## Perspectives on the Theory and Practice of Belief Functions (1990)

Venue: | International Journal of Approximate Reasoning |

Citations: | 89 - 7 self |

### BibTeX

@ARTICLE{Shafer90perspectiveson,

author = {Glenn Shafer},

title = {Perspectives on the Theory and Practice of Belief Functions},

journal = {International Journal of Approximate Reasoning},

year = {1990},

volume = {4},

pages = {323--362}

}

### Years of Citing Articles

### OpenURL

### Abstract

The theory of belief functions provides one way to use mathematical probability in subjective judgment. It is a generalization of the Bayesian theory of subjective probability. When we use the Bayesian theory to quantify judgments about a question, we must assign probabilities to the possible answers to that question. The theory of belief functions is more flexible; it allows us to derive degrees of belief for a question from probabilities for a related question. These degrees of belief may or may not have the mathematical properties of probabilities; how much they differ from probabilities will depend on how closely the two questions are related. Examples of what we would now call belief-function reasoning can be found in the late seventeenth and early eighteenth centuries, well before Bayesian ideas were developed. In 1689, George Hooper gave rules for combining testimony that can be recognized as special cases of Dempster's rule for combining belief functions (Shafer 1986a). Similar rules were formulated by Jakob Bernoulli in his Ars Conjectandi, published posthumously in 1713, and by Johann-Heinrich Lambert in his Neues Organon, published in 1764 (Shafer 1978). Examples of belief-function reasoning can also be found in more recent work, by authors

### Citations

7319 | Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference - Pearl - 1988 |

4058 |
Pattern Classification and Scene Analysis
- Duda, Hart
- 1973
(Show Context)
Citation Context ...hand, there has always been a role for probability in AI applications. Standard probabilistic and statistical methods have long been used in AI work in pattern recognition and learning (Nilsson 1965, =-=Duda and Hart 1973-=-), and recently probabilistic ideas and techniques have been used in expert systems (Spiegelhalter 1986). The drive to make logic a foundation for AI has faltered in recent years. At the beginning of ... |

2815 |
A Robust Layered Control System for a Mobile Robot
- Brooks
- 1986
(Show Context)
Citation Context ...abilities. In others, it is more efficient to give more reliable sources of information priority in a hierarchical system of control than to weigh the probabilities of each report from these sources (=-=Brooks 1986-=-, Cohen 1987). Since the ideal picture of probability involves frequencies, probability is most easily applied when relevant frequencies are available. Artificial intelligence is no exception to the r... |

2367 |
A mathematical theory of evidence
- Shafer
- 1976
(Show Context)
Citation Context ...f belief to that proposition and its consequences, and zero degree of belief to all other propositions. Arbitrarily complex belief functions can be built up by combining such simple belief functions (=-=Shafer 1976-=-a, p. 200), but in some cases we may want to produce complex belief functions more directly, in order to represent evidence that conveys a complex or mixed message but cannot be broken down into indep... |

1763 | Foundations of Statistics - Savage - 1954 |

1322 | Local computations with probabilities on graphical structures and their application to expert systems (with discussion - Lanritzen, Spiegelhalter - 1988 |

620 | Upper and lower probabilities induced by a multivalued mapping - Dempster - 1967 |

437 |
Random Set and Integral Geometry
- Matheron
- 1975
(Show Context)
Citation Context ...Nguyen (1985), and Shafer, Shenoy, and Mellouli (1987). It is convenient for advanced mathematical exposition, because the idea of a random subset is well established among mathematical probabilists (=-=Matheron 1975-=-). The Axiomatic Approach. Another approach is to characterize belief functions directly in terms of their mathematical properties. We simply list a set of axioms that a belief function Bel must satis... |

426 | Probability logic - Nilsson - 1986 |

387 | Fusion, Propagation, and Structuring in Belief Networks - Pearl - 1986 |

367 | Possibility theory: an approach to computerized processing of uncertainty - Dubois, Prade - 1988 |

346 |
Fuzzy sets as a basis for a theory of possibility
- Zadeh
- 1978
(Show Context)
Citation Context ...rguments such as those of Savage. It is not a task that can be disposed of by finding the right alternative calculus, such as the theory of belief functions or Zadeh's theory of possibility measures (=-=Zadeh 1978-=-). It is a task that must be dealt with in the context of each application. 3. The Basic Ideas of Belief Functions The theory of belief functions is based on two ideas: the idea of obtaining degrees o... |

331 |
Complexity of finding embedings in a k-tree
- Arnborg, Corneil, et al.
- 1987
(Show Context)
Citation Context ...e manageable tree. How to embed collections of clusters in trees in the most efficient way is the subject of a growing literature (Rose 1970, 19sBertele and Brioschi 1972, Tarjan and Yannakakis 1984, =-=Arnborg et al. 1987-=-, Mellouli 1987, Zhang 1990). 10. Implementing Belief Functions in Artificial Intelligence Belief functions have been implemented in a wide variety of expert systems, and I am not prepared to evaluate... |

326 |
Measure Theory
- Halmos
- 1950
(Show Context)
Citation Context ...r Probability. The idea of deriving minimal degrees of belief for some sets from probabilities for other sets has long been familiar in abstract probability theory in the context of “inner measures” (=-=Halmos 1950-=-) or “inner probabilities” (Neveu 1965). With attention to a few technicalities, we can relate belief functions to the idea of inner measure or inner probability. It is easiest to explain this using t... |

251 | Theory of Capacities. Annales de l’Institut Fourier 1953–1954 - Choquet |

250 | The Enterprise of Knowledge - Levi - 1980 |

236 | Sparse Distributed Memory - Kanerva - 1998 |

181 |
Nonserial dynamic programming
- Bertele, Brioschi
- 1972
(Show Context)
Citation Context ...6a,b) gives another way, which usually results in a more manageable tree. How to embed collections of clusters in trees in the most efficient way is the subject of a growing literature (Rose 1970, 19s=-=Bertele and Brioschi 1972-=-, Tarjan and Yannakakis 1984, Arnborg et al. 1987, Mellouli 1987, Zhang 1990). 10. Implementing Belief Functions in Artificial Intelligence Belief functions have been implemented in a wide variety of ... |

155 |
Hugin - a shell for building bayesian belief universes for expert systems
- Andersen, Olesen, et al.
- 1989
(Show Context)
Citation Context ...eory practically requires that the relation between evidence and questions of interest should be unique to each application. Many probabilistic systems—such as the HUGIN system for medical diagnosis (=-=Anderson et al. 1989-=-)—apply the same conditional independence structure and, for the most part, the same numerical judgments to each new case. This means relating the entire structure of the evidence in each case to the ... |

128 |
Triangulated graphs and the elimination process
- Rose
- 1970
(Show Context)
Citation Context ...his; Kong (1986a,b) gives another way, which usually results in a more manageable tree. How to embed collections of clusters in trees in the most efficient way is the subject of a growing literature (=-=Rose 1970-=-, 19sBertele and Brioschi 1972, Tarjan and Yannakakis 1984, Arnborg et al. 1987, Mellouli 1987, Zhang 1990). 10. Implementing Belief Functions in Artificial Intelligence Belief functions have been imp... |

117 | The Algebra of Probable Inference - Cox - 1961 |

109 | Probability propagation - Shafer, Shenoy - 1990 |

82 |
Bayesian and non-bayesian evidential updating
- Kyburg
- 1987
(Show Context)
Citation Context ...hem all on A, and then take lower bounds over the resulting conditional probabilities. The lower-probability approach produces weaker degrees of belief than the belief-function approach (Shafer 1981, =-=Kyburg 1987-=-). Though applying (8) to an arbitrary class of probability distributions p does not always produce a belief function, it does produce a belief function surprisingly often (Wasserman 1990). Moreover, ... |

82 | A Simple View of DempsterShafer Theory of Evidence and its Implication for the Rule of Combination. The AI Magazine
- Zadeh
- 1986
(Show Context)
Citation Context ...may also happen when normalization is required, as in case 1 above, but we cannot count on this. In general, a probability-bound interpretation of belief functions is inconsistent with normalization (=-=Zadeh 1986-=-). Probability bounds do provide another way to use the ideal picture of probability in subjective judgment. I have called this the lower-probability approach in order to distinguish it from the belie... |

72 | Clinical versus actuarial judgment - Dawes, Faust, et al. - 1989 |

67 | Computational methods for a mathematical theory of evidence - Barnett - 1981 |

67 | Allocations of probability - Shafer - 1979 |

56 |
Upper and lower probabilities generated by a random closed interval
- Dempster
- 1968
(Show Context)
Citation Context ...t of possible values for a single unknown probability is a continuous interval. Dempster's early work included a treatment of the mathematics of belief functions generated by random closed intervals (=-=Dempster 1968-=-a). This topic was also treated by Strat (1984). In general, the mathematical study of belief functions on continuous spaces will employ various regularity conditions, analogous to countable additivit... |

54 |
A Valuation-Based Language for Expert Systems
- Shenoy
(Show Context)
Citation Context ...ese include Gister (Lowrance, Garvey, and Strat 1986), Russell Almond's program (Almond 1988), DELIEF (Zarley et al. 1988), AUDITOR'S ASSISTANT (Shafer, Shenoy, and Srivastava 1988), and MacEvidence (=-=Hsia and Shenoy 1989-=-). These systems help human users build and evaluate belief networks. They require the user to make the judgments of independence that justify the network and to provide the numerical judgments of sup... |

54 |
Simple Linear Time Algorithms to Test Chordality of Graphs, Test Acyclicity of Hypergraphs, and Selectively Reduce Acyclic Hypergraphs
- Tarjan, Yannakakis
- 1984
(Show Context)
Citation Context ...ich usually results in a more manageable tree. How to embed collections of clusters in trees in the most efficient way is the subject of a growing literature (Rose 1970, 19sBertele and Brioschi 1972, =-=Tarjan and Yannakakis 1984-=-, Arnborg et al. 1987, Mellouli 1987, Zhang 1990). 10. Implementing Belief Functions in Artificial Intelligence Belief functions have been implemented in a wide variety of expert systems, and I am not... |

52 | On random sets and belief functions - Nguyen - 1978 |

50 | A framework for evidential-reasoning systems - Lowrance, Garvey, et al. - 1986 |

49 | A method for managing evidential reasoning in a hierarchical hypothesis space - GORDON, SHORTLIFFE - 1985 |

47 | Linear utility theory for belief functions - Jaffray - 1989 |

43 | Implementing Dempsters rule for Hierarchical Evidence - Shafer, Logan - 1987 |

43 |
R.: The bayesian and belief-function formalisms a general perspective for auditing
- Shafer, Srivastava
- 1990
(Show Context)
Citation Context ...pretation of belief-function degrees of belief as lower bounds over classes of probabilities (p. ix). In later articles, I have amplified, emphasized, and repeated this disavowal (Shafer 1981, 1987b, =-=Shafer and Srivastava 1990-=-). Dempster and other proponents of belief functions have seconded the disavowal (Dempster 1982, Ruspini 1987, Smets 1988). 6. The Semantics of Belief Functions As I noted in Section 4, the use of bel... |

42 | Languages and designs for probability judgment
- SHAFER, TVERSKY
- 1985
(Show Context)
Citation Context ...umber of questions that are related to the question of interest, then we may have a chance to find a successful analogy between the ideal picture and our evidence for at least one of these questions (=-=Shafer and Tversky 1985-=-). Which of these many different ways of using probability is most important in practice? Fisherian and Neyman-Pearson applications are by far the most common and most important. Bayesian applications... |

40 | Entropy and specificity in a mathematical theory of evidence - Yager - 1983 |

39 | Upper and lower probability inferences based on a sample from a finite univariate population - Dempster - 1967 |

38 | Multivariate Belief Functions and Graphical Models - Kong - 1986 |

36 |
New methods for reasoning towards posterior distributions based on sample data
- Dempster
- 1966
(Show Context)
Citation Context ...meters. These probability distributions depend, however, on prior subjective opinions as well as on the sample data (Savage 1961, Lindley 1972). 14sDempster, in his original work on belief functions (=-=Dempster 1966-=-, 1967a,b, 1968a,b, 1969, 1972), was motivated by the desire to obtain probability judgments based only on sample data, without dependence on prior subjective opinion. His work, together with later wo... |

36 | The Dempster-Shafer theory of evidence - GORDON, SHORTLIFFE - 1984 |

36 |
Towards a frequentist theory of upper and lower probability
- Walley, Fine
- 1982
(Show Context)
Citation Context ...eneral set functions. Going in one direction, this takes us to the lower-probability theory discussed in Section 5 and even to versions of lower-probability theory that generalize probability bounds (=-=Walley and Fine 1982-=-, Wasserman and Kadane 1990). Going in another direction, away from probability but towards a variety of rules of combination, we enter the vast literature on fuzzy sets (Dubois and Prade 1986, 1988).... |

33 | Uncertainty Models for Knowledge-Based Systems - Goodman, Nguyen - 1985 |

31 | Epistemic logics, probability and the calculus of evidence
- Ruspini
- 1987
(Show Context)
Citation Context ...have amplified, emphasized, and repeated this disavowal (Shafer 1981, 1987b, Shafer and Srivastava 1990). Dempster and other proponents of belief functions have seconded the disavowal (Dempster 1982, =-=Ruspini 1987-=-, Smets 1988). 6. The Semantics of Belief Functions As I noted in Section 4, the use of belief functions in practical problems requires metaphors that can guide us in relating practical problems to th... |

29 |
Non-Additive Probabilities in the Work of Bernoulli and Lambert." Archive for History of Exact Sciences
- Shafer
- 1978
(Show Context)
Citation Context ...nctions (Shafer 1986a). Similar rules were formulated by Jakob Bernoulli in his Ars Conjectandi, published posthumously in 1713, and by JohannHeinrich Lambert in his Neues Organon, published in 1764 (=-=Shafer 1978-=-). Examples of belief-function reasoning can also be found in more recent work, by authors who were unaware of the seventeenth and eighteenth century work. For example, Per Olof Ekelöf, a Swedish lega... |

28 |
A generalization of Bayesian inference (with discussion
- Dempster
- 1968
(Show Context)
Citation Context ...t of possible values for a single unknown probability is a continuous interval. Dempster's early work included a treatment of the mathematics of belief functions generated by random closed intervals (=-=Dempster 1968-=-a). This topic was also treated by Strat (1984). In general, the mathematical study of belief functions on continuous spaces will employ various regularity conditions, analogous to countable additivit... |

26 | The measurement of belief - Suppes |

25 |
Belief functions and parametric models (with commentary
- Shafer
- 1982
(Show Context)
Citation Context ...talk about causal models and conditional probabilities and populations of repetitions associated with them. But it is very hard to go beyond this talk and define these populations, even conceptually (=-=Shafer 1982-=-a). They usually do not provide a good starting point either for Bayesian or belief-function analyses. I also do not think that purely statistical problems are the most important domain of application... |

21 |
A set-theoretic view of belief functions
- Dubois, Prade
- 1986
(Show Context)
Citation Context ...as an introduction of frames intermediate between the S and T of Section 4. In other cases, the probabilistic basis of belief functions is lost, and the rationale for the generalization is not clear (=-=Dubois and Prade 1986-=-). 20sFinally, we can generalize the class of belief functions to more general set functions. Going in one direction, this takes us to the lower-probability theory discussed in Section 5 and even to v... |