( 6 1 ) = 0 . 9 5 m 2 (6» 1 , 0 2 ) = O . O 5 The degree of c o n f l i c t between m,(A) and m 2 (A) i s 0 . 9 4 0 5 . Thi s s i t u a t i o n can be cons ide red to be s t r o n g l y c o n f l i c t i n g . Three cases w i l l again be cons ide red as i n example 5 . 3 and s ince the procedure i s unchanged, only the r e s u l t a n t b e l i e f func t ion a f t e r d i s c o u n t i n g and combining i s l i s t e d for each case Case 1. m ^ A ) i s d i scoun ted by d iscount ra te a wh i l e m 2 (A) remains unchanged 74 The combined b e l i e f func t ion a f t e r d i s c o u n t i n g m,(A) i s given by 0.01+0.94a m(0, ) = 1-0.9405(1-a) 0.0495(1-a) m(0 2)= (5.15) 1-0.9405(1-a) 0.05a m(6,,02) = 1-0.9405(1-a) Case 2. m,(A) remains unchanged wh i l e m 2 (A) i s d i scoun ted by d iscount ra te a The combinat ion of d i scounted m 2 (A) wi th m,(A) w i l l y i e l d 0.01 m(0,) = 0.0595+0.9405a (5.16) 0.0495+0.9405a m(0 2)= 0.0595+0.9405a Case 3. Both m,(A) and m 2 (A) are d i scoun ted s imul taneous ly Again i t w i l l be assumed they are d i scoun ted by the same d i scount ra te a . A f t e r combining the d i scounted b e l i e f func t ions by Dempster 's r u l e of combina t ion , the new b e l i e f func t ion i s 0.01(1-a)+0.95a(1-a) m(0,)= : 1-0 .9405(1-a) 2 0.0495(1-a) 2 +0.99a(1-a) m(0 2)= (5.17) 1-0 .9405(1-a) 2 75 0.05a(1-a)+a2) m(0,,02.) = 1-0.9405(1-a)2 The re la t i onsh ip between m(A) and a, expressed by Eq. 5.15 through Eq . 5.17, are drawn in Figure 5.2(a) through Figure 5.2(c) respect ive ly . It can be seen from Figure 5.2(a) to Figure 5.2(c) that , in the s i tua t ion in which the two be l i e f functions are highly c o n f l i c t i n g , the divergence of Dempster-Shafer approach from Bayesian approach w i l l be very sens i t ive to the change of a when a i s very smal l . In other words, the Dempster-Shafer resu l t s vary with a subs tant ia l ly non- l inear ly when a i s smal l . It i s also seen that in the early stage when a i s smal l , the Dempster-Shafer resul t s converge more rap id ly than the l inear convergence, as was discussed in Example 5.3, on the non discounted be l i e f function (or vacuous b e l i e f function in case 3). As a becomes l a r g e r , the divergence increases more gradually or l i n e a r l y . Note that th i s resu l t i s true for a l l three cases. The two e n t i r e l y d i f f erent s i tuat ions described in Examples 5.3 and 5.4 a lso reveal the fact that when the two b e l i e f functions- are discounted at the same time, the Dempster-Shafer resul t s w i l l converge on the complete ignorance ( i . e . vacuous b e l i e f function) as a progress ively increases . The two examples a lso indicate that the divergence curve is more complex in the case where the two b e l i e f functions are discounted simultaneously. This i s e spec ia l ly true when the two b e l i e f functions are strongly c o n f l i c t i n g . While i t i s d i f f i c u l t to assess a value for a, 76 the o b s e r v a t i o n s from above two examples may be u s e f u l in judg ing the a p p r o p r i a t e va lue for a i n p r a c t i c e . 5.5 SUMMARY When an i n f e r e n t i a l problem can be f i t t e d i n t o the s t a t i s t i c a l s p e c i f i c a t i o n model , and the p r i o r ev idence can be expres sed as a Bayes ian b e l i e f f u n c t i o n , both Dempster -Shafer theory and Bayes ian theory are a p p r o p r i a t e . The two t h e o r i e s w i l l y i e l d the same r e s u l t s . In the s i t u a t i o n where one has some doubt c o n c e r n i n g e i t h e r the Bayes ian b e l i e f f u n c t i o n or the consonant b e l i e f f u n c t i o n , the b e l i e f f u n c t i o n may be d i s c o u n t e d by a d i s c o u n t r a t e a . The r e s u l t s of a Dempster -Shafer approach based on the d i s c o u n t e d b e l i e f f u n c t i o n s w i l l d i v e r g e from those o b t a i n e d u s i n g a Bayes ian a p p r o a c h . The degree of the d i v e r g e n c e w i l l depend on the va lue of the d i s c o u n t r a t e a and the degree of c o n f l i c t between the p r i o r and e v i d e n t i a l b e l i e f f u n c t i o n s . When these two b e l i e f f u n c t i o n s are h i g h l y c o n f l i c t i n g and a i s c l o s e to z e r o , the d i v e r g e n c e of the r e s u l t s from the two methods i s very s e n s i t i v e to the a v a l u e . T h i s a l s o i n d i c a t e s t h a t the c o m b i n a t i o n of two h i g h l y c o n f l i c t i n g b e l i e f f u n c t i o n s w i l l be very s e n s i t i v e to a s l i g h t e r r o r in e i t h e r b e l i e f f u n c t i o n . Whi le Bayes ian theory f a i l s to d e a l wi th t h i s problem at a l l , S h a f e r ' s d i s c o u n t i n g method toge ther w i t h Dempster ' s r u l e of combinat ion i s at l e a s t ab le to i n c o r p o r a t e the i n f l u e n c e of doubt about the e v i d e n c e . As mentioned e a r l i e r , however, the c o r r e c t v a l u e 77 of a may not be easy to e s t a b l i s h i n p r a c t i c e . I t should be noted that the d i s c o u n t i n g of a Bayesian b e l i e f func t ion w i l l become a non-Bayesian b e l i e f func t ion wh i l e the d i s c o u n t i n g of a consonant b e l i e f func t ion s t i l l remains a consonant b e l i e f f u n c t i o n . Though the d i s c u s s i o n s i n t h i s chapter are based on the s imple two element frame of discernment , i t seems to be i n t u i t i v e l y reasonable to extend the c o n c l u s i o n s obta ined from these d i s c u s s i o n s to more complex frames. 1 - T „ 0.9 -a V S 0.8 -0 0.2 0.4 0.6 0.8 1 Discount rate Q. Figure 5.1(a) m(A) vs. a when the two belief functions are mildly conflicting and only prior belief function is discounted a £> o ca 0.2 0.4 0.6 0.8 Discount rate Q, Figure 5.1(b) The same as in Fig. 5.1(a) but only the consonant belief function is discounted 0 0.2 0.4 0.6 0.8 1 Discount rate CL Figure 5.1(c) The same as in Fig. 5.1(a) but the two belief functions are discounted simultaneously a a! a) O 03 0.6 0.8 Discount rate a. Figure 5.2(b) The same as in Fig. 5.2(a) but only the consonant belief function is discounted Discount rate Q, Figure 5.2(c) The same as in Fig. 5.2(a) but the two belief functions are discounted simultaneously 6. CONCEPTUAL DIFFERENCE BETWEEN THE TWO THEORIES AND REPRESENTATION OF EVIDENCE 6.1 INTRODUCTION So f a r , the d i s c u s s i o n s about in fe rence based on Bayesian and Dempster-Shafer theory have been under the assumption that the b e l i e f func t ion i s obta ined p e r f e c t l y once a p iece of evidence i s o b t a i n e d . L i t t l e has been s a i d about the evidence i t s e l f and the r e l a t i o n s h i p between the types of evidence and the forms of b e l i e f f u n c t i o n s . The ques t ion does not a r i s e i n a pu re ly t h e o r e t i c a l comparis ior i of the two t h e o r i e s . However, i n eng ineer ing p r a c t i c e , c o l l e c t i n g , rea r ranging and c o r r e c t l y express ing the evidence through a mathematical b e l i e f func t ion are very important steps in the i n f e r e n t i a l p rocess . In p r a c t i c e , c o n s i d e r a t i o n must be given to the sources of evidence , to the evidence i t s e l f and to i t s express ion through a b e l i e f f u n c t i o n . Before d i s c u s s i n g evidence and i t s r e p r e s e n t a t i o n , a d e t a i l e d d i s c u s s i o n about the conceptual d i f f e r e n c e between Bayesian theory and Dempster-Shafer theory i s presented i n Sec. 6 .2 . The advantages of Dempster-Shafer theory over Bayesian theory are assessed from t h i s v i e w p o i n t . In Sec. 6 .3 , genera l d i s c u s s i o n s about the c h a r a c t e r i s t i c of evidence are presented . In Sec. 6 .4 , the i ssue of r ep resen t ing evidence by b e l i e f func t ions w i l l be d i s c u s s e d . F i n a l l y , i n l a s t s e c t i o n of t h i s chapter , some genera l 81 82 c o n c l u s i o n s are presented . 6.2 CONCEPTUAL DIFFERENCE BETWEEN THE TWO THEORIES 6.2.1 TWO TYPES OF UNCERTAINTIES There are va r ious k inds of u n c e r t a i n t i e s a s s o c i a t e d w i t h a g iven q u e s t i o n . Some examples may be a) What w i l l be the weather tomorrow: r a i n y , sunny or c loudy? b) What w i l l be the r e s u l t of t o s s i n g a f a i r co in? c) What i s the depth of the c l a y l a y e r on the s i t e of an ear th dam? d) What i s going to be the streamflow on a r i v e r for the next month? e) What are the c o r r e c t va lues for some parameters i n a h y d r o l o g i c a l model? As was mentioned e a r l i e r i n Chapter 2, the u n c e r t a i n t i e s a s s o c i a t e d wi th the above ques t ions can be c l a s s i f i e d i n t o two types , one i s a natural (or inherent) type of uncertainty {NU) and the other i s informational (or statistical) type of uncertainty (JU)[22] . NU i s always encountered when one needs to p r e d i c t randomly o c c u r r i n g , random magnitude events . I U , on the other hand, i s e n t i r e l y due to inadequancy of the sampling in format ion concern ing 83 h i s t o r i c even ts . In the above examples, u n c e r t a i n t i e s about a ) , b) and d) belong to the NU catagory wh i l e u n c e r t a i n t i e s about c) and e) belong to the IU ca tagory . The i n f e r e n t i a l o b j e c t i v e s a r i s i n g for the two types of u n c e r t a i n t i e s are s l i g h t l y d i f f e r e n t . For NU, one 's best o b j e c t i v e can only be to f i n d the c o r r e c t chance d e n s i t y func t ion which governs the outcome of a future event . The p o s s i b l e outcomes w i l l form a mutua l ly e x c l u s i v e and exhaust ive set which may be c a l l e d the set for NU and denoted as X={x^, i=1, 2 , . . . m } . I f enough p ieces of evidence are ob ta ined , the f i n a l i n f e r e n t i a l r e s u l t w i l l be a c o r r e c t chance d e n s i t y func t ion (or p r o b a b i l i t y mass func t ion) on X . For the case of IU, the o b j e c t i v e w i l l be to t r y to f i n d the t r u t h from seve ra l p o s s i b i l i t i e s , such as the s p e c i f i c depth of a sub- laye r somewhere underground. These p o s s i b i l i t i e s w i l l a l s o form a mutua l ly e x c l u s i v e and exhaus t ive s e t . Th i s set may be c a l l e d the set for IU and i s denoted by 0={0j, j = 1 , 2 , . . . n } . For set 0, i f a reasonable amount of evidence i s a v a i l a b l e , the i n f e r e n t i a l r e s u l t w i l l tend to focus on a s i n g l e element wi th h i g h b e l i e f v a l u e . F o r t u n a t e l y , the two types of u n c e r t a i n t i e s can both be expressed w i t h i n the same framework of u n c e r t a i n t y . For example, i n order to f i n d the chance d e n s i t y func t ion which governs the outcome x of a c o i n - t o s s , one may assume a set of p o s s i b i l i t i e s 0={0j, j=1,2 , . .m} wi th each element 6^ represents a p o s s i b l e chance d e n s i t y func t ion q ^ j ( x ) . Because there i s only one c o r r e c t chance d e n s i t y f u n c t i o n , 84 the set 0 w i l l be c o n s i s t e n t w i t h the I U . Thus, f i n d i n g the c o r r e c t chance dens i t y func t ion q^(x) which governs the exper imenta l outcomes X i s equ iva l en t to f i n d i n g the c o r r e c t parameter 6 i n set 0 . Because of t h i s r e l a t i o n s h i p , the i n f e r e n t i a l problem about NU may be conver ted to the i n f e r e n t i a l problem about I U . The d i s t i n c t i o n between the two types of u n c e r t a i n t i e s i s not unimportant . In f a c t , as w i l l be seen i n Sec. 6 . 2 . 2 , the d i s t i n c t i o n may he lp one to be t te r understand the conceptua l d i f f e r e n c e between the Bayesian theory and Dempster-Shafer theory . 6 .2 .2 CONCEPTUAL DIFFERENCE BETWEEN THE TWO THEORIES R e c a l l that the o r i g i n a l Bayesian theory i s based on the s t a t i s t i c a l s p e c i f i c a t i o n model ( q^ (x )} , which i s designed to f i n d the c o r r e c t chance dens i t y func t ion for an experiment . In such a model, the set of exper imenta l r e s u l t s X={x^, i=1, 2 , . . .m} has NU a s s o c i a t e d wi th i t , wh i l e the set of p o s s i b l e chance dens i t y func t ions ©={0j , j = 1 , 2 , . . . n } has IU a s s o c i a t e d wi th i t . Because the p r o b a b i l i t y model p rov ides the a b i l i t y to incorpora te both types of u n c e r t a i n t i e s , i t i s the re fore p o s s i b l e to use Bayesian theory i n both kinds of i n f e r e n t i a l s i t u a t i o n s . In f a c t , i f the i n f e r e n t i a l problem i s to f i n d the t rue chance d e n s i t y func t ion over set X , the Bayesian theory needs one to imagine a set &={6^} w i t h each s i n g l e element 8^ r ep resen t ing a d i f f e r e n t chance dens i t y f u n c t i o n ; i f the 85 problem i s . to f i n d the t rue element out of a p o s s i b l e set 0 , the Bayesian theory needs one to th ink about a pretended experiment whose outcomes are the set X={x^}, and each s i n g l e element in 0 represents a chance d e n s i t y func t ion on set X={x^}. Once the two se ts 0 and X are determined, Bayesian theory r equ i r e s one to f i n d a p r i o r c o n v e n t i o n a l p r o b a b i l i t y d i s t r i b u t i o n on set 0 based on the p r i o r evidence and a set of chance dens i t y func t ions on set X (from which the sample l i k e l i h o o d i s d e r i v e d ) . The in fe rence based on Bayesian theory can then be undertaken a f t e r an exper imenta l obse rva t ion x has been ob ta ined , as was d i scussed i n Chapter 2 . While the o r i g i n a l Bayesian theory i s based on the s t a t i s t i c a l s p e c i f i c a t i o n model, the i n f e r e n t i a l problems may not always f i t i n t o t h i s p r o b a b i l i t y model p e r f e c t l y . In that case , i f one chooses Bayesian theory for the i n f e r ence , the i n f e r e n t i a l problem must be forced i n t o the context of a s t a t i s t i c a l s p e c i f i c a t i o n model. In other words, a d e c i s i o n has to be made as to which of the a v a i l a b l e p ieces of in fo rmat ion should be used as as " p r i o r " evidence and which as "new" ev idence . As w i l l be d i s cus sed i n d e t a i l i n Sec . 6.3, the c o l l e c t i v e evidence may inc lude va r ious forms, v a r y i n g not on ly i n t h e i r degree of accuracy but a l s o i n t h e i r re levance to the i n f e r e n t i a l problem of i n t e r e s t . Furthermore the cho ice of "new" evidence on which the sample l i k e l i h o o d func t ion i s to be based may of ten not be i n the form of exper imenta l outcomes as the o r i g i n a l Bayesian 86 p r o b a b i l i t y model r e q u i r e s . Once the p a r t i t i o n i n g to " o l d " evidence and "new" evidence i s determined, the " p r i o r " p r o b a b i l i t y d i s t r i b u t i o n must be obta ined from the " p r i o r " evidence and sample l i k e l i h o o d must be obta ined from the "new" ev idence . Note that i n both cases the evidence must be t r a n s l a t e d i n t o the form of a conven t iona l p r o b a b i l i t y d i s t r i b u t i o n . Since e i t h e r the " o l d " evidence or the "new" evidence may not be s u f f i c i e n t to e x a c t l y spec i fy these conven t iona l p r o b a b i l i t y d i s t r i b u t i o n s , the t r a n s l a t i o n may there fore a l s o i n v o l v e s u b j e c t i v e judgements. I t i s c l e a r that Bayesian theory presumes a pe r fec t t r a n s l a t i o n i s p o s s i b l e and thus ignores the q u a l i t y of the ev idence , i . e . i t does not care whether or not the evidence i s s u f f i c i e n t enough to p rov ide a s p e c i f i c conven t iona l p r o b a b i l i t y d i s t r i b u t i o n . Though the a p p l i c a t i o n of Bayesian theory i n t h i s case i s q u i t e d i f f e r e n t from i t s o r i g i n a l sense, such an ex tens ion of the a p p l i c a t i o n of Bayesian theory should be cons idered appropr ia te only i f a l l the p r o b a b i l i t y judgements can be made wi th reasonable conf idence . Since a great number of s u b j e c t i v e judgements may be i n v o l v e d e x p e c i a l l y when the i n f e r e n t i a l problem becomes b i g , i t may sometimes be very d i f f i c u l t to ob t a in the appropr ia te p r o b a b i l i t y judgements[20 ] [21 ]. Bayesian theory has long been the predominant method for inexact i n f e r ence . R e c a l l t h a t , in Bayesian theory , the p r o b a b i l i t y judgements on a set 0 and on a set X both take 87 the form of f r e q u e n c y - l i k e conven t iona l p r o b a b i l i t y d i s t r i b u t i o n s . A concern about the conven t iona l p r o b a b i l i t y d i s t r i b u t i o n i s that i t expresses the p r o b a b i l i t y judgements i n an i n t e r n a l l y c o n f l i c t i n g way, i . e . i t supports d i s j o i n t conc lu s ions at the same t ime . As was mentioned e a r l i e r , the set X has NU a s s o c i a t e d w i t h i t and the set 0 has IU a s s o c i a t e d w i t h i t . For the set X , s ince each element may be the t r u t h w i th some degree of p o s s i b i l i t y , i t seems to be reasonable to express the p r o b a b i l i t y judgements by us ing a f r e q u e n c y - l i k e conven t iona l p r o b a b i l i t y d i s t r i b u t i o n . However, for the set 0, s ince there i s on ly one element which i s the t r u t h and the r e s t must be wrong, i t seems to be i n t u i t i v e l y more appropr i a t e to express the p r o b a b i l i t y judgements on set 0 i n a concordant form ins tead of the i n h e r e n t l y i n t e r n a l l y c o n f l i c t i n g conven t iona l p r o b a b i l i t y d i s t r i b u t i o n . The p r o b a b i l i t y judgements for both types of u n c e r t a i n t i e s have- to be made i n s i m i l a r ways i n Bayesian theory s imply because there i s no other c h o i c e . Th i s may he lp to e x p l a i n the fact t h a t , i n most of the l i t e r a t u r e d e a l i n g w i t h u n c e r t a i n t y problems, the two types of u n c e r t a i n t i e s are mixed and t r ea t ed i d e n t i c a l l y (for example, see Ref . [ 25 ] ) . As a suggested g e n e r a l i z a t i o n of Bayesian t h e o r y [ l 8 ] , Dempster-Shafer theory does not need the i n f e r e n t i a l problem to be conf ined to the s t a t i s t i c a l s p e c i f i c a t i o n model. As a s p e c i a l case , when the i n f e r e n t i a l problem can be put i n t o t h i s p r o b a b i l i t y model, the Dempster-Shafer theory can s t i l l 88 be a p p r o p r i a t e l y used. In f a c t , i t has a l ready been shown i n Chapter 4 that i f the Dempster-Shafer theory i s used i n t h i s case , the p r i o r b e l i e f func t ion can be represented e i t h e r as a Bayesian b e l i e f func t ion or as a non-Bayesian b e l i e f f u n c t i o n . I f the former cho ice i s made then i t i s equ iva len t to Bayesian theo ry . In the more genera l case , the Dempster-Shafer theory r equ i r e s one to c a l c u l a t e the b e l i e f func t ions from the a v a i l a b l e p ieces of ev idence . (The i s sue of express ing evidence by b e l i e f func t ion w i l l be d i s cus sed i n Sec. 6 . 4 ) The inference i s then undertaken by combining a l l of the b e l i e f func t ions together through the Dempster 's r u l e of combina t ion . In Dempster-Shafer theory , the b e l i e f func t ions ob ta ined from the a v a i l a b l e p ieces of in fo rmat ion may take v a r i o u s forms i n c l u d i n g the consonant b e l i e f func t ions and the i n t e r n a l l y c o n f l i c t i n g conven t iona l p r o b a b i l i t y d i s t r i b u t i o n s ( i . e . Bayesian b e l i e f f u n c t i o n s ) . S ince Dempster -Shafer ' s b e l i e f func t ion i s much more f l e x i b l e than the conven t iona l p r o b a b i l i t y d i s t r i b u t i o n s i n express ing ev idence , i t i s therefore p o s s i b l e to represent v a r i o u s k inds of evidence i n b e l i e f f u n c t i o n s . In other words, the concept of a b e l i e f func t ion makes i t p o s s i b l e for one to represent the evidence more f a i t h f u l l y accord ing to the q u a l i t y of the evidence and i t s o r i g i n . R e c a l l i n g that i n Bayesian theory , any k ind of ev idence , r ega rd le s s of i t s q u a l i t y , must be represented i n the form of a conven t iona l 8 9 p r o b a b i l i t y d i s t r i b u t i o n , i t should then be concluded that the Dempster-Shafer theory i s . a much more f l e x i b l e and p o t e n t i a l l y c o n v i n c i n g method. The inference based on Dempster-Shafer theory i s undertaken on the frame of discernment 0 . Since there i s on ly one element which i s the t r u t h and the r es t must be wrong, the u n c e r t a i n t i e s a s s o c i a t e d wi th the frame 0 should belong to the IU category accord ing to the d e f i n i t i o n s given i n Sec. 6 . 2 . 1 . As was d i scus sed e a r l i e r i n t h i s s e c t i o n , the p r o b a b i l i t y judgements on the frame 0 based on a p iece of evidence should be made i n a more consonant way. Therefore , i t may be concluded that w i th Dempster-Shafer theory , the consonant b e l i e f func t ion would be g e n e r a l l y the more appropr i a t e way to represent the in format ion provided by evidence a s s o c i a t e d wi th 0 . Even though i t seems a t t r a c t i v e to use a consonant b e l i e f f unc t i on to represent the evidence i n Dempster-Shafer theory , some people may s t i l l p re fe r c o n v e n t i o n a l , f r e q u e n c y - l i k e p r o b a b i l i t y judgements. As was mentioned e a r l i e r , the main concern about t h i s k ind of p r o b a b i l i t y judgement i s that i t a l l o w s b e l i e f va lues de r ived from a s i n g l e p iece of evidence to be ass igned to d i s j o i n t p o s s i b i l i t i e s even though the s i n g l e p iece of evidence should i t s e l f be consonant. Th i s enigma may be so lved by i n t e r p r e t i n g such p r o b a b i l i t y judgements i n a d i f f e r e n t way: to i n t e r p r e t i t us ing the odds c o n c e p t [ l 6 ] . For example, the depth of a sub- laye r somewhere underground l i e s i n one of 9 0 the three p o s s i b i l i t i e s 0,=4O f t , 0 2 = 50 f t , 0 3 = 6O f t . Assume that the conven t iona l p r o b a b i l i t y judgements are p (0 1 )=O.3 , p (0 2 )=O.6 , and p (0 3 )=O.1 . The odds r a t i o can then be expressed as : p ( 0 , ) : p ( 0 2 ) : p ( 6 3 ) = 3 : 6 : 1 which can be i n t e r p r e t e d a s : i t i s two t imes as l i k e l y tha t w i l l be the r i g h t va lue as i t i s 0 ^ and s i x times as l i k e l y that 0 2 w i l l be the r i g h t va lue as i t i s 0^. Thought of t h i s way, the conven t iona l p r o b a b i l i t y d i s t r i b u t i o n can be cons idered as an e f f e c t i v e and reasonable way to express ev idence . Neve r the l e s s , t h i s i n t e r p r e t a t i o n would not s a t i s f y those who i n s i s t on the consonant way of r ep resen t ing ev idence . 6.3 GENERAL CONSIDERATION OF EVIDENCE The foregoing d i s c u s s i o n s have concent ra ted on the conceptual d i f f e r e n c e between Bayesian theory and Dempster-Shafer theory . L i t t l e has been s a i d about the evidence i t s e l f on which both the Dempster-Shafer ' s b e l i e f func t ion and the Bayesian conven t iona l p r o b a b i l i t y d i s t r i b u t i o n are based. In t h i s and the f o l l o w i n g s e c t i o n s , genera l c o n s i d e r a t i o n of evidence and i t s r ep re sen ta t ion through a b e l i e f func t ion w i l l be d i s c u s s e d . Any in fo rmat ion which i s r e l evan t to the ques t ion one i s i n t e r e s t e d i n r e s o l v i n g should be cons idered as evidence bear ing on that q u e s t i o n . For an i n f e r e n t i a l problem, the evidence comes from two d i f f e r e n t types of source . The f i r s t source i s from p r a c t i c a l obse rva t ions such as observed h i s t o r i c da t a , exper imenta l r e s u l t s e t c . . The evidence thus 91 obta ined may be re fered to as objective evidence. The second source i s from exper ts who p rov ide the p r o b a b i l i t y judgements, not only accord ing to t h e i r past exper ience and t h e i r pe r sona l knowledge, but a l s o somewhat accord ing to t h e i r d e s i r e to c o l l a b o r a t e and even to i n f l uence the f i n a l outcome. Th i s type of evidence may be c a l l e d subjeel i ve evidence. In p r a c t i c e , the evidence , e i t h e r from some observa t ions or from expert judgements, may not alone be s u f f i c i e n t to provide a b e l i e f f u n c t i o n . In f a c t , i t may be necessary that a s i n g l e b e l i e f func t ion be determined by an expert based j o i n t l y on observed data and h i s pe r sona l judgement. Furthermore, s ince the observed data w i l l vary both i n i t s re levance to the problem, and i n i t s a b i l i t y to p rov ide numer ica l b e l i e f va lue s , the de te rmina t ion of a b e l i e f func t ion may have to be based on m u l t i p l e obse rva t ions ins tead of on ly one i n d i v i d u a l o b s e r v a t i o n . In other words, i f a set of p ieces of evidence i s ob ta ined , i t i s not necessary that a s p e c i f i c b e l i e f func t ion be obta ined based on each of the obse rva t i ons . I t i s ra ther more l i k e l y tha t s e v e r a l p ieces of evidence together support one b e l i e f f u n c t i o n . As i t w i l l be seen l a t e r on i n t h i s s e c t i o n , the way of de termining a b e l i e f func t ion by pe rsona l judgements based on some p r a c t i c a l observa t ions p l ays an e s s e n t i a l par t i n r ep resen t ing evidence by numerica l b e l i e f f u n c t i o n s . Once a set of p ieces of evidence are ob ta ined , d i f f e r e n t t h e o r i e s may be chosen for the i n f e r e n c e . The way the set of p ieces of evidence w i l l be dea l t w i th v a r i e s 92 accord ing to the chosen theo ry . As was mentioned in S e c . 6 . 2 , i f one choose Bayesian theory , the evidence has to be grouped i n t o two p a r t s , one ac t s as "new" evidence and the other as " o l d " (or p r i o r ) ev idence . The Bayesian p r i o r p r o b a b i l i t y judgements based on the " o l d " evidence and the sample l i k e l i h o o d func t ion based on the "new" evidence are then made and the inference i s then completed us ing Bayesian theory . In doing t h i s , the " o l d " evidence should be independent of the "new" ev idence . There are no r e s t r i c t i v e r u l e s to govern the d i v i s i o n of the p ieces of ev idence , but the general somewhat s e l f - s e r v i n g p r i n c i p l e [ 2 0 ] i s that i t should . be easy and e f f e c t i v e to judge the p r i o r p r o b a b i l i t i e s from the " o l d " evidence and to judge the sample l i k e l i h o o d func t ion from the "new" ev idence . I t i s c l e a r that the d i v i s i o n i s ra ther s u b j e c t i v e and d i f f e r e n t d i v i s i o n s are p o s s i b l e . I f the more genera l Dempster-Shafer theory i s chosen, a set of b e l i e f func t ions should be obta ined which correspond to the set of p ieces of ev idence . As was mentioned e a r l i e r i n t h i s s e c t i o n , each p iece of evidence a lone w i l l not n e c e s s a r i l y spec i fy a b e l i e f f u n c t i o n . I t i s there fore necessary to group the p ieces of evidence so tha t each group i s s u f f i c i e n t l y s t rong and s p e c i f i c enough for one to b u i l d a b e l i e f func t ion wi th resonable conf idence . The way to group the p ieces of evidence i s more s u b j e c t i v e than t h e o r e t i c a l . However some genera l r u l e s should be f o l l o w e d . The p ieces of ev idence , which are dependent should be 93 grouped toge the r . The more vague and l e s s r e l evan t evidence should be grouped w i t h more s p e c i f i c and more r e l evan t p ieces of ev idence . The grouped items of evidence should be independent of each o the r . Once the b e l i e f func t ions are obta ined from the grouped items of ev idence , the inference should then be completed by combing the b e l i e f func t ions through Dempster 's r u l e of combina t ion . Choosing d i f f e r e n t approaches for the same i n f e r e n t i a l problem and d i f f e r e n t t reatments of the c o l l e c t i v e evidence w i l l l ead to d i f f e r e n t i n f e r e n t i a l r e s u l t s . Instead of adopt ing a s i n g l e approach, a comparative approach might be worth c o n s i d e r i n g i f i t i s not compu ta t iona l ly p r o h i b i t i v e . The genera l c o n s i d e r a t i o n s of the evidence have been d i scus sed i n t h i s s e c t i o n . In the next s e c t i o n , the s p e c i f i c problem of express ing evidence through a b e l i e f func t ion w i l l be d i s c u s s e d . 6 . 4 EXPRESSING EVIDENCE THROUGH A BELIEF FUNCTION Once an item of evidence i s ob ta ined , s u b j e c t i v e judgements w i l l be needed to e s t a b l i s h the b e l i e f func t ion which represents t h i s item of ev idence . Accord ing to the theory of c o n s t r u c t i v e p r o b a b i l i t y proposed by Shafer and o the r s t14] [19] [20] the exper t , who makes the s u b j e c t i v e p r o b a b i l i t y judgements, needs to compare the evidence wi th a scale of canonical examples and p i ck the c a n o n i c a l example which matches the evidence bes t . Because the form of the b e l i e f func t ion for the choice of the c a n o n i c a l example has 94 a l ready been determined, the b e l i e f func t ion which represents the evidence i s thus s p e c i f i e d . I t i s seen that comparing an item of evidence wi th a c a n o n i c a l example i s not only q u a l i t i v e but a l s o q u a n t i t a t i v e . I t should be noted tha t the c a n o n i c a l examples are only dev ices by which one can e f f e c t i v e l y ob ta in the b e l i e f func t ion to represent some ev idence . I t i s not necessary tha t the evidence be s i m i l a r to the c a n o n i c a l example i n a l l r e spec t s . For a g iven item of evidence, a d i f f e r e n t expert may choose d i f f e r e n t types of c a n o n i c a l examples, or make d i f f e r e n t numer ica l judgements even w i t h i n the same choice of c a n o n i c a l example. Th i s w i l l obv ious ly l ead to d i f f e r e n t i n f e r e n t i a l r e s u l t s , fu r ther r e f l e c t i n g the i n e v i t a b l e s u b j e c t i v i t i e s i n express ing evidence by b e l i e f f u n c t i o n . There are many p o s s i b l e forms of b e l i e f func t ions for a g iven frame of discernment , t h i s makes i t very d i f f i c u l t to choose the r i g h t b e l i e f func t ion to express the ev idence . Th i s d i f f i c u l t y , however, can be minimized by r e c o g n i z i n g t h a t , i n eng ineer ing p r a c t i c e , the engineers or other exper ts would p re fe r the evidence to be expressed i n the more meaningful and s impler forms of b e l i e f f u n c t i o n s . Indeed, i t seems to be d i f f i c u l t to i n t e r p r e t the p r a c t i c a l i m p l i c a t i o n s of a genera l b e l i e f func t ion where b e l i e f va lues are ass igned to ( p o t e n t i a l l y at l e a s t ) a l l j o i n t p r o p o s i t i o n s and s i n g l e t o n s . I t would be very hard to r e l a t e such a genera l b e l i e f func t ion to an item of ev idence . In f a c t , i f there i s an i tem of evidence which i s so complex as 95 to r equ i r e a genera l b e l i e f f u n c t i o n , i t might be p re fe rab le to decompose i t i n t o s e v e r a l s impler and e a s i l y understandable p ieces of ev idence , each of them being represented by a s impler form of b e l i e f f u n c t i o n . A f i n a l complex genera l b e l i e f f unc t i on can be obta ined only a f t e r combining these s impler b e l i e f f u n c t i o n s . I t appears to be h i g h l y d e s i r a b l e there fore to have c l a s s e s of b e l i e f func t ions which , on the one hand, have s imple forms of express ion and, on the other hand, are meaningful i n r ep re sen t ing types of evidence encountered f r equen t ly i n eng ineer ing p r a c t i c e . In Chapter 3 , s e v e r a l c l a s s e s of b e l i e f func t ions have a l r eady been mentioned. Fur ther c h a r a c t e r i s t i c s of the evidence which may be expressed by these c l a s s e s of b e l i e f func t ions w i l l be d i scussed i n the f o l l o w i n g . 1. The frequency form of evidence and conven t iona l p r o b a b i l i t y judgement R e c a l l that i n Bayesian theory , both the p r i o r p r o b a b i l i t i e s and the sample l i k e l i h o o d func t ion have the form of conven t iona l p r o b a b i l i t y d i s t r i b u t i o n s . S ince a c o n v e n t i o n a l p r o b a b i l i t y d i s t r i b u t i o n can be i n t e r p r e t e d by the frequency concept , the evidence which supports a c o n v e n t i o n a l p r o b a b i l i t y d i s t r i b u t i o n may be r e f e r r e d to as the frequency form of evidence. The genera l c h a r a c t e r i s t i c s of the frequency form of ev idence , i s that i t p rov ides odds for each element i n the 96 frame of discernment . A c a n o n i c a l example for t h i s type of evidence i s g iven as f o l l o w s Assume that the frame of discernment for the depth of a sub- laye r somewhere underground i s ©={0 1 =5Oft , 0 2 =6Oft, 0 3 =7Oft}. A p iece of evidence may t e l l you that there i s o v t imes more l i k e l y that 62 i s the t r u t h than 0 , , and there i s o 2 t imes more l i k e l y that d3 i s the t r u t h than 0 , , then, the Bayesian p r o b a b i l i t y assignments might be 1 O T o 2 P(0,)= P(0 2 )= P(0 3 )= 1+0!+02 1+0!+02 1+0!+02 Another c a n o n i c a l example a p p l i e s where an event i s d i r e c t l y governed by some chance. By comparing the evidence w i t h t h i s c a n o n i c a l example, the p r o b a b i l i t y judgement P(0^)=p^ may be i n t e r p r e t e d as evidence which supports 6^ by p r o b a b i l i t y p^ . Th i s i s equ iva l en t to some knowledge which supports the t r u t h i n the c a n o n i c a l example e x a c t l y p^ of the t i m e [ 2 0 ] . Any i tem of evidence which i s compat ible w i t h the above c a n o n i c a l examples should be represented by a conven t iona l p r o b a b i l i t y d i s t r i b u t i o n . The p r o b a b i l i t y va lues should be determined on ly a f t e r choosing the c a n o n i c a l example which matches the evidence bes t . 2. Complete ignorance and the vacuous b e l i e f func t ion 97 The evidence which corresponds to the vacuous b e l i e f func t ion i s complete ignorance. The c h a r a c t e r i s t i c of such evidence i s that i t conf i rms that the t r u t h i s i n the frame of discernment but i t does not say any more. While t h i s appears to be a more s a t i s f a c t o r y r ep re sen t a t i on of ignorance than i s p rov ided by the uniform Bayesian p r i o r , care in d e f i n i n g the frame of discernment i s s t i l l e s s e n t i a l . For the sub- laye r example presented above, i f the evidence can only t e l l that the t rue depth l i e s i n the frame 0, but no th ing more, then i t can be represented by the vacuous b e l i e f f u n c t i o n . 3. C lea r but doubt fu l evidence and the s imple b e l i e f func t ion The s imple b e l i e f func t ion i s featured by a s s i g n i n g par t of one ' s b e l i e f to a s i n g l e p r o p o s i t i o n Ae0 and the r e s t to the whole frame 0. A c a n o n i c a l example for the evidence which corresponds to the s imple b e l i e f func t ion i s as f o l l o w s Using the same sub - l aye r example as before , suppose a s i t e i n v e s t i g a t i o n i s conducted and i t t e l l s you that the t rue depth of the sub- l aye r i s w i t h i n the range 50ft to 60ft 1. e. l i e s i n a subset A of 0 . Suppose a l s o that t h i s i n v e s t i g a t i o n i s only r e l i a b l e a p ropo r t i on p of the t ime . Then the b e l i e f func t ion corresponding to an obse rva t ion of the experiment r e s u l t i s g iven by a simple b e l i e f func t ion and can be expressed as 98 m(A)=p m(G)=1-p Any evidence which i s compatable wi th the above c a n o n i c a l example can be represented by a s imple b e l i e f f u n c t i o n . In p r a c t i c e , the feature of the evidence which may be represented by a s imple b e l i e f func t ion i s that the measurement and i t s i m p l i c a t i o n are c l e a r , but the r e l i a b i l i t y of the measurement i s i n ques t i on t20]. 4. Concordant evidence and the consonant b e l i e f func t ion The most a t t r a c t i v e feature of a consonant b e l i e f func t ion i s that i t a l l o w s one to a s s ign p r o b a b i l i t i e s without any i n t e r n a l disagreement or c o n f l i c t . The ev idence , which can be expressed by a consonant b e l i e f func t ion therefore may be re fe red to as concordant evidence. As was d i scussed i n Chapter 3 ( a l so see Ref . [5]) one de te rmina t ion of a consonant b e l i e f func t ion r equ i re s only the s p e c i f i c a t i o n of a b e l i e f value on one s i n g l e element and the p l a u s i b i l i t y va lues for a l l e lements . Therefore , a c a n o n i c a l example for the concordant evidence may take the f o l l o w i n g form Assume the same sub- l aye r example as above, i . e . the depth of a sub- laye r somewhere underground i s w i t h i n the frame of discernment 0 = { 8 , , 6 2 , 6 3 } . Suppose one can t e l l , based on a source of ev idence , that 0, i s the most l i k e l y t rue element, 92 i s l e s s l i k e l y and 83 i s the l e a s t l i k e l y t rue element. Suppose one can a l s o t e l l , based on the same 99 source of ev idence , that the maximum p o s s i b i l i t y . ( i . e . the p l a u s i b i l i t y ) of each element being the t r u t h i s P l ( 0 ^ ) for each 0^ r e s p e c t i v e l y . S ince 0, i s the most l i k e l y candidate for the t r u t h , the maximum p o s s i b i l i t y P1(0 , ) (but not the p r o b a b i l i t y ) should be 1.0. O b v i o u s l y , P1(0^) should s a t i s f y P I ( 9 , ) > P 1 ( 0 2 ) ^ P 1 ( 9 2 ) . Th i s source of evidence can then be represented through a consonant b e l i e f func t ion as f o l l o w s B e K f i , )=1-P1(0 2 ) B e l(0, , 0 2 ) = 1 - P 1 ( 0 3 ) B e l(0, , 0 2 , 0 3 ) = 1 - 0 Any ev idence , which i s compat ible to the above example can be expressed through a consonant b e l i e f f u n c t i o n . 6.5 SUMMARY The conceptual d i f f e r e n c e between Bayesian theory and Dempster-Shafer theory i s that Bayesian theory , which r equ i r e s the i n f e r e n t i a l problem to be f i t t e d i n t o the s t a t i s t i c a l s p e c i f i c a t i o n model, s t r e s se s more the s a t i s f a c t i o n of the p r o b a b i l i t y model and pays l e s s a t t e n t i o n to the q u a l i t y of the evidence wh i l e Dempster-Shafer theory emphasizes more the q u a l i t y of evidence i t s e l f . That i s , Bayesian theory r equ i r e s the c o l l e c t i v e p ieces of evidence be expressed e i t h e r as the p r i o r p r o b a b i l i t i e s or the sample l i k e l i h o o d s i n the form of conven t iona l p r o b a b i l i t y d i s t r i b u t i o n s whi le Dempster-Shafer 1 00 b e l i e f func t ion concept makes i t p o s s i b l e to represent va r ious forms of evidence accord ing to the q u a l i t y of the ev idence . S ince the p ieces of evidence i n p r a c t i c e do not n e c e s s a r i l l y e x p l i c i t l y s p e c i f y conven t iona l p r o b a b i l i t y d i s t r i b u t i o n s , Dempster-Shafer theory appears to be a more r e a l i s t i c and more c o n v i n c i n g method for eng ineer ing p r a c t i c e than does the Bayesian method. The d i s t i n c t i o n between the IU and NU types of u n c e r t a i n t i e s were d i s c u s s e d . Th i s d i s c u s s i o n leads to the choice of choosing the consonant b e l i e f f unc t i on as a b e l i e f s t r u c t u r e i n the Dempster-Shafer framework. While there i s a s t rong case for adopt ing the consonant b e l i e f f u n c t i o n , i t s t i l l does not preclude the adopt ion of a Bayesian b e l i e f func t ion ( i . e . a conven t iona l p r o b a b i l i t y d i s t r i b u t i o n ) to represent evidence i f warranted. The evidence i n p r a c t i c e v a r i e s both i n i t s q u a l i t y and i n i t s re levance to the i n f e r e n t i a l problem of i n t e r e s t . While a genera l b e l i e f func t ion i s of ten far too complex to be used to represent ev idence , the cons t r a ined and s i m p l i f i e d c l a s s e s of b e l i e f func t ions d i s cus sed above seem to be p e r t i n e n t i n r ep resen t ing much of the evidence which occur r s i n p r a c t i c e . Of ten , a s i n g l e p iece of evidence may not be s u b s t a n t i a l or complete enough to e x p l i c i t l y spec i fy a b e l i e f f u n c t i o n , the ques t ion of grouping of evidence then a r i s e s . Vague and incomplete p ieces of evidence shou ld , wherever p o s s i b l e , be combined i n t o s i n g l e i tem of evidence and then be expressed by the corresponding c l a s s of b e l i e f 101 f u n c t i o n s . Converse ly , the more compl ica ted p iece of evidence should a l s o be decomposed i n t o s e v e r a l s impler items of evidence which can then be expressed by the cor responding c l a s s e s of b e l i e f f u n c t i o n s . 7 . DEMSTER-SHAFER DECISION MAKING IN WATER RESOURCES ENGINEERING 7 .1 INTRODUCTION A d e t a i l e d comparison between Dempster-Shafer theory and Bayesian theory has been the p r i n c i p a l t o p i c s i n Chapters 4,5, and 6. A genera l c o n c l u s i o n down from t h i s comparison i s that Bayesian theory , which conf ines i t s e l f to a s p e c i a l case of the s t a t i s t i c a l s p e c i f i c a t i o n model, i s a s p e c i a l case of the more genera l Dempster-Shafer theory . The g e n e r a l i z i n g aspect of Dempster-Shafer theory r e so lves va r ious aspects of Bayesian theory which are o v e r l y r e s t r i c t i v e in many s i t u a t i o n s commonly a r i s i n g i n p r a c t i c e . Desp i te i t s l i m i t a t i o n s , Bayesian theory has been s u c c e s s f u l l y used i n d e a l i n g w i t h u n c e r t a i n t i e s i n v a r i o u s p r a c t i c a l s i t u a t i o n s i n c i v i l eng ineer ing i n c l u d i n g water resources eng inee r ing . The use of Bayesian theory i n water resources eng inee r ing , as was d i scus sed i n Chapter 2 , has l a r g e l y been to reduce the i n f o r m a t i o n a l u n c e r t a i n t i e s a s s o c i a t e d w i t h the e v a l u a t i o n of some parameters of a model and the s e l e c t i o n of the model, and making eng ineer ing des ign ( i . e . d e c i s i o n making under u n c e r t a i n t y ) based on the i n f e r e n t i a l r e s u l t s . As a more genera l theory of i n f e r ence , the Dempster-Shafer method appears to be a s e r ious candidate as an a l t e r n a t i v e to Bayesian method i n d e a l i n g w i t h u n c e r t a i n t i e s i n p r a c t i c e . In f a c t , renewed a t t e n t i o n has been g iven i n recent years by researchers i n a t tempt ing to 102 1 03 apply Dempster-Shafer theory i n t h e i r own f i e l d s of i n t e r e s t , most of the work being s t i m u l a t e d by the a p p l i c a t i o n of expert systems. A few of the a p p l i c a t i o n s of t h i s new theory are g iven i n references [2 ] [5 ] [10] [11] [12] [15 ] [25 ] . In s p i t e of these e f f o r t s , no major breakthrough has been repor ted so f a r , and most of the work i s s t i l l at the stage of s o l v i n g elementary problems. In c i v i l eng inee r ing , Case l ton et al. [5] f i r s t pa id a t t e n t i o n to Dempster-Shafer theory . They have t r i e d to u t i l i z e the new theory i n c o n s t r u c t i o n eng ineer ing w i t h i n an exper t system framework. But t h i s work s t i l l remains at the fundamental s tage . In t h i s Chapter , the use of Dempster-Shafer theory as an a l t e r n a t i v e to Bayesian theory i n d e c i s i o n making i n water resources eng ineer ing w i l l be d i s c u s s e d . The issue of d e c i s i o n making based on Dempster-Shafer scheme was f i r s t mentioned by Dempster[8] . Such d e c i s i o n making, s ince i t i s based on Dempster-Shafer theory , w i l l be r e fe red to here as Dempst er-Shafer decision theory. The bas ic idea of Dempster-Shafer d e c i s i o n theory i s presented i n Sec . 7 . 2 . In Sec 7 . 3 , a r e a l , i f e lementary, problem from water resources eng ineer ing p r a c t i c e i s presented to demonstrate an a p p l i c a t i o n of t h i s new theo ry . A summary of the d i s c u s s i o n s of Dempster-Shafer d e c i s i o n theory and i t s a p p l i c a t i o n i s presented i n Sec. 7 .4 . 1 04 7.2 DEMPSTER-SHAFER DECISION THEORY Suppose there i s a d e c i s i o n problem i n v o l v i n g a random outcome which has p o s s i b l e values which form a frame of discernment 0={0,, d2, . . . . 9 n ) . The d e c i s i o n set i s represented by D, D={d,, d 2 , . . . . d }. The u t i l i t y f u n c t i o n , which g ives the l o s s va lue for a s e l ec t ed d e c i s i o n dj and for a p o s s i b l e outcome 9^, i s def ined as u=U ( t9^,dj) . The problem i s to minimize the expected u t i l i t y . I f the p r o b a b i l i t y d i s t r i b u t i o n p(0) over 0 i s known, the expecta t ion can be c a l c u l a t e d by E(u/d.)=Z U ( e . f d . ) - p ( 0 . ) (7.1) J ^ J * Note that the p r o b a b i l i t y d i s t r i b u t i o n P(0^) i n E q . 7.1 i s assumed to be known. Bayesian d e c i s i o n theory i n v o l v e s * choosing the opt imal d e c i s i o n d so tha t : E(u/d*)=Min E ( u / d . ) a, J Assume now that there i s a b e l i e f func t ion B e l ( A ) ins tead of a conven t iona l p r o b a b i l i t y d i s t r i b u t i o n on 0. The t rue convent iona l p r o b a b i l i t i e s of s i n g l e t o n elements are not known i n t h i s s i t u a t i o n but the d e f i n i t i o n s of b e l i e f and p l a u s i b i l i t y con ta in the p r o b a b i l i t i e s as fo l l ows [28 ] (7 .2) 105 Where Pj^  i s the p r o b a b i l i t y of 6^ and ZP^ = 1.0. Note that the p r o b a b i l i t i e s {P^, i = 1 , 2 , . . .n} are a set of unknown v a r i a b l e s which s a t i s f y the c o n s t r a i n t s g iven in Eq . 7 . 2 . Assuming the same u t i l i t y funct ion U ( 0 ^ , d j ) , a p p l i e s the expected u t i l i t y for a d e c i s i o n dj can s t i l l be expressed along the same l i n e s as E q . 7 . 1 , i . e . E(u/d. )=L U ( 0 i , d - ) - P i (7.3) i S ince the P^, i = 1 , 2 , . . . n , i n Eq. 7.3 are subject to known c o n s t r a i n t s , but might otherwise be cons idered as random v a r i a b l e s , i t i s imposs ib le to ob t a in a determinate expected value for any given d e c i s i o n . But i t i s p o s s i b l e to obta in both maximum and minimum p o s s i b l e expected u t i l i t y values for d e c i s i o n d j . The maximum i s c a l l e d the upper it expected value and i s denoted as E ( u / d j ) ; the minimum i s c a l l e d the lower expected value and i s denoted as E * ( u / d j ) . The c a l c u l a t i o n s of upper and lower expected values i n v o l v e f i n d i n g the sets of P^ values which y i e l d these extremes whi le meeting the above s p e c i f i e d c o n s t r a i n t s . Therefore , t h i s c a l c u l a t i o n can be formulated as two conven t iona l l i n e a r programs which have the same form of o b j e c t i v e funct ion and c o n s t r a i n t s but one i s to maximize and another minimize the o b j e c t i v e f u n c t i o n . In t h i s f o rmu la t i on , the P^ are the unknown v a r i a b l e s of the l i n e a r program. The formulat ions of the two l i n e a r programs for c a l c u l a t i n g the upper and lower expected values are 106 Objec t ive Min/Max E(u/d. )=Z U ( $ i , d . ) - P I Subject to ( 7 . 4 ) 1 .0 and P ^ O i = 1,2 f • • • n Since the d e c i s i o n problem i s to minimize the expected l o s s , the appropr ia te d e c i s i o n c r i t e r i o n might be to choose the d e c i s i o n d^eD that y i e l d s a minimum upper expected u t i l i t y . Dempster c a l l e d t h i s the mini upper decision against Bel[8], The d e c i s i o n making dj based on such c r i t e r i o n i s not n e c e s s a r i l l y the g l o b a l l y optimum d e c i s i o n . In f a c t , the true expected value determined by the t rue p r o b a b i l i t i e s , which are not known, i s somewhere between the upper and lower bounds. I t i s p o s s i b l e , t he re fo re , to have another d e c i s i o n a c t i o n d^ which w i l l g ive a smal le r expected l o s s than that g iven by d j . Never the le s s , the d e c i s i o n choice dj based on the miniupper d e c i s i o n c r i t e r i o n can be cons idered as min imiz ing the maximum l i m i t cost which the system may face. In other words, a l l the p o s s i b l e expected cos t s w i l l f a l l below t h i s upper l i m i t . Such a d e c i s i o n i s the safes t or most conse rva t ive one, and might be s u i t a b l e where the expected cost must be minimized under a l l c i rcumstances . For the s i t u a t i o n where the d e c i s i o n problem i s to maximize the expected b e n e f i t s , a l o g i c a l d e c i s i o n c r i t e r i o n might be to 107 choose the a c t i o n dj which g ives the maximum lower expected v a l u e . Th i s expected value i s c a l l e d the maxi lower decision agai nst Bel . Dempster-Shafer d e c i s i o n theory i s the use of Dempster-Shafer theory w i t h i n the framework of conven t iona l d e c i s i o n theory . The advantages of Dempster-Shafer theory over Bayesian theory are e s s e n t i a l l y the advantages of Dempster-Shafer d e c i s i o n theory over Bayesian d e c i s i o n theory . Bayesian d e c i s i o n theory r equ i re s the i n f e r e n t i a l problem to be conf ined to a s p e c i a l case of the s t a t i s t i c a l s p e c i f i c a t i o n model i n which the p r i o r evidence has to be expressed by a conven t iona l p r o b a b i l i t y d i s t r i b u t i o n and the new evidence expressed by i t s sample l i k e l i h o o d v a l u e s . As d e c i s i o n s must be made i n c i v i l eng ineer ing p r a c t i c e under a wide range of c i rcumstances , the Bayesian s t a t e of a f f a i r s appears to be unduly r e s t r i c t i v e . On the other hand, Dempster-Shafer d e c i s i o n theory does not r equ i r e the i n f e r e n t i a l problem to be conf ined to the s t a t i s t i c a l s p e c i f i c a t i o n model. In f a c t , i t only r equ i r e s one to e s t a b l i s h a b e l i e f func t ion from each p iece of evidence and then combine them accord ing to Dempster 's r u l e of combinat ion to produce a r e s u l t a n t b e l i e f f u n c t i o n . The d e c i s i o n a n a l y s i s can then be undertaken by a p p l y i n g the Dempster-Shafer d e c i s i o n theory i n con junc t ion w i t h the r e s u l t a n t b e l i e f f u n c t i o n . Jus t as Bayesian theory i s a s p e c i a l case of Dempster-Shafer theory , i t can be demonstrated that Bayesian d e c i s i o n theory i s a s p e c i a l case 108 of the more genera l Dempster-Shafer d e c i s i o n theory . Under the same ci rcumstance of Bayesian b e l i e f , both w i l l y i e l d i d e n t i c a l r e s u l t s . That i s , the miniupper and the maxilower d e c i s i o n s are equal and c o i n c i d e w i th the op t ima l Bayesian d e c i s i o n . 7.3 THE APPLICATION OF DEMPSTER-SHAFER DECISION THEORY The f o l l o w i n g water resources example i s taken from R . J . McAni f f et al. [17 ] , The problem was o r i g i n a l l y designed to demonstrate the use of Bayesian d e c i s i o n theo ry . I t w i l l be used here to show how Dempster-Shafer d e c i s i o n theory can be used to improve the d e c i s i o n making i n a l e s s r e s t r i c t i v e s i tuat i o n . 7.3.1 DESCRIPTION OF THE ORIGINAL PROBLEM An a g r i c u l t u r a l producer i s concerned about the future cost of an i r r i g a t i o n system. A d e c i s i o n set D={dj , j=1 , . .5} represents the p o s s i b l e i r r i g a t i o n systems between which the producer can choose. The cost w i l l depend on the cho ice of i r r i g a t i o n system and a future energy p r i c e l e v e l . This energy p r i c e l e v e l i s a random v a r i a b l e and i t s p o s s i b l e va lues are denoted by ©={0^, i = 1 , 2 , . . . 1 0 } . The p r i o r p r o b a b i l i t i e s P(0) over 0 are known and are g iven i n Table 7 . 1 . New informat ion i s obta ined from e s t i m a t i n g the future energy p r i c e l e v e l s us ing a f o r e c a s t i n g model . The es t imated future energy l e v e l i s denoted as Z ^ . The sample l i k e l i h o o d of Z, g iven va r ious parameters 0- , i = 1 , 2 . . . l 0 , i s g iven in 109, Table 7 .2 . The u t i l i t y func t ion t H f l ^ d j ) s p e c i f i e s -the cost of choosing i r r i g a t i o n system d -Table 7.1 Prior p r o b a b i l i t i e s and u t i l i t i e s (after R.J. McAniff et al.) U t i l i t i e s (costs) i n dollars Range Possi- of Average Gated ble price price Prior Travel- pipe price in- in- proba- Centre ing with Open Dead level crease crease b i l i t i e s pivot t r i c k l e return ditch l e v e l • 6. l % % P ( * i ) d, d 2 d 3 d , d 5 <0 <0 0.11 268,190 269,750 273,780 287,430 299,780 e 2 0-3 1.5 0.07 280,020 280,930 284,440 306,540 313,300 e, 3-6 4.5 0.09 308,620 307,540 310,050 331,370 330,720 8, 6-9 7.5 0.11 344,760 341,510 342,810 371,150 358,800 9S 9-12 10.5 0.12 382,080 385,320 385,060 421,980 394,810 9s 12-15 13.5 0.12 452,790 441,870 439,530 488,280 441,610 9y 15-18 16.5 0.11 531,180 515,060 509,990 573,170 501,540 0S 18-21 19.5 0.09 646,880 609,830 601,250 683,540 579,410 99 21-24 22.5 0.07 764,400 726,960 719,420 826,540 680,290 9 1 0 >24 0.11 903,500 862,160 844,350 977,600 787,020 Table 7.2 Sample likel i h o o d of 2 k= 1 0.5% (after R.J. McAniff et al.) 9 i 92 9' 0, 8s 0.031 0.041 0.071 0.143 0.408 e. i 9s 97 eB 9s 9 i o i ( z k / * . > 0.143 0.071 0.041 0.031 0.020 and ob t a in ing energy p r i c e l e v e l 9^ i s a l s o given in Table 7 .1 . The problem i s to choose the i r r i g a t i o n system so that the expected cost w i l l be min imized . 1 10 Since the p r i o r p r o b a b i l i t y d i s t r i b u t i o n P(0^) and sample l i k e l i h o o d L{Z^/9^) are given e x p l i c i t l y , Bayesian p o s t e r i o r s can be obtained d i r e c t l y . The d e c i s i o n a n a l y s i s us ing Bayesian d e c i s i o n theory, based on these p o s t e r i o r s , can then be undertaken. The expected cos t s based on the p o s t e r i o r s for va r ious d e c i s i o n a l t e r n a t i v e s are given i n Table 7 .3 . Table 7.3 Expected costs based on Bayesian posteriors Expe^tedN^ costs (dollars) Decision Choice Center pivot d, Travel-l i n g t r i c k l e d 2 Gated pipe with return d 3 Open ditch d, Dead le v e l d 5 418,881 410,129 409,046* 451,278 416,102 I t i n d i c a t e s that the i r r i g a t i o n system choice "Gated Pipe wi th Return" should be chosen to minimize the c o s t s . Now, cons ider the s i t u a t i o n in which there i s some doubt about the p r i o r p r o b a b i l i t y d i s t r i b u t i o n . Th i s d i s t r i b u t i o n has to be d iscounted accord ing to S h a f e r ' s d i s coun t i ng method desc r ibed i n Chapter 5. Bayesian d e c i s i o n theory i s no longer app rop r i a t e , and Dempster-Shafer d e c i s i o n theory has to be used. Th i s i s demonstrated i n Sec. 7 .3 .2 111 7 .3 .2 DEMPSTER-SHAFER DECISION ANALYSIS L i k e the Bayesian d e c i s i o n theory , Dempster-Shafer d e c i s i o n theory needs one f i r s t to poo l a l l the a v a i l a b l e in format ion together and c a l c u l a t e a f i n a l b e l i e f f u n c t i o n . In our problem, the p r i o r b e l i e f func t ion i s obtained from p r i o r in format ion by d i s c o u n t i n g the p r i o r p r o b a b i l i t y d i s t r i b u t i o n P(d^) by a fac to r a. Therefore , the p r i o r b e l i e f func t ion can be expressed by i t s bas ic p r o b a b i l i t y assignment as m , ( 5 . ) = ( l - a ) P ( « i ) m,(0)=a i = 1 , 2 , . . . l 0 As was d i scussed in Chapter 4, the Dempster-Shafer approach to d e a l i n g wi th the i n f e r e n t i a l evidence i s to de r ive a consonant b e l i e f func t ion from such evidence. The bas ic p r o b a b i l i t y assignment m 2(A) of the consonant b e l i e f f u n c t i o n , based on the sample l i k e l i h o o d given in Table 2, i s g iven i n Table 7 .4 . Table 7.4 Basic probability assignment of a consonant b e l i e f function derived from i n f e r e n t i a l evidence A 05 8 4 . . 8 s 8 3 . • 9 7 8 2 ' • 9 a 9 , . . 9 9 9 i . . 9 , o m2 (A) 0. 650 0.175 0.075 0.025 0.025 0.050 Once the p r i o r b e l i e f func t ion expressed by m,(A) and the consonant b e l i e f func t ion expressed by m 2(A) are ob ta ined , they can then be combined to g ive a f i n a l b e l i e f 1 1 2 func t ion B e l ( A ) . The combination w i l l obv ious ly depend on the d i s c o u n t i n g fac tor a. In our example, a i s ranged over s eve ra l va lues , i . e . a=0.0, 0 .2 , 0 .8 , 1.0. The l a rge r the a i s , the greater the doubt being expressed about the p r i o r p r o b a b i l i t y d i s t r i b u t i o n . Thus a=0.0 i n d i c a t e s that the p r i o r p r o b a b i l i t y d i s t r i b u t i o n i s based on s o l i d i n fo rma t ion , and a=1.0 i n d i c a t e s that the p r i o r in format ion i s no be t te r than complete ignorance . In t h i s l a t t e r case , Table 7.5 Combined basic p r o b a b i l i t y ass ignments i i ^  's 1 s »7 'a ' 9 ' io ' a -' 3 . » 2 - -' l • >1 • Discounting factor a 0 0 0. 2 0.8 1. 0 0 031 0. 016 0. 002 0 0 026 0 013 0 002 0 0 058 0 030 0 004 0 0 142 0 074 0 009 0 0 444 0 543 0 637 0 650 0 155 0 081 0 010 0 0 071 0 037 0 005 0 0 033 0 017 0 002 0 0 019 0 010 0 001 0 0 .020 0 011 0 001 0 0 0 084 0 164 0 175 0 0 036 0 070 0 075 8 0 0 .012 0 023 0 .025 9 0 0 .012 0 023 0 .025 10 0 0 .024 0 047 0 .050 the combined b e l i e f func t ion i s determined s o l e l y by the consonant b e l i e f f unc t i on , i . e . the i n f e r e n t i a l evidence i s the only e f f e c t i v e informat ion source . The r e s u l t i n g bas ic p r o b a b i l i t y assignments m(A) for d i f f e r e n t a values are 1 1 3 given in Table 7 . 5 . The b e l i e f and p l a u s i b i l i t y values for s ing le tons de r ived from these r e s u l t s are given in Table 7 .6 . Table 7.6 B e l i e f and p l a u s i b i l i t y values of singletons a= 0. 0 = 0. 2 a= 0. B a= 1.0 Bel(A) pl(A) Bel(A) pl(A) Bel(A) pl(A) Bel(A) pl(A) 0 031 0 .031 0 016 0 052 0 002 0 072 0 0.075 0 026 0 .026 0 013 0 062 0 002 0 09 5 0 0.100 0 058 0 .058 0 030 0 114 0 004 ' 0 168 0 0. 175 fl, 0 142 0 . 142 0 074 0 242 0 009 0 337 0 0. 350 05 0 444 0 .444 0 543 0 .711 0 637 0 965 0.650 1.000 0s 0 155 0 .155 0 081 0 249 0 010 0 338 0 0. 350 07 0 071 0 .071 0 037 0 121 0 005 0 168 0 0.175 08 0 033 0 .033 0 017 0 .065 0 002 0 096 0 0.100 0 9 0 019 0 .019 0 010 0 .046 0 001 0 071 0 0.075 0 1 0 0 020 0 .020 0 011 0 .035 0 001 0 048 0 0.050 Once the b e l i e f s and p l a u s i b i l i t i e s are obtained for the s i n g l e elements, the upper and lower expected u t i l i t i e s can be computed for each d e c i s i o n a c t i o n d j . This was repeated for each value of the d i s coun t ing factor a. As was d iscussed i n Sec. 7 . 1 , the computation invo lves fo rmula t ion and s o l u t i o n of a l i n e a r program. For example, consider the s i t u a t i o n i n which a=0.2 and the d e c i s i o n cho ice dj=d,="Centre P i v o t " . The l i n e a r program formulat ion for c a l c u l a t i n g the upper and lower expected losses are O b j e c t i v e : Min/Max Z = 268 1 90P , +280020P2+3 08620P3+3 44760P,+382 08 0P 5 1 1 4 +452790P 6+531180P 7+646880P 8+764400P 9+903500P, 0 Subject t o : (1) The upper and lower l i m i t s of the p r o b a b i l i t i e s of s i n g l e t o n s de f ined by the supports and p l a u s i b i l i t i e s are 0.016<P 1<0.052 0.013<P 2<0.062 0.030<P 3<0.114 0 .074<P„<0.242 0.543<P 5^0.711 0.081<P 6<0.249 0.037<P 7<0. 121. 0.017<PB<0.065 0.010<P 9<0.046 0.011<P, O<0.035 (2) The c o n s t r a i n t on the summation of the p r o b a b i l i t i e s Z P ^ l .0 (3) The nonnega t i v i t y c o n s t r a i n t s P ^ O i = 1 , 2 , . . . 1 0 The c a l c u l a t e d r e s u l t s for the v a r i o u s combined a values and d i f f e r e n t d e c i s i o n cho ices dj are g iven i n Table 7 .7 . Both miniupper and maxilower expected cos t s are shown for a l l d e c i s i o n c h o i c e s . From Table 7 .7 , i t i s seen that the greater the d i scoun t f ac to r a, the l a r g e r the d i f f e r e n c e between the expected upper and lower v a l u e s . 1 1 5 Table 7.7 The upper and lower expected costs Decision set Travelling Gated pipe Discount Center pivot trickle with return rate d 3 a * * * E , E i * E 2 E 3 E 3 * 0.0 418881 418881 410129 410129 409046* 409046 0.2 448906 375115 442286 374340 440159* 374513 0.8 478430 353430 470135 355206 367121 356039 1.0 481905 350480 473419 352605 470303 353529 Table 7.7 (continuation) Decision set Discount Open ditch Dead level rate d i d 5 a * * E « *' 0.0 451278 451278 416102 416102 0.2 488780 409673 441943 386083 0.8 512281 387505 464895* 370444 1.0 525117 384492 467604* 368319 When a=1.0, i . e . the Bayesian s i t u a t i o n , the expected miniupper and maxilower cos t s c o i n c i d e as there i s no "ignorance" a s soc i a t ed wi th the p r i o r d i s t r i b u t i o n . When a=1.0, i . e . the p r i o r in format ion i s equivalen to complete ignorance, the c a l c u l a t e d upper and lower expected va lues are e n t i r e l y dependent on the "new" i n f e r e n t i a l evidence Z^ provided by the f o r e c a s t i n g model. D e c i s i o n d 3 y i e l d s the 1 16 smal les t miniupper expected cos t and, i n t h i s con t ex t , would be the appropr ia te c h o i c e . When the d i s c o u n t i n g ra tes a i s 0 .2 , suggest ing that the d e c i s i o n maker s t i l l has reasonable confidence i n the p r i o r p r o b a b i l i t y d i s t r i b u t i o n , the a g r i c u l t u r e producer should s t i l l choose d e c i s i o n a c t i o n d 3 . When the d i s c o u n t i n g ra te a increases to 0.8 or 1.0 i . e . when the d e c i s i o n maker expresses s t rong doubts about the p r i o r p r o b a b i l i t y d i s t r i b u t i o n , the d e c i s i o n a c t i o n d 5 becomes the op t imal c h o i c e . Maxilower expected cos t s are a l s o shown i n Table 7.7 for completeness but does not have any bear ing on the op t ima l d e c i s i o n choice i n t h i s example. 7.4 SUMMARY S i m i l a r to Bayesian d e c i s i o n theory , Dempster-Shafer d e c i s i o n theory i s the a p p l i c a t i o n of Dempster-Shafer theory w i t h i n the framework of conven t iona l d e c i s i o n theory . Therefore , the advantages of Dempster-Shafer d e c i s i o n theory over Bayesian d e c i s i o n theory are e s s e n t i a l l y the advantages of Dempster-Shafer theory over Bayesian t heo ry . Since d e c i s i o n making i n c i v i l eng inee r ing p r a c t i c e i s undertaken under a wide v a r i e t y of c i rcumstances , Dempster-Shafer d e c i s i o n theory seems to be much more a t t r a c t i v e i n p r a c t i c a l a p p l i c a t i o n than Bayesian . dec i s ion t heo ry . An example of a p p l y i n g Dempster-Shafer d e c i s i o n theory i n s o l v i n g a d e c i s i o n problem i n water resources eng ineer ing shows that i t does improve the d e c i s i o n r e s u l t s under 1 17 c i rcumstances which Bayesian d e c i s i o n theory cannot accommodate. While l i t t l e work has been done i n app ly ing Dempster-Shafer theory i n s o l v i n g r e a l wor ld p r a c t i c a l problems, the preceding d i s c u s s i o n of the Dempster-Shafer d e c i s i o n theory togther w i t h the example g i v e n , i n d i c a t e s a very promis ing new s t a r t i n g po in t i n the a p p l i c a t i o n of t h i s new theory i n p r a c t i c e . O b v i o u s l y , f u l l acceptance and adopt ion of Dempster-Shafer theory i n p r a c t i c e s t i l l r equ i r e s a l a rge amount of work. 8 . CONCLUSIONS Th i s research has concent ra ted on the comparison of Bayesian theory and Dempster-Shafer theory and, to a l i m i t e d ex t en t , the a p p l i c a t i o n of Dempster-Shafer theory to some elementary water resources eng ineer ing problems. The equiva lence between the two t heo r i e s was d i scussed i n Chapter 4. The i n v e s t i g a t i o n of divergence of r e s u l t s from Bayesian and Dempster-Shafer approaches and the s e n s i t i v i t y a n a l y s i s were presented i n Chapter 5. In Chapter 6, the conceptua l d i f f e r e n c e betwen the two t heo r i e s and the i s sues of s o r t i n g c o l l e c t i v e sources of evidence and represen t ing them through b e l i e f func t ions were argued. F i n a l l y i n Chapter 7, the Dempster-Shafer d e c i s i o n theory was presented . I t s a p p l i c a t i o n i n water resources eng ineer ing p r a c t i c e were demonstrated through a r e a l water resources des ign example. The c o n c l u s i o n s of t h i s research can be summarized as f o l l o w s . Bayesian theory , as a s p e c i a l case of Dempster-Shafer theory , r equ i r e s one to th ink of the i n f e r e n t i a l problem i n terms of the s t a t i s t i c a l s p e c i f i c a t i o n model, though c l e a r l y not a l l of the i n f e r e n t i a l problems s a t i s f y the requirements of t h i s model. Th i s i n turn r equ i r e s that the evidence be grouped i n t o two p a r t s : namely the " o l d " evidence and the "new" ev idence . Bayesian theory then r equ i re s one to spec i fy e x p l i c i t l y the p r i o r p r o b a b i l i t i e s from the " o l d " evidence and the sample l i k e l i h o o d s from the "new" ev idence . I f a l l of t h i s f a i t h f u l l y r e f l e c t s the r e a l s i t u a t i o n then Bayesian 1 1 8 1 19 theory can be cons idered to be the proper approach. But often the evidence may not be s p e c i f i c enough to spec i fy or support these p r o b a b i l i t i e s . In t h i s case, though s u b j e c t i v e judgements may be used to reso lve the d i f f i c u l t i e s , one may not f e e l comfortable or conf ident about i n f e r e n t i a l r e s u l t s based on those p r o b a b i l i t y judgements. Dempster-Shafer theory addresses the more genera l case and pays much more a t t e n t i o n to the q u a l i t y and charac te r of the evidence than the p r o b a b i l i t y model i t s e l f . I t does not need the i n f e r e n t i a l problem to be f i t t e d i n t o the s t a t i s t i c a l s p e c i f i c a t i o n model. But i f the i n f e r e n t i a l problem does f i t i n t o the s t a t i s t i c a l s p e c i f i c a t i o n model, then Dempster-Shafer theory accommodates the s i t u a t i o n p r o p e r l y . In fact through the use of d i s c o u n t i n g , i t a l s o accommodates smal l d e v i a t i o n from the s t a t i s t i c a l s p e c i f i c a t i o n model w i t h ease. The h igh s e n s i t i v i t y of the r e s u l t s to d i s c o u n t i n g under c e r t a i n c o n d i t i o n , which has been demonstrated i n Chapter 5, i s d i s c o n c e r t i n g i n the face of the n e c e s s a r i l y p r e c i s e view taken by the Bayesian approach. The p r i o r p r o b a b i l i t y judgements may be expressed e i t h e r i n the form of a Bayesian b e l i e f func t ion or as a genera l b e l i e f f u n c t i o n , the former being i d e n t i c a l to a Bayesian approach. The more genera l Dempster-Shafer theory r equ i r e s one to cons t ruc t a b e l i e f func t ion from each p iece of evidence and then combine a l l of the b e l i e f functons by the Dempster 's r u l e of combina t ion . I t i s therefore seen that there are at l e a s t two advantages of Dempster-Shafer 120 theory over Bayesian theory : the Dempster-Shafer theory does not need the i n f e r e n t i a l problem to be conf ined to the s t a t i s t i c a l s p e c i f i c a t i o n model; and i t p rov ides the concept of a b e l i e f func t ion which may express the evidence i n a much more r e a l i s t i c and f a i t h f u l way than the very r e s t r i c t i v e conven t iona l express ion of p r o b a b i l i t y i n Bayesian theory . There are e s s e n t i a l l y two types of u n c e r t a i n t i e s , i n f o r m a t i o n a l u n c e r t a i n t y or IU and n a t u r a l unce r t a in ty or NU. The p r o b a b i l i t y judgements based on a p iece of evidence for the set a s s o c i a t e d w i t h NU are r e a l i s t i c a l l y expressed by the f r e q u e n c y - l i k e conven t iona l p r o b a b i l i t y d i s t r i b u t i o n wh i l e for the set a s s o c i a t e d wi th IU these are more r e a l i s t i c a l l y expressed by a consonant b e l i e f f u n c t i o n . In Dempster-Shafer theory , s ince the u n c e r t a i n t i e s a s s o c i a t e d w i t h the frame of discernment 0 belong to the IU ca tegory , the more appropr i a t e b e l i e f s t r u c t u r e on the frame 0 i s the consonant type of b e l i e f f u n c t i o n . Though i t i s more reasonable to adopt a consonant b e l i e f f u n c t i o n , the conven t iona l p r o b a b i l i t y d i s t r i b u t i o n i s not prec luded as an e f f e c t i v e way of express ing ev idence . C o l l e c t i n g and rea r rang ing p ieces of ev idence , and express ing them through b e l i e f f unc t i ons , p a r a l l e l very important aspects of eng inee r ing p r a c t i c e . So f a r , l i t t l e or no work has been undertaken to fo rma l i ze the processes i n v o l v e d . The d i s c u s s i o n s i n Sec. 6 . 3 and Sec . 6 . 4 d i d suggest some p o s s i b l e components of a fo rma l i zed s t r u t u r e . 121 With the v a r i e t y of the evidence t y p i c a l l y bear ing on a frame of discernment , i t may not be necessary or appropr i a t e that each p iece of evidence be expressed by i t s own e x p l i c i t b e l i e f func t ion and the evidence may have to be grouped. Some p o s s i b l e g u i d e l i n e s are t h a t : the vague p ieces of evidence should be put together w i th the more s p e c i f i c ev idence; the interdependent p ieces of evidence should be grouped toge ther ; the more compl ica ted evidence should be decomposed i n t o s eve ra l s imple p ieces of evidence i f i t i s d i f f i c u l t to express such evidence by a complex genera l b e l i e f f u n c t i o n . In a d d i t i o n , j u s t a few c l a s s e s of b e l i e f func t ions appear to represent the major i ty types of evidence o c c u r r i n g i n eng ineer ing p r a c t i c e . The obta ined p ieces of evidence should be grouped i n t o the types of evidence which correspond to the c l a s s e s of b e l i e f f u n c t i o n s . The i m p l i c a t i o n s of the q u i d e l i n e s mentioned here are not yet f u l l y undestood and fur ther i n v e s t i g a t i o n i s e s s e n t i a l to the development of a s u i t a b l e scheme for express ing domain s p e c i f i c evidence i n p r a c t i c e . Express ing evidence through a b e l i e f func t ion can a l s o be a s s i s t e d by comparing the evidence w i t h a s ca l e of c a n o n i c a l examples and adopt ing the example which matches the evidence bes t . The framework for the b e l i e f f unc t i on p rov ided for t h i s example can then be used as the b e l i e f func t ion for the n a t u r a l p iece of ev idence . S u b j e c t i v i t y of course enters when an expert i s asked to choose the c a n o n i c a l example and make the numer ica l b e l i e f judgements. 122 C l e a r l y the cho ices of d i f f e r e n t exper ts w i l l l e ad to d i f f e r e n t i n f e r e n t i a l r e s u l t s for the same i n f e r e n t i a l problem. Bounding and s e n s i t i v i t y methods may the re fo re be e s s e n t i a l i n g r e d i e n t s i n a p r a c t i c a l b e l i e f ent ry system. Dempster-Shafer d e c i s i o n theory , i . e . the a p p l i c a t i o n of Dempster-Shafer theory i n con junc t ion w i t h conven t iona l d e c i s i o n theory , f u l l y e x p l o i t s the advantages of Dempster-Shafer theory over Bayesian theo ry . I t can be regarded as a g e n e r a l i z a t i o n of Bayesian d e c i s i o n theory a l s o . Under those c o n d i t i o n s i n which Dempster-Shafer theory and Bayesian theory are i d e n t i c a l , i t i s reasoning that Dempster-Shafer d e c i s i o n theory and Bayesian d e c i s i o n theory are a l s o i d e n t i c a l . However, Dempster-Shafer d e c i s i o n theory can be used i n the more genera l s i t u a t i o n i n which Bayesian d e c i s i o n theory i s o v e r l y r e s t r i c t i v e and u n s a t i s f a c t o r y . The s imple example of an a p p l i c a t i o n of t h i s d e c i s i o n theory i n r e s o l v i n g a r e a l i s t i c water resources eng ineer ing des ign problem demonstrates the p o t e n t i a l of the Dempster-Shafer d e c i s i o n theory to improve the d e c i s i o n a n a l y s i s framework. While Dempster-Shafer theory gains some s u b s t a n t i a l advantages over Bayesian theory , these are not achieved without cos t and we may a n t i c i p a t e some d i f f i c u l t i e s i n i t s a p p l i c a t i o n . The d i f f i c u l t i e s may be summarised i n the f o l l o w i n g c a t e g o r i e s . F i r s t l y , i n Bayesian theory , both the p r i o r p r o b a b i l i t i e s based on p r i o r evidence and the sample l i k e l i h o o d based on the i n f e r e n t i a l evidence are expressed 1 23 i n the form of f a m i l i a r , conven t iona l p r o b a b i l i t y d i s t r i b u t i o n s . In implementing Dempster-Shafer theory , b e l i e f i s expressed i n the u n f a m i l i a r format of a b e l i e f f u n c t i o n . A l s o , s ince the b e l i e f va lue can be ass igned to any grouping of p r o p o s i t i o n s , one faces a much greater number of p o t e n t i a l cho ices of b e l i e f assignment i n Dempster-Shafer theory . Th i s increased cho ice w i l l make i t d i f i c u l t for one to choose the appropr i a t e l e v e l of p r o p o s i t i o n s and the appropr i a t e b e l i e f assignments for any g iven p iece of ev idence . Even though s p e c i a l c l a s s e s of b e l i e f func t ions can be adopted, as d i scussed i n Chapter 6 , the d i f f i c u l t y s t i l l remains about which type of b e l i e f func t ion should be choosen and how the evidence should be expressed by a numerica l b e l i e f f u n c t i o n . In Chapter 6 , the idea of c a n o n i c a l example was presented i n order to f a c i l i t a t e the process of express ing evidence by a b e l i e f f u n c t i o n . Even though such an idea i s t h e o r e t i c a l l y a t t r a c t i v e , fur ther work s t i l l needs to be done to f a c i l i t a t e i t s adopt ion i n eng ineer ing p r a c t i c e . Secondly , the Dempster-Shafer b e l i e f func t ion i s l e s s managable than the conven t iona l p r o b a b i l i t y d i s t r i b u t i o n i n Bayesian theory . In Bayesian theory , the p r o b a b i l i t y d i s t r i b u t i o n i s often mathemat ica l ly w e l l expressed and the i m p l i c a t i o n s of such a p r o b a b i l i t y d i s t r i b u t i o n broadly unders tood. In Dempster-Shafer theory , however, the express ion of general b e l i e f func t ion i s more complex, s ince i t i s de f ined on the whole frame of discernment the f u l l 1 24 i m p l i c a t i o n s of the b e l i e f func t ion are much more d i f f i c u l t to i n t e r p r e t , t h i s being e s p e c i a l l y t rue when the frame becomes l a r g e . F i n a l l y , r e c a l l that Bayesian theory can be expressed i n cont inuous form. Since the v a r i a b l e s i n many s i t u a t i o n s i n p r a c t i c e are cont inuous v a r i a b l e s , the cont inuous express ion of Bayesian theory i s often p r e f e r r e d . However, most of the d i s c u s s i o n of Dempster-Shafer theory so far has been conf ined to the d i s c r e t e form and very l i t t l e work has been done for the cont inuous s i t u a t i o n . T . M . S t r a t [ 2 3 ] s tud ied t h i s problem based on the assumption t h a t , for a continuous random v a r i a b l e , on ly the p r o p o s i t i o n which i s formed by a set of cont iguous va lues i s of s i g n i f i c a n c e . For example, a random v a r i a b l e x has a set of p o s s i b l e , a scend ing ly ordered , d i sc re t e - va lues x , , . . . x 9 . Then p r o p o s i t i o n s l i k e { x 3 , x „ , x 5 } w i l l be of s i g n i f i c a n c e and p r o p o s i t i o n s l i k e { x 2 , x t t , x 5 } w i l l be cons ide red to be i m p o s s i b l e . The adopt ion of t h i s assumption reduces the s i z e of the frame of discernment to l e s s than h a l f of i t s o r i g i n a l s i z e . The cont inuous frame of discernment i s obta ined by reducing the i n t e r v a l of the d i s c r e t e element to i t s l i m i t . A cont inuous b e l i e f func t ion based on a cont inuous frame of discernment can then be def ined i f a p iece of evidence i s ob ta ined . S t r a t d i d d e r i v e the genera l mathematical formulas for the cont inuous b e l i e f func t ion and Dempster 's r u l e of combina t ion . These formulas turned out to i nvo lve very complex and i n t r a c t i b l e exp re s s ions . 125 Never the l e s s , i t seems to p rov ide a good s t a r t i n g p o i n t . I t i s worthwhile to note the way the assumption i s incorpora ted and the way the problem i s s t r e t ched to accommodate the cont inuous frame. In c o n c l u s i o n , the Dempster-Shafer scheme appears to o f f e r many s i g n i f i c a n t advantages beyond the Bayesian scheme under the c o n d i t i o n s of u n c e r t a i n t y exper ienced i n c i v i l e n g i n e e r i n g . As the a l r eady e s t a b l i s h e d Bayesian scheme occupies the p o s i t i o n of s p e c i a l case w i t h i n the Dempster-Shafer scheme, i n both inference and d e c i s i o n a n a l y s i s , no l o s s of c a p a b i l i t y i s exper ienced when op t ing for the Dempster-Shafer scheme (except p o s s i b l y i n some cases where cont inuous v a r i a b l e s are i n v o l v e d ) . While implementat ion of the Dempster-Shafer scheme w i l l c l e a r l y i n v o l v e overcoming a number of d i f f i c u l t i e s , which can range from lack of f a m i l i a r i t y w i t h the l e s s conven t iona l theory i n v o l v e d to numerica l i n t r a c t i b i l i t y w i th l a rge frames, none would appear se r ious enough to obs t ruc t the implementat ion. The s u b s t a n t i a l improvements promised by the Dempster-Shafer scheme i n the r a t i o n a l i z a t i o n and f o r m a l i z a t i o n of p r a c t i c a l d e c i s i o n making under u n c e r t a i n t y would appear to provide more than adequate i n c e n t i v e to overcoming these d i f f i c u l t i e s . REFERENCES 1. Ang, A . H-S . and Tang, W . H . , Probability Concepts in Engineering Planning and Design, W i l e y , New York , 1975. 2. Barneyy, J . A . , "Computational Methods for A Mathemat ical Theory of Ev idence , " Proceedings Seventh National Joint Conference on Artifical Intelligence, Vancouver, B . C . , pp. 319-325, Aug. 1981 . 3. Benjamin, J . R . " and C o r n e l l , C . A . , Probability, S t a t i s t i c s , and Decision for Civil Engineers, M c G r a w - H i l l , New York , 1970. 4. Bodo, B . and Unny, T . E . , "Model U n c e r t a i n t y i n F lood Frequency A n a l y s i s and Frequency-Based D e s i g n , " Water Resour. Res., 12(6) , pp. 1109-1117, 1976. 5. C a s e l t o n , W . F . , Froese , T . M . , R u s s e l l , A . D . and Luo, W., " B e l i e f Input Procedures for Dempster-Shafer Based Expert System," Artificial Intelligence in Engineering: Robotics and Processes, Computat ional Mechanics P u b l i c a t i o n s , Southampton, pp. 351-370, 1988. 6. D a v i s , D . R . , K i s i e l , C . C . and D u c k s t e i n , L . , "Bayesian D e c i s i o n Theory A p p l i e d to Design i n Hydro logy , " Water Resour. Res., 8 ( 1 ) , 33-41, 1972. 7. Dempster, A . P . , "A G e n e r a l i z a t i o n of Bayesian In fe rence , " Journal of the Royal Statistical Society, Ser i e s B, 30, pp. 205-247. 8. Dempster, A . P . and Kong, A . , "Comment," Statistical Science, V o l . 2, No. 1, pp. 32-36. 1 26 127 9. Duda, R . O . , H a r t , P . E . and N i l s s o n , N . J . , "Sub jec t ive Bayesian Methods for Rule-based Inference Systems," AFIPS Conf. Proc., 45, pp. 1075-1082, 1976. 10. Garvey, T . D . , Lowrance, J . D . and F i s c h l e r , M . A . , "An Inference Technique for I n t e l l i g e n c e Knowledge from Dispara te Sources , " Proceedings Seventh International Joi nt • Conf er e nee on Artificial Intelligence, Vancouver, B . C . , pp. 319-325, Aug. 1981. 11. Gordon, J . and S h o r t l i f f e , E . H . , "A Method for Managing E v i d e n t i a l Reasoning i n A H i e r a c h i c a l Hypothes is Space," Artificial Intelligence, V o l . 26, pp. 323-357, J u l y 1985. 12. Gordon, J . and S h o r t l i f f e , E . H . , "The Dempster-Shafer Theory of E v i d e n c e , " i n : B . G . Buchanan and E . H . S h o r t l i f f e ( e d s . ) , Rule-based Expert Systems: The MYCIN Experiments of the Stanford-Heuristic Programming Project, (Addison-Wesley, Reading, MA), pp. 272-292, 1984. 13. J e f f r e y , R . C . , The Logic of Decision, M c G r a w - H i l l , New York , 1965. 14. K r a n t z , D . H . and Miyamoto, J . , "Prors and L i k e l i h o o d Ra t io s as E v i d e n c e , " Journal of the American Statistical Association, 78, pp. 418-423, 1983. 15. L u , S . Y . and Stephanou, H . E . , "A S e t - t h e o r e t i c Framework for the Process ing of Uncer t a in Knowledge," Proceedings Fourth National Conference on Artificial Intelligence, A u s t i n , TX, pp. 308-313, 1984. 16. Moore, P . G . and Thomas, H . , "Measuring Omega, V o l . 3, No. 6, pp. 657-672, 1975. U n c e r t a i n t y , " 128 17. M c A n i f f , R . J . , F l u g , M. and Wade, J . , "Bayesian A n a l y s i s of Energy P r i c e s on I r r i g a t i o n , " Journal of the Water Resources Planning and Management Division, ASCE, 106(WR2), pp. 401-408. 18. Shafer , G . , A Mathematical Theory of Evidence, P r i n c e t o n U n i v e r s i t y P ress , P r i n c e t o n , New Je r s ey , 1976. 19. Shafer , G . , " B e l i e f Func t ions and Parametr ic Mode l s , " Journal of the Royal Statistical Society, Se r i e s B, 44, pp. 322-352, 1982. 20. Shafer , G. and Tversky , A . , "Languages and Designs for P r o b a b i l i t y Judgement," Cognitive Science, 9, pp. 309-339, 1983. 21 . Shafer , G . , " P r o b a b i l i t y Judgement i n A r t i f i c i a l I n t e l l i g e n c e and Exper t System," Statistical Science, V o l . 2, No. 1, pp. 3-44, 1987. 22. Shane, R . M . and Gaver, D . P . , " S t a t i s t i c a l D e c i s i o n Theory Techniques for the R e v i s i o n of Mean Flood Flow Regress ion E s t i m a t e s , " Water Resour. Res., 6 ( 6 ) , 1649-1654, 1970. 23. S t r a t , T . M . , "Continuous B e l i e f Func t ions for E v i d e n t i a l Reason ing ," Proceedings Fourth National Conference on Artificial Intelligence, A u s t i n , Texas, pp. 308-313, Aug. 1984. 24. Tschanne r l , G . , "Designing R e s e r v o i r s wi th Short Streamflow Records , " Water Resour. Res., 7 ( 4 ) , 827-833, 1 971 . 25. Tversky , A . And Kahneman, D . , "Judgement under U n c e r t a i n t y : H e u r i s t i c s and B i a s e s , " Science, V o l . 185, pp. 1124-131, 1974. 129 26. V i c e n s , G . J . - , R o d r i g u e z - I t u r b e , I . and Schaake, J . C . "A Bayesian Framework for the Use of Regiona l Informat ion i n Hydro logy , " Water Resour. Res., 11(3) , 405-414, 1972. 27. Wood, E . F . and R o d r i g u e z - I t u r b e , I . , "Bayesian Inference and D e c i s i o n Making for Extreme Hydro log ic E v e n t s , " Water Resour. Res., 11(4) , 533-542, 1975. 28. Yager, R . R . , "On the Dempster-Shafer Framework," Art i fi ci al Sciences, V o l . 41, pp. 93-137, 1987. 

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