a generalization of the dempster-shafer theory - IJCAI
A G E N E R A L I Z A T I O N OF T H E DEMPSTER-SHAFER THEORY J . W . G u a n , D . A . Bell Department of Information Systems University of Ulster at Jordanstown BT37 OQB, Northern Ireland, U.K. E-mail: [email protected]
Abstract T h e Dempster-Shafer theory gives a solid basis for reasoning a p p l i c a t i o n s characterized by uncertainty. A key feature of the theory is t h a t propositions are represented as subsets of a set which represents a hypothesis space. T h i s power set along w i t h the set operations is a Boolean algebra. Can we generalize the theory to cover a r b i t r a r y Boolean algebras ? We show t h a t the answer is yes. T h e theory then covers, for e x a m p l e , i n f i n i t e sets. T h e practical advantages of generalization are t h a t increased f l e x i b i l i t y of representation is a l lowed and t h a t the performance of evidence acc u m u l a t i o n can be enhanced. In a previous paper we generalized the Dempster-Shafer o r t h o g o n a l sum o p e r a t i o n to s u p p o r t p r a c t i c a l evidence p o o l i n g . In the present paper we provide the theoretical u n d e r p i n n i n g of t h a t procedure, by systematically considering f a m i l i a r evidential functions in t u r n . For each we present a "weaker f o r m " and we look at the relationships between these variations of the f u n c t i o n s . T h e relationships are not so s t r o n g as for the conventional functions. However, when we specialize to the fam i l i a r case of subsets, we do indeed get the wellk n o w n relationships.
1
Introduction
U n c e r t a i n t y is a feature of our experience a n d observat i o n of the w o r l d . F i n d i n g suitable means of represent a t i o n and m a n i p u l a t i o n o f u n c e r t a i n t y o f i n f o r m a t i o n and knowledge [Bell 1992] is a challenge which w i l l have to be m e t if c o m p u t e r i z e d d e c i s i o n - m a k i n g based on i m perfect i n p u t is to be c o n t e m p l a t e d . An u n d e r s t a n d i n g of the effect of u n c e r t a i n t y on evidence appraisal, and u l t i m a t e l y on the behavior a n d properties of agents is essential. T h i s paper contributes to this u n d e r s t a n d i n g and to the p r a c t i c a l h a n d l i n g of evidence. It addresses the extension, in b o t h p r a c t i c a l and theoretical t e r m s , of a n u m e r i c a l system w h i c h enables c o m p u t e r a p p l i c a t i o n s to reflect some aspects of u n c e r t a i n t y .
592
Knowledge Representation
T h e theory of evidence which o r i g i n a t e d w i t h D e m p ster and Shafer underpins a m e t h o d which has been shown to be a p r o m i s i n g t o o l for m a k i n g judgements when confronted w i t h u n c e r t a i n t y in n u m e r i c a l evidence. It generalizes Bayesian theory which is itself a popular theory of u n c e r t a i n t y . T h e generalization of evidence theory in t u r n is the subject of this paper. It involves m o v i n g away f r o m the s t a n d a r d finite set based d e r i v a t i o n of theoretical and c o m p u t a t i o n a l results u n d e r p i n n i n g the DempsterShafer approach. C o n v e n t i o n a l l y propositions are represented as subsets of a collection of all possible values of a target variable. This p a r t i c u l a r representation of the hypothesis space, is not the only way to represent propositions. M o s t obviously we can t h i n k of leaving the propositions as they are, a v o i d i n g their t r a n s f o r m a t i o n i n t o subsets. T h i s is of i m m e d i a t e interest in reasoning a p p l i c a t i o n s , because propositions are f a m i l i a r and can be used to represent a r g u m e n t s , hypotheses, etc. If this were done we w o u l d s t i l l be dealing w i t h a structure which has an i m p o r t a n t s i m i l a r i t y to the previous space — b o t h are Boolean algebras. T h i s leads to the question: can we generalize evidence theory to general Boolean algebras ? If we can, this allows us to choose a representation — we can use subsets, propositions, and other means to represent hypotheses and their relationships, as a p p r o p r i a t e , in the hypothesis space. It can also allow us to establish a theory which covers infinite hypothesis spaces and evidence spaces by this extension. T h i s representational and theoretical advantage of the generalization is our focus of a t t e n t i o n in this paper. However we have argued elsewhere [Guan & Bell 1993a], and supported our a r g u m e n t s by defining operations, t h a t m a n y a p p l i c a t i o n s can achieve i m p r o v e d performance t h r o u g h using more a p p r o p r i a t e representations. We d e m o n s t r a t e d t h a t by generalizing the orthogonal sum o p e r a t i o n so t h a t hypotheses could be represented d i r e c t l y as p r o p o s i t i o n s , such a performance enhancem e n t could accrue. Using this representation, all the subsets of the hypothesis space , i.e., subsets, need not be considered (as they w o u l d in s t a n d a r d DempsterShafer t h e o r y ) . By focusing on relevant propositions only, the t i m e c o m p l e x i t y m a y be reduced to well below the previous time. To these advantages of representational and m a n i p u lative flexibility and efficiency for a p p l i c a t i o n s , we can
add the advantage of developing a theory which covers infinite hypothesis spaces and evidence spaces, and ext e n d i n g our u n d e r s t a n d i n g of the Dempster-Shafer technique. In section 2 we define evidential functions which are based on Boolean algebras. In p a r t i c u l a r we introduce weak versions of Bayesian functions, belief functions, and other e v i d e n t i a l f u n c t i o n s . We establish relationships between these weak f u n c t i o n s and the f a m i l i a r corresponding f u n c t i o n s f r o m evidence theory, showing t h a t the results for power sets do not carry over to Boolean algebras in the general case. In section 3 we discuss nested evident i a l f u n c t i o n s . We show in section 4 t h a t we can o b t a i n f a m i l i a r relations for the p a r t i c u l a r case of the power set. T h e w e l l - k n o w n inversions between the most conspicuous evidential functions are derived. T h e n we complete the paper by s u m m a r i z i n g the relationships between the weak evidential f u n c t i o n s o b t a i n e d .
2
E v i d e n t i a l functions
In evidence theory, evidence is described in terms of evid e n t i a l f u n c t i o n s . There are several functions c o m m o n l y used in the theory — mass functions, belief functions, c o m m o n a l i t y f u n c t i o n s , d o u b t functions, and plausibili t y functions. N o r m a l l y they are defined over finite sets. Here we generalize evidential functions to Boolean algebras. T h e significance of this is t h a t conventional evident i a l functions are defined over the power set of a frame of discernment, b u t Boolean algebras include other interesting spaces, such as the space of propositions.
Guan and Bell
593
594
Knowledge Representation
Guan and Bell
595
596
Knowledge Representation
Proc. of the 2nd I n t . Conf. on Computers and A p p l i cations, Peking, 1987. [Guan and Lesser, 1987b] G u a n , J. W. ; Lesser, V R. 1987b, " On the evidence c o m b i n a t i o n scheme in a hierarchical hypothesis space ", Proceedings of T E N C O N 87, Seoul, I E E E Region 10 Conference 1987. [Guan et al., 1989] G u a n , J. W ; P a v l i n , J . ; Lesser, V. R. 1989, " C o m b i n i n g evidence in the extended Dempster-Shafer theory " , P r o c . of the 2nd I r i s h Con/. on AI and Cognitive Science, D u b l i n , 1989. Smeaton and M c D e r m o t t (Eds.) Springer-Verlag, 1990, 163178.
5
SUMMARY
Most i m p o r t a n t spaces in a r t i f i c i a l intelligence are Boolean algebras, for example, power sets and propos i t i o n sets. T h e Dempster-Shafer theory originally addressed o n l y the power sets. T h i s paper generalizes the theory to Boolean algebras. We investigate all the most i m p o r t a n t kinds of belief functions on an algebra to enable us to choose the most suitable belief f u n c t i o n to represent evidence, according to the p a r t i c u l a r s i t u a t i o n presented. The generalizat i o n enables us to choose the most suitable algebra to represent knowledge and reason efficiently. We i n t r o d u c e weak Bayesian ( p r o b a b i l i s t i c ) functions, Bayesian ( p r o b a b i l i s t i c ) functions, weak belief functions, and belief functions. We show t h a t Bayesian functions are weak Bayesian f u n c t i o n s ; and Bayesian functions and belief f u n c t i o n s are weak belief functions. Mass f u n c t i o n s , c o m m o n a l i t y functions, p l a u s i b i l i t y f u n c t i o n s , and d o u b t f u n c t i o n s are also introduced. In the case where is a finite set, we show t h a t weak Bayesian functions are Bayesian functions a n d vice versa. Moreover, weak belief functions are then belief f u n c t i o n s and vice versa, and weak nested belief f u n c t i o n s are nested belief functions and vice versa.
References [ B a r n e t t , 198l] B a r n e t t , J. A. 1981, " C o m p u t a t i o n a l m e t h o d s for a m a t h e m a t i c a l theory of evidence ", Proc. I J C A I - 8 1 , 1981, 868-875. [Bell et a/., 1992] B e l l , D. A . ; G u a n , J. W . ; Lee, S. K. 1993, " Efficient generalized union and projection operations for p o o l i n g uncertain and imprecise i n f o r m a t i o n ", I n f o r m a t i c s Technical Reports, U n i v . of Ulster.
[Guan et al., 1991] G u a n , J. W. ; Bell, D. A. ; Lesser, V. R. 1991, " E v i d e n t i a l reasoning and rule strengths in expert systems ", In McTear, M. F. ; Creany, N. (eds) A r t i f i c i a l Intelligence and Cognitive Science ' 90, pp. 378-390, 1991. [Guan et a l , 1990] G u a n , J. W. ; Bell, D. A. ; P a v l i n , J. ; Lesser, V. R. 1990, " T h e Dempster-Shafcr theory and rule strengths in expert systems ", I E E Colloquium on " Reasoning under Uncertainty ", 1990. [Guan and Bell, 199l] G u a n , J. W. ; Bell, D. A. 1991, " Evidence theory and its applications ", Volume 1, Studies in Computer Science and A r t i f i c i a l I n t e l l i gence 7, Elsevier, N o r t h - H o l l a n d . [Guan and Bell, 1992] G u a n , J. W. ; Bell, D. A. 1992, " Evidence theory and its applications ", Volume 2, Studies in Computer Science and A r t i f i c i a l I n t e l l i gence 8, Elsevier, N o r t h - H o l l a n d . [Guan and Bell, 1993a] G u a n , J. W. ; Bell, D. A. 1993, " Generalizing the Dempster-Shafer rule of combinat i o n t o Boolean algebras " , Proc. o f I E E E / A C M Int. Conf. on Developing and M a n a g i n g Intelligent Systems Projects, W a s h i n g t o n , March 1993. [Guan and B e l l , 1993b] G u a n , J. W. ; B e l l , D. A. 1993, " On some kinds of belief functions in E x p e r t control ", Proc. of I F S I C C 9 3 , U S A , M a r c h 1993. [Prade, 1985] Prade, H. 1985, " A c o m p u t a t i o n a l approach to a p p r o x i m a t e and plausible reasoning w i t h applications to expert systems ", I E E E Trans. Vol. P A M I - 1 , 260-283. [Shafer, 1976] Shafer, G. 1976, " A m a t h e m a t i c a l theory of evidence ", P r i n c e t o n University Press, P r i n c e t o n , New Jersey, 1976. [Shortliffe, 1976] Shortliffc, E. H. 1976, " Computerbased medical consultations: M Y C I N ", Elsevier Publishing Company, New Y o r k .
[ G o r d o n and Shortliffc, 1985] G o r d o n , J. ; Shortliffc, E. H. 1985 ," A m e t h o d for m a n a g i n g evidential reasoni n g in a hierarchical hypothesis space ", A r t i f i c i a l I n telligence, 26(1985) 323-357.
[Shortliffe and B u c h a n a n , 1975] Shortliffe, E. H. ; Buchanan, B. G. 1975, " A model of inexact reasoning in medicine ", M a t h e m a t i c a l Biosciences, 23, 1975, 351-379.
[ G u a n et al., 1985] G u a n , J. W. ; X u , Y. and others, 1985, " M o d e l expert system M E S ", Proc. of I J C A I 85, 397-399.
[Yen, 1989] Yen, J. 1989, " GERT1S: A Dempster-Shafer approach to diagnosing hierarchical hypotheses ", C A C M 5 vol. 32, 1989, pp.573-585.
[ G u a n and Lesser, 1987a] G u a n , J. W. ; Lesser, V. R. 1987a, " T h e c o m p u t a t i o n a l formulae of evidence comb i n a t i o n scheme in a hierarchical hypothesis space ",
Guan and Bell
597
Abstract T h e Dempster-Shafer theory gives a solid basis for reasoning a p p l i c a t i o n s characterized by uncertainty. A key feature of the theory is t h a t propositions are represented as subsets of a set which represents a hypothesis space. T h i s power set along w i t h the set operations is a Boolean algebra. Can we generalize the theory to cover a r b i t r a r y Boolean algebras ? We show t h a t the answer is yes. T h e theory then covers, for e x a m p l e , i n f i n i t e sets. T h e practical advantages of generalization are t h a t increased f l e x i b i l i t y of representation is a l lowed and t h a t the performance of evidence acc u m u l a t i o n can be enhanced. In a previous paper we generalized the Dempster-Shafer o r t h o g o n a l sum o p e r a t i o n to s u p p o r t p r a c t i c a l evidence p o o l i n g . In the present paper we provide the theoretical u n d e r p i n n i n g of t h a t procedure, by systematically considering f a m i l i a r evidential functions in t u r n . For each we present a "weaker f o r m " and we look at the relationships between these variations of the f u n c t i o n s . T h e relationships are not so s t r o n g as for the conventional functions. However, when we specialize to the fam i l i a r case of subsets, we do indeed get the wellk n o w n relationships.
1
Introduction
U n c e r t a i n t y is a feature of our experience a n d observat i o n of the w o r l d . F i n d i n g suitable means of represent a t i o n and m a n i p u l a t i o n o f u n c e r t a i n t y o f i n f o r m a t i o n and knowledge [Bell 1992] is a challenge which w i l l have to be m e t if c o m p u t e r i z e d d e c i s i o n - m a k i n g based on i m perfect i n p u t is to be c o n t e m p l a t e d . An u n d e r s t a n d i n g of the effect of u n c e r t a i n t y on evidence appraisal, and u l t i m a t e l y on the behavior a n d properties of agents is essential. T h i s paper contributes to this u n d e r s t a n d i n g and to the p r a c t i c a l h a n d l i n g of evidence. It addresses the extension, in b o t h p r a c t i c a l and theoretical t e r m s , of a n u m e r i c a l system w h i c h enables c o m p u t e r a p p l i c a t i o n s to reflect some aspects of u n c e r t a i n t y .
592
Knowledge Representation
T h e theory of evidence which o r i g i n a t e d w i t h D e m p ster and Shafer underpins a m e t h o d which has been shown to be a p r o m i s i n g t o o l for m a k i n g judgements when confronted w i t h u n c e r t a i n t y in n u m e r i c a l evidence. It generalizes Bayesian theory which is itself a popular theory of u n c e r t a i n t y . T h e generalization of evidence theory in t u r n is the subject of this paper. It involves m o v i n g away f r o m the s t a n d a r d finite set based d e r i v a t i o n of theoretical and c o m p u t a t i o n a l results u n d e r p i n n i n g the DempsterShafer approach. C o n v e n t i o n a l l y propositions are represented as subsets of a collection of all possible values of a target variable. This p a r t i c u l a r representation of the hypothesis space, is not the only way to represent propositions. M o s t obviously we can t h i n k of leaving the propositions as they are, a v o i d i n g their t r a n s f o r m a t i o n i n t o subsets. T h i s is of i m m e d i a t e interest in reasoning a p p l i c a t i o n s , because propositions are f a m i l i a r and can be used to represent a r g u m e n t s , hypotheses, etc. If this were done we w o u l d s t i l l be dealing w i t h a structure which has an i m p o r t a n t s i m i l a r i t y to the previous space — b o t h are Boolean algebras. T h i s leads to the question: can we generalize evidence theory to general Boolean algebras ? If we can, this allows us to choose a representation — we can use subsets, propositions, and other means to represent hypotheses and their relationships, as a p p r o p r i a t e , in the hypothesis space. It can also allow us to establish a theory which covers infinite hypothesis spaces and evidence spaces by this extension. T h i s representational and theoretical advantage of the generalization is our focus of a t t e n t i o n in this paper. However we have argued elsewhere [Guan & Bell 1993a], and supported our a r g u m e n t s by defining operations, t h a t m a n y a p p l i c a t i o n s can achieve i m p r o v e d performance t h r o u g h using more a p p r o p r i a t e representations. We d e m o n s t r a t e d t h a t by generalizing the orthogonal sum o p e r a t i o n so t h a t hypotheses could be represented d i r e c t l y as p r o p o s i t i o n s , such a performance enhancem e n t could accrue. Using this representation, all the subsets of the hypothesis space , i.e., subsets, need not be considered (as they w o u l d in s t a n d a r d DempsterShafer t h e o r y ) . By focusing on relevant propositions only, the t i m e c o m p l e x i t y m a y be reduced to well below the previous time. To these advantages of representational and m a n i p u lative flexibility and efficiency for a p p l i c a t i o n s , we can
add the advantage of developing a theory which covers infinite hypothesis spaces and evidence spaces, and ext e n d i n g our u n d e r s t a n d i n g of the Dempster-Shafer technique. In section 2 we define evidential functions which are based on Boolean algebras. In p a r t i c u l a r we introduce weak versions of Bayesian functions, belief functions, and other e v i d e n t i a l f u n c t i o n s . We establish relationships between these weak f u n c t i o n s and the f a m i l i a r corresponding f u n c t i o n s f r o m evidence theory, showing t h a t the results for power sets do not carry over to Boolean algebras in the general case. In section 3 we discuss nested evident i a l f u n c t i o n s . We show in section 4 t h a t we can o b t a i n f a m i l i a r relations for the p a r t i c u l a r case of the power set. T h e w e l l - k n o w n inversions between the most conspicuous evidential functions are derived. T h e n we complete the paper by s u m m a r i z i n g the relationships between the weak evidential f u n c t i o n s o b t a i n e d .
2
E v i d e n t i a l functions
In evidence theory, evidence is described in terms of evid e n t i a l f u n c t i o n s . There are several functions c o m m o n l y used in the theory — mass functions, belief functions, c o m m o n a l i t y f u n c t i o n s , d o u b t functions, and plausibili t y functions. N o r m a l l y they are defined over finite sets. Here we generalize evidential functions to Boolean algebras. T h e significance of this is t h a t conventional evident i a l functions are defined over the power set of a frame of discernment, b u t Boolean algebras include other interesting spaces, such as the space of propositions.
Guan and Bell
593
594
Knowledge Representation
Guan and Bell
595
596
Knowledge Representation
Proc. of the 2nd I n t . Conf. on Computers and A p p l i cations, Peking, 1987. [Guan and Lesser, 1987b] G u a n , J. W. ; Lesser, V R. 1987b, " On the evidence c o m b i n a t i o n scheme in a hierarchical hypothesis space ", Proceedings of T E N C O N 87, Seoul, I E E E Region 10 Conference 1987. [Guan et al., 1989] G u a n , J. W ; P a v l i n , J . ; Lesser, V. R. 1989, " C o m b i n i n g evidence in the extended Dempster-Shafer theory " , P r o c . of the 2nd I r i s h Con/. on AI and Cognitive Science, D u b l i n , 1989. Smeaton and M c D e r m o t t (Eds.) Springer-Verlag, 1990, 163178.
5
SUMMARY
Most i m p o r t a n t spaces in a r t i f i c i a l intelligence are Boolean algebras, for example, power sets and propos i t i o n sets. T h e Dempster-Shafer theory originally addressed o n l y the power sets. T h i s paper generalizes the theory to Boolean algebras. We investigate all the most i m p o r t a n t kinds of belief functions on an algebra to enable us to choose the most suitable belief f u n c t i o n to represent evidence, according to the p a r t i c u l a r s i t u a t i o n presented. The generalizat i o n enables us to choose the most suitable algebra to represent knowledge and reason efficiently. We i n t r o d u c e weak Bayesian ( p r o b a b i l i s t i c ) functions, Bayesian ( p r o b a b i l i s t i c ) functions, weak belief functions, and belief functions. We show t h a t Bayesian functions are weak Bayesian f u n c t i o n s ; and Bayesian functions and belief f u n c t i o n s are weak belief functions. Mass f u n c t i o n s , c o m m o n a l i t y functions, p l a u s i b i l i t y f u n c t i o n s , and d o u b t f u n c t i o n s are also introduced. In the case where is a finite set, we show t h a t weak Bayesian functions are Bayesian functions a n d vice versa. Moreover, weak belief functions are then belief f u n c t i o n s and vice versa, and weak nested belief f u n c t i o n s are nested belief functions and vice versa.
References [ B a r n e t t , 198l] B a r n e t t , J. A. 1981, " C o m p u t a t i o n a l m e t h o d s for a m a t h e m a t i c a l theory of evidence ", Proc. I J C A I - 8 1 , 1981, 868-875. [Bell et a/., 1992] B e l l , D. A . ; G u a n , J. W . ; Lee, S. K. 1993, " Efficient generalized union and projection operations for p o o l i n g uncertain and imprecise i n f o r m a t i o n ", I n f o r m a t i c s Technical Reports, U n i v . of Ulster.
[Guan et al., 1991] G u a n , J. W. ; Bell, D. A. ; Lesser, V. R. 1991, " E v i d e n t i a l reasoning and rule strengths in expert systems ", In McTear, M. F. ; Creany, N. (eds) A r t i f i c i a l Intelligence and Cognitive Science ' 90, pp. 378-390, 1991. [Guan et a l , 1990] G u a n , J. W. ; Bell, D. A. ; P a v l i n , J. ; Lesser, V. R. 1990, " T h e Dempster-Shafcr theory and rule strengths in expert systems ", I E E Colloquium on " Reasoning under Uncertainty ", 1990. [Guan and Bell, 199l] G u a n , J. W. ; Bell, D. A. 1991, " Evidence theory and its applications ", Volume 1, Studies in Computer Science and A r t i f i c i a l I n t e l l i gence 7, Elsevier, N o r t h - H o l l a n d . [Guan and Bell, 1992] G u a n , J. W. ; Bell, D. A. 1992, " Evidence theory and its applications ", Volume 2, Studies in Computer Science and A r t i f i c i a l I n t e l l i gence 8, Elsevier, N o r t h - H o l l a n d . [Guan and Bell, 1993a] G u a n , J. W. ; Bell, D. A. 1993, " Generalizing the Dempster-Shafer rule of combinat i o n t o Boolean algebras " , Proc. o f I E E E / A C M Int. Conf. on Developing and M a n a g i n g Intelligent Systems Projects, W a s h i n g t o n , March 1993. [Guan and B e l l , 1993b] G u a n , J. W. ; B e l l , D. A. 1993, " On some kinds of belief functions in E x p e r t control ", Proc. of I F S I C C 9 3 , U S A , M a r c h 1993. [Prade, 1985] Prade, H. 1985, " A c o m p u t a t i o n a l approach to a p p r o x i m a t e and plausible reasoning w i t h applications to expert systems ", I E E E Trans. Vol. P A M I - 1 , 260-283. [Shafer, 1976] Shafer, G. 1976, " A m a t h e m a t i c a l theory of evidence ", P r i n c e t o n University Press, P r i n c e t o n , New Jersey, 1976. [Shortliffe, 1976] Shortliffc, E. H. 1976, " Computerbased medical consultations: M Y C I N ", Elsevier Publishing Company, New Y o r k .
[ G o r d o n and Shortliffc, 1985] G o r d o n , J. ; Shortliffc, E. H. 1985 ," A m e t h o d for m a n a g i n g evidential reasoni n g in a hierarchical hypothesis space ", A r t i f i c i a l I n telligence, 26(1985) 323-357.
[Shortliffe and B u c h a n a n , 1975] Shortliffe, E. H. ; Buchanan, B. G. 1975, " A model of inexact reasoning in medicine ", M a t h e m a t i c a l Biosciences, 23, 1975, 351-379.
[ G u a n et al., 1985] G u a n , J. W. ; X u , Y. and others, 1985, " M o d e l expert system M E S ", Proc. of I J C A I 85, 397-399.
[Yen, 1989] Yen, J. 1989, " GERT1S: A Dempster-Shafer approach to diagnosing hierarchical hypotheses ", C A C M 5 vol. 32, 1989, pp.573-585.
[ G u a n and Lesser, 1987a] G u a n , J. W. ; Lesser, V. R. 1987a, " T h e c o m p u t a t i o n a l formulae of evidence comb i n a t i o n scheme in a hierarchical hypothesis space ",
Guan and Bell
597
