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## A Unified Treatment of Uncertainties (1993)

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### Citations

8904 |
Probabilistic reasoning in intelligent systems: networks of plausible inference
- Pearl
- 1988
(Show Context)
Citation Context ... operations in situations where different types of uncertainty are involved ([31]). 2.3 The Bayesian approach The Bayesian approach for uncertain reasoning is characterized by the following features (=-=[16]-=-): 1. probability is interpreted as degree of belief, based on available evidence; 2. current knowledge is represented by a (first-order, real-valued) probability distribution on a proposition space; ... |

6240 |
Fuzzy sets
- Zadeh
- 1965
(Show Context)
Citation Context ...nting the uncertainty of a proposition by a real number in [0, 1]. To simplify our discussion, let's assume the proposition have the form "b is A", so the number is the grade of membership o=-=f b in A ([35]-=-). The problem of using such an approach in IRS is: fuzziness is not properly analyzed and interpreted in fuzzy logic (see [31] for a detailed discussion). According to Zadeh, membership functions are... |

3134 |
A Mathematical Theory of Evidence
- Shafer
- 1976
(Show Context)
Citation Context ...ion and a wide application domain. However, it cannot be used by IRS, for the following reasons: ffl As in the case of fuzziness, how the degree of belief is related to weight of evidence is unclear (=-=[21]-=-). ffl Due to insufficient knowledge and resources, it is usually impossible for IRS to maintain a consistent probability assignment on its knowledge base ([9]). ffl As discussed in [28], conditionali... |

2584 |
Judgment under uncertainty: heuristics and biases
- Tversky, Kahneman
- 1974
(Show Context)
Citation Context ...hey are "reasonable errors" in the sense that they are similar to human mistakes under the same situation (for example, some phenomena happened in NARS are like what Tversky and Kahneman dis=-=cussed in [25] as "-=-heuristics and biases"). Therefore, NARS is not only proposed as an engineering model for solving certain practical problems under certain situations, but also as a cognitive model to explain int... |

1774 | Some philosophical problems from the standpoint of artificial intelligence.
- McCarthy, Hayes
- 1969
(Show Context)
Citation Context ...rigin of the uncertainty will be lost, and it may lead to a false sense of accuracy. Moreover, it is hard to find a natural mapping between the numbers and the verbal expressions in natural language (=-=[14, 24, 27]-=-). I'll discuss two typical non-numerical approaches: Non-Monotonic Logics and Endorsement Theory. There are several formal systems within the category of Non-Monotonic Logics. Though built differentl... |

786 |
Family resemblances: Studies in the internal structure of categories.
- Rosch, Mervis
- 1975
(Show Context)
Citation Context ..." (oldest man, biggest bird). The second type fuzziness mainly happens with nouns and verbs, where the membership is a similarity between an instance to be judged and a prototype or a known insta=-=nce ([20]), so memb-=-ership measurement is reduced to similarity measurement. In NARS, the inheritance relation is exactly what usually referred as asymmetric similarity, that is, "S inherit P 's property". The ... |

609 |
Fuzzy sets as a basis for theory of possibility, Fuzzy
- Zadeh
- 1978
(Show Context)
Citation Context ...o types of approaches that can be called "fuzzy": one is using linguistic variables to represent uncertainty, and the other is using grade of membership (or its variations, such as possibili=-=ty, as in [37]-=-) to do it. The former falls into the category of non-numerical approaches, so the previous discussion applies, that is, though such an approach may work well for the communication language, it is not... |

263 |
On the internal structure of perceptual and semantic categories. In
- Rosch
- 1973
(Show Context)
Citation Context ... open system, the notion of prototypes and natural kinds have to be defined and maintained according to quantitative information. Concretely, prototype or default are based on the "central tenden=-=cy" ([19]), but wit-=-h constantly coming evidence, whether a tendency is "central" is usually a matter of degree. On the other hand, all knowledge is revisible, but with different sensitivity or stability, which... |

149 | Change in View.
- Harman
- 1986
(Show Context)
Citation Context ...lated to weight of evidence is unclear ([21]). ffl Due to insufficient knowledge and resources, it is usually impossible for IRS to maintain a consistent probability assignment on its knowledge base (=-=[9]-=-). ffl As discussed in [28], conditionalization cannot be referred as a general way to symmetrically combine evidence from different sources. ffl With all the efforts to improve its efficiency, the re... |

131 |
Fuzzy Sets and Systems,
- Dubois, Prade
- 1980
(Show Context)
Citation Context ...ew interpretation of fuzziness (see [31] for a detailed discussion). Here I only discuss how two simple 2 For a comparison, in fuzzy logic the set may be translated into f0, 0.25, 0.5, 0.75, 1g as in =-=[4]-=-, or into five 4-tuples as in [3]. types of fuzziness are interpreted and represented in NARS. The first type of fuzziness mainly happens with adjectives and adverbs, and appears in sentences with the... |

99 |
Nonmonotonic Reasoning',
- Reiter
- 1987
(Show Context)
Citation Context ... accepted. 3. When there are competing guesses generated by different default rules, there is no universal way to make the choice. This is the so-called 'Multiple extensions problem". Reiter wrot=-=e in [18]: "No-=-nmonotonic reasoning is necessary precisely because the information associated with such settings requires that certain conventions be respected." Since IRS is defined as a system that open to al... |

97 |
The logical foundations of statistical Inference.
- Kyburg
- 1974
(Show Context)
Citation Context ... and it is logical in the sense that it is interpreted as a logical relation between a judgment and available evidence. Therefore, it is related to all the three major interpretations of probability (=-=[7, 10]-=-), but identical to none of them. NARS processes both extensional factors and intensional factors by which a proposition become uncertain, while traditionally probability theory is used for extensions... |

92 | The Estimation of Probabilities.
- Good
- 1965
(Show Context)
Citation Context ... and it is logical in the sense that it is interpreted as a logical relation between a judgment and available evidence. Therefore, it is related to all the three major interpretations of probability (=-=[7, 10]-=-), but identical to none of them. NARS processes both extensional factors and intensional factors by which a proposition become uncertain, while traditionally probability theory is used for extensions... |

90 |
Bayesian and non-Bayesian evidential updating
- Kyburg
- 1987
(Show Context)
Citation Context ...sure ignorance is to use an interval, rather than a point, to represent the probability of a proposition, and interpret the interval as the lower bound and upper bound of the "objective probabili=-=ty" ([2, 8, 11, 13, 33]-=-). In this way, ignorance can be represented by the width of the interval. When the system know nothing about a proposition, the interval is [0, 1], that is, the objective probability can be anywhere;... |

74 |
A fuzzy-set-theoretic interpretation of linguistic hedges,”
- Zadeh
- 1972
(Show Context)
Citation Context ...(see [31] for a detailed discussion). According to Zadeh, membership functions are subjective and context-dependent, therefore, there is no general method to determine them by experiment or analysis (=-=[36]-=-). One problem caused by this attitude to fuzziness is: the operations, typically the max and min functions for union and interjection of fuzzy sets, lack a cognitive justification. Such an approach m... |

51 |
The Bayesian and Belief-Function Formalisms: A General Perspective for Auditing
- Shafer, Srivastava
- 1990
(Show Context)
Citation Context ...r quantities is necessary to represent the uncertainty of f and c. As mentioned previously, the intuition behind confidence is similar to whose of Yager's credibility ([34]) and Shafer's reliability (=-=[22]-=-). What differs confidence from them is its explicitly defined relations with the other uncertainty measurement, such as weight of evidence, ignorance, and the range the frequency will be in the near ... |

45 | A theory of higher order probabilities.
- Gaifman
- 1986
(Show Context)
Citation Context ...uncertain, we can measure the uncertainty by assigning a probability q to it. To the original proposition A, q is its second-order probability. Mathematically, this can been done, and have been done (=-=[5, 6, 15]-=-), but its interpretation and practical benefit is doubtful ([12, 16]). From the point of view of IRS, we can (at least) find the following two reasons that are against this approach: ffl Though secon... |

45 |
On the justification of Dempster’s rule of combination
- Voorbraak
- 1991
(Show Context)
Citation Context ... due to insufficient knowledge and resources. ffl Dempster's rule need unrealistic assumptions about the independence of the evidence to be combined, at least under the message-coding interpretation (=-=[26]-=-). ffl Dempster-Shafer theory lack well-developed inference rules ([1]). This is partly caused by the above problem, because Dempster-Shafer theory cannot be treated as a generalization of probability... |

40 |
Comparing the calibration and coherence of numerical and verbal probability scales
- Wallsten, Budescu, et al.
- 1993
(Show Context)
Citation Context ...rigin of the uncertainty will be lost, and it may lead to a false sense of accuracy. Moreover, it is hard to find a natural mapping between the numbers and the verbal expressions in natural language (=-=[14, 24, 27]-=-). I'll discuss two typical non-numerical approaches: Non-Monotonic Logics and Endorsement Theory. There are several formal systems within the category of Non-Monotonic Logics. Though built differentl... |

33 | From inheritance relation to nonaxiomatic logic
- Wang
- 1994
(Show Context)
Citation Context ...o the source of uncertainty. To solve such problems, some other method can be used as a supplement, rather than a replacement, of the numerical approach. For example, such a mechanism is described in =-=[30]-=- for detecting correlated evidence in revision. Though non-numerical approach is unsuitable in IRS, their motivations need to be respected, that is, the numerical measurement should have a natural int... |

28 | On the validity of Dempster-Shafer theory - Dezert, Wang, et al. - 2012 |

25 |
Summarizing and propagating uncertain information with triangular norms
- Bonissone
- 1987
(Show Context)
Citation Context ...easurements form a consistent foundation for the NARS project. 4.5 The interval approach The interval form of truth value in NARS shares similar intuitions with the "probability interval" ap=-=proaches ([2, 11, 33]-=-). For example, ignorance can be represented by the width of the interval. However, in NARS the interval is defined as the range the frequency will be in the near future, rather than in the infinite f... |

23 |
An inequality paradigm for prob abilistic knowledge.
- Grosof
- 1986
(Show Context)
Citation Context ...sure ignorance is to use an interval, rather than a point, to represent the probability of a proposition, and interpret the interval as the lower bound and upper bound of the "objective probabili=-=ty" ([2, 8, 11, 13, 33]-=-). In this way, ignorance can be represented by the width of the interval. When the system know nothing about a proposition, the interval is [0, 1], that is, the objective probability can be anywhere;... |

21 | Belief revision in probability theory
- Wang
- 1993
(Show Context)
Citation Context ...e is unclear ([21]). ffl Due to insufficient knowledge and resources, it is usually impossible for IRS to maintain a consistent probability assignment on its knowledge base ([9]). ffl As discussed in =-=[28]-=-, conditionalization cannot be referred as a general way to symmetrically combine evidence from different sources. ffl With all the efforts to improve its efficiency, the resources expense of Bayesian... |

21 |
A methodology for uncertainty in knowledgebased systems, volume 419
- Weichselberger, Pöhlmann
- 1990
(Show Context)
Citation Context ...sure ignorance is to use an interval, rather than a point, to represent the probability of a proposition, and interpret the interval as the lower bound and upper bound of the "objective probabili=-=ty" ([2, 8, 11, 13, 33]-=-). In this way, ignorance can be represented by the width of the interval. When the system know nothing about a proposition, the interval is [0, 1], that is, the objective probability can be anywhere;... |

17 |
Quantitative meanings of verbal probability expressions
- Reagan, Mosteller, et al.
- 1989
(Show Context)
Citation Context ...0.6, 0.8], and [0.8, 1], respectively (similar methods are discussed in [27]). 2 It is also possible to determine the meanings of everyday verbal uncertainty expressions by psychological experiments (=-=[17, 27]-=-). These two forms don't have one-to-one mappings with the three forms defined above, but they are good for communication in the situations where naturalness and simplicity are weighed more than accur... |

12 |
Varieties of Ignorance and the Need for Well-Founded Theories',
- Smets
- 1991
(Show Context)
Citation Context ...easoning system defined as above (henceforth IRS), there are all kinds of uncertainties in the system, such as randomness, fuzziness, ignorance, imprecision, incompleteness, inconsistence, and so on (=-=[23]-=-). Therefore, uncertainty should be taken into consideration when the above operations are carried out. In this paper, I'll concentrate on the representation and interpretation of uncertainty. By repr... |

12 | Non-axiomatic reasoning system (version 2.2
- Wang
- 1993
(Show Context)
Citation Context ...versally accepted definition for intelligent reasoning system, I want to give a definition to make it clear that what type of system I'm talking about. Why such a definition is chosen is explained in =-=[32]-=-. By reasoning system, I mean an information processing system that has the following components: a representation language which is defined by a formal grammar, and used for the internal representati... |

10 |
Higher order probabilities
- Kyburg
- 1987
(Show Context)
Citation Context ...to it. To the original proposition A, q is its second-order probability. Mathematically, this can been done, and have been done ([5, 6, 15]), but its interpretation and practical benefit is doubtful (=-=[12, 16]-=-). From the point of view of IRS, we can (at least) find the following two reasons that are against this approach: ffl Though second order probability q can be used to represent the uncertainty in a p... |

8 |
Second order probabilities for uncertain and con icting evidence
- Paa
- 1991
(Show Context)
Citation Context ...uncertain, we can measure the uncertainty by assigning a probability q to it. To the original proposition A, q is its second-order probability. Mathematically, this can been done, and have been done (=-=[5, 6, 15]-=-), but its interpretation and practical benefit is doubtful ([12, 16]). From the point of view of IRS, we can (at least) find the following two reasons that are against this approach: ffl Though secon... |

8 |
An endorsement-based plan recognition program
- Sullivan, Cohen
- 1985
(Show Context)
Citation Context ...rigin of the uncertainty will be lost, and it may lead to a false sense of accuracy. Moreover, it is hard to find a natural mapping between the numbers and the verbal expressions in natural language (=-=[14, 24, 27]-=-). I'll discuss two typical non-numerical approaches: Non-Monotonic Logics and Endorsement Theory. There are several formal systems within the category of Non-Monotonic Logics. Though built differentl... |

7 |
Credibility discounting in the theory of approximate reasoning
- Yager
- 1991
(Show Context)
Citation Context ...re other attempts to measure ignorance by introduce a number which is at a "higher level" in certain sense, but not a "probability of probability" as defined above. For examples, Y=-=ager's credibility ([34]) and Shafer's relia-=-bility ([22]) both evaluate the uncertainty of a probability assignment, where 0 is interpreted as "unknown", rather than "impossible". However, these approaches still lack clear i... |

6 |
Metaprobability and Dempster-Shafer in evidential reasoning
- Fung, Chong
- 1986
(Show Context)
Citation Context ...uncertain, we can measure the uncertainty by assigning a probability q to it. To the original proposition A, q is its second-order probability. Mathematically, this can been done, and have been done (=-=[5, 6, 15]-=-), but its interpretation and practical benefit is doubtful ([12, 16]). From the point of view of IRS, we can (at least) find the following two reasons that are against this approach: ffl Though secon... |

4 |
Handling uncertain information
- Bhatnagar, Kanal
- 1986
(Show Context)
Citation Context ...umerical and non-numerical. The basic difference between them is: the former attach one or several numbers to each piece of knowledge to represent the degree of uncertainty, while the latter doesn't (=-=[1, 3]). Though -=-using numbers to express "degree of uncertainty" seems to be a natural idea, there are still sound objections against it: Human knowledge is usually represented in natural language, where un... |

4 |
Selecting uncertain calculi and granularity
- Bonissone, Decker
- 1986
(Show Context)
Citation Context ...umerical and non-numerical. The basic difference between them is: the former attach one or several numbers to each piece of knowledge to represent the degree of uncertainty, while the latter doesn't (=-=[1, 3]). Though -=-using numbers to express "degree of uncertainty" seems to be a natural idea, there are still sound objections against it: Human knowledge is usually represented in natural language, where un... |

3 | Interval-based decisions for reasoning systems
- Loui
- 1986
(Show Context)
Citation Context |

2 | order probabilities for uncertain and con icting evidence - Paa - 1991 |

1 | Quantitative meaningsofverbal probability expressions - Reagan, Mosteller, et al. - 1989 |

1 | On the justi cation of Dempster's rule of combination - Voorbraak - 1991 |