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## On the Use of Imprecise Probabilities in Reliability (2004)

Venue: | Quality and Reliability Engineering International |

Citations: | 11 - 5 self |

### Citations

2962 |
Robust statistics
- Huber, Ronchetti
- 2009
(Show Context)
Citation Context ...al differing concepts [2, 34]. Statistical inference based on imprecise probability offers a wide range of modelling opportunities, among these are well-known models from robust (Bayesian) statistics =-=[3, 18]-=-, which however have a different interpretation from imprecise probability perspective, with differences between upper and lower bounds for inferences fundamentally quantifying indeterminacy in the in... |

1068 |
Statistical reasoning with imprecise probabilities
- Walley
- 1991
(Show Context)
Citation Context ...erences to such concepts, and discussion from the perspective of generalized probability, see [32, 33]. One such a concept is called 'imprecise probability' [32], also known as 'interval probability' =-=[34, 35]-=-, and in recent years this has particularly been a growing area of research, as it is realized that it provides a consistent theory for generalized uncertainty quantification (which to some extend coi... |

858 |
Bayesian Networks and Decision Graphs
- Jensen, Nielsen
- 2009
(Show Context)
Citation Context ...sticians have successfully developed models allowing very many random quantities, with careful representation of dependence structures between these quantities. For example, Bayesian graphical models =-=[15, 19]-=- have proven successful for large scale applications [26, 36]. Typically, the elicitation task for such models is enormous, as it is exponential in the number of quantities included in the model and t... |

771 |
Probabilistic Networks and Expert Systems
- Cowell, Dawid, et al.
- 1999
(Show Context)
Citation Context ...sticians have successfully developed models allowing very many random quantities, with careful representation of dependence structures between these quantities. For example, Bayesian graphical models =-=[15, 19]-=- have proven successful for large scale applications [26, 36]. Typically, the elicitation task for such models is enormous, as it is exponential in the number of quantities included in the model and t... |

542 |
Statistical Models & Methods for Lifetime Data
- Lawless
- 1982
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Citation Context ...s. Although there is some overlap with the issues relating to expert judgements, some probability distributions can be assumed based on physical properties, e.g. the shape of the Weibull distribution =-=[22]-=- for the lifetime of a unit might be assumed a known constant based on knowledge of the ageing process of a unit, e.g. shape parameter 1, giving the exponential distribution, relates to no ageing effe... |

321 |
Statistical Methods for Reliability Data
- Meeker, Escobar
- 1998
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Citation Context ...y taking observations at points where large imprecision may prevent clear decisions to be taken. In reliability, such research could for example be relevant for design of accelerated life tests (e.g. =-=[24]-=-), where imprecision might realistically reflect indeterminacy with regard to modelling of the acceleration. There is no doubt that actual applications of imprecise probability methods in reliability ... |

131 |
Credal networks
- Cozman
(Show Context)
Citation Context ...time constraints, so methods are needed that can leave many probabilities unspecified, or at best only partially specified. Such methods have been suggested during the past few years, see e.g. Cozman =-=[16]-=-, and are being developed further, where main problems involve development of fast optimisation algorithms combined with the algorithms for learning in such models. Studying imprecision in such models... |

79 |
Experts in Uncertainty,
- Cooke
- 1991
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Citation Context ...n is often highly subjective, even if there is a fair amount of relevant data available there are still many possible distributions which are equally well supported by the data. Several authors (e.g. =-=[1, 5, 25]-=-) have suggested that data are often sparse in reliability applications, if available at all, so quantification of uncertain aspects should be based on subjective information, i.e. experts' judgements... |

67 |
Markov Chain Monte Carlo,
- Gamerman
- 1997
(Show Context)
Citation Context ...cent work by Utkin and co-authors has provided great progress on this aspect, yet much more needs to be done. For example, modern Bayesian methods often require simulation-based computational methods =-=[17]-=-, and it is not immediately clear, from theoretical perspective, how such methods could be generalized to allow imprecise probabilities, let alone how to actually develop algorithms for such computati... |

64 |
Elementare Grundbegriffe einer Allgemeineren Wahrscheinlichkeitsrechnung I. Intervallwahrscheinlichkeit als Umfassendes Konzept. (Elementary Foundations of a Generalized Probability Calculus. I: Interval Probability as a Comprehensive Framework.)
- Weichselberger
- 2001
(Show Context)
Citation Context ...y random quantities, with careful representation of dependence structures between these quantities. For example, Bayesian graphical models [15, 19] have proven successful for large scale applications =-=[26, 36]-=-. Typically, the elicitation task for such models is enormous, as it is exponential in the number of quantities included in the model and these models normally add unobservable random quantities ('par... |

50 |
Robust Bayesian analysis: Sensitivity to the prior.
- Berger
- 1990
(Show Context)
Citation Context ...al differing concepts [2, 34]. Statistical inference based on imprecise probability offers a wide range of modelling opportunities, among these are well-known models from robust (Bayesian) statistics =-=[3, 18]-=-, which however have a different interpretation from imprecise probability perspective, with differences between upper and lower bounds for inferences fundamentally quantifying indeterminacy in the in... |

36 |
Nonparametric predictive inference and interval probability.
- Augustin, Coolen
- 2004
(Show Context)
Citation Context ...onal probability, which e.g. plays a crucial role in Bayesian statistics, does not have a unique generalization to imprecise probability, where conditioning can be done via several differing concepts =-=[2, 34]-=-. Statistical inference based on imprecise probability offers a wide range of modelling opportunities, among these are well-known models from robust (Bayesian) statistics [3, 18], which however have a... |

22 |
Interval-valued finite Markov chains,
- Kozine, Utkin
- 2002
(Show Context)
Citation Context ...ies, let alone how to actually develop algorithms for such computations. Interesting theoretical results on Markov chains with imprecise probabilities have recently been presented by Kozine and Utkin =-=[21]-=-, and these may well provide a step in the direction towards such simulation-based computational methods. In addition, for many reliability applications Markov models have been used successfully, e.g.... |

14 |
New reliability models based on imprecise probabilities, World Scientific:
- Utkin, Gurov
- 2001
(Show Context)
Citation Context ...presented by Coolen [7]. 3.3 Systems In the previous subsection we briefly mentioned reliability results for systems corresponding to the distribution classes by Utkin and Gurov [30]. Utkin and Gurov =-=[28, 29]-=-, Kozine and Filimonov [20], and Utkin and Kozine [31] have presented a variety of useful results on system reliability with imprecise probability, mostly by 0.8 0.6 0.4 0.2 Upper bound Lower bound ..... |

13 |
Quantifying uncertainty under a predictive, epistemic approach to risk analysis,”
- Apeland, Aven, et al.
- 2002
(Show Context)
Citation Context ...n is often highly subjective, even if there is a fair amount of relevant data available there are still many possible distributions which are equally well supported by the data. Several authors (e.g. =-=[1, 5, 25]-=-) have suggested that data are often sparse in reliability applications, if available at all, so quantification of uncertain aspects should be based on subjective information, i.e. experts' judgements... |

13 |
Bayesian reliability analysis with imprecise prior probabilities.
- Coolen, Newby
- 1994
(Show Context)
Citation Context ...nd defining separate such classes on each interval. Nonparametric predictive methods [9] enable lifetime inferences with a minimum of subjective assumptions added to data. For example, Coolen and Yan =-=[12]-=- present predictive upper and lower survival functions for a future lifetime Tn+, based on observations of n such previous lifetimes, which may include right-censored observations. This gives an alter... |

13 |
Imprecise reliability of general structures,
- Utkin, Gurov
- 1999
(Show Context)
Citation Context ...presented by Coolen [7]. 3.3 Systems In the previous subsection we briefly mentioned reliability results for systems corresponding to the distribution classes by Utkin and Gurov [30]. Utkin and Gurov =-=[28, 29]-=-, Kozine and Filimonov [20], and Utkin and Kozine [31] have presented a variety of useful results on system reliability with imprecise probability, mostly by 0.8 0.6 0.4 0.2 Upper bound Lower bound ..... |

9 |
Bayesian nonparametric survival analysis
- Berliner, Hill
- 1988
(Show Context)
Citation Context ...hich may include right-censored observations. This gives an alternative to the well-known product-limit estimate by Kaplan and Meier (see e.g. [22]), and to the predictive method by Berliner and Hill =-=[4]-=-, which is based on similar foundations as the Coolen-Yan method, but cannot deal with exact censoring information. For example, the following data are part of an example discussed by Coolen and Yan [... |

7 | Imprecise reliability for some new lifetime distribution classes
- Utkin, Gurov
(Show Context)
Citation Context ...10]. This is generally discussed by Walley [32]. A particularly interesting class of lifetime distributions, leading to imprecise reliability inferences, has recently been proposed by Utkin and Gurov =-=[30]-=-. For a non-negative random lifetime T, with cumulative hazard function H(t) = fot h(x)dx, where h(.) is the hazard rate, they define the class of distributions 7-/(r, s), for 0 _, as those distributi... |

6 |
Managing the uncertainties of software testing: a Bayesian approach
- Rees, Coolen, et al.
- 2001
(Show Context)
Citation Context ...y random quantities, with careful representation of dependence structures between these quantities. For example, Bayesian graphical models [15, 19] have proven successful for large scale applications =-=[26, 36]-=-. Typically, the elicitation task for such models is enormous, as it is exponential in the number of quantities included in the model and these models normally add unobservable random quantities ('par... |

5 |
Bayesian enhanced strategic decision making for reliability
- Percy
(Show Context)
Citation Context ...n is often highly subjective, even if there is a fair amount of relevant data available there are still many possible distributions which are equally well supported by the data. Several authors (e.g. =-=[1, 5, 25]-=-) have suggested that data are often sparse in reliability applications, if available at all, so quantification of uncertain aspects should be based on subjective information, i.e. experts' judgements... |

4 |
2003a, ‘Nonparametric predictive inference for grouped lifetime data’, Reliability Engineering and System Safety 80
- Coolen, Yan
(Show Context)
Citation Context ...est systems. We address this issue in Subsection 3.3. There are more reasons suggesting benefits of using imprecise probability in reliability, e.g. the nature of data collection such as grouped data =-=[7, 13]-=-, but we reckon that the ones we discuss explicitly in this paper are strong enough to emphasize the possible benefits of using imprecise probabilities in reliability. In Subsection 3.4 we briefly add... |

4 |
Bayesian Poisson models for the graphical combination of dependent expert information, J.R.Statis
- Smith, Faria
- 2000
(Show Context)
Citation Context ...imprecise probabilities may not take into account that the information of some of the experts may not be independent. For precise probabilities some progress has been made on dealing with this aspect =-=[27]-=-, it may well be that imprecise probabilities are again better suited to deal with this but, as far as we are aware, this has not yet been reported in the literature. Imprecise probabilities also natu... |

3 |
Statistical modelling of experts opinions using imprecise probabilities.
- Coolen
- 1994
(Show Context)
Citation Context ...link random quantities on future observations to past observations, which can be achieved by a post-data assumption related to exchangeability, leading to so-called nonparametric predictive inference =-=[2, 9, 14]-=-, brief examples of such inference are presented in Subsections 3.2 and 3.4. Finally, with ever growing computational powers, statisticians have successfully developed models allowing very many random... |

3 | Nonparametric predictive inference for age replacement with a renewal argument. Quality and Reliability Engineering
- Coolen-Schrijner, Coolen
- 2004
(Show Context)
Citation Context ...link random quantities on future observations to past observations, which can be achieved by a post-data assumption related to exchangeability, leading to so-called nonparametric predictive inference =-=[2, 9, 14]-=-, brief examples of such inference are presented in Subsections 3.2 and 3.4. Finally, with ever growing computational powers, statisticians have successfully developed models allowing very many random... |

3 |
Filimonov (2000). Imprecise reliabilities: experiences and advances
- Kozine
(Show Context)
Citation Context ...s, but upper probability 0.7 for the same event, which is not acceptable for obvious reasons. Such combination rules are discussed in more detail by, for example, Walley [32] and Kozine and Filimonov =-=[20]-=-. Coolen [6] also discusses the possibility of taking weighted averages of individual lower (upper) probabilities as the combined lower (upper) probability, where weights could reflect the expertise o... |

3 |
Discussion of Walley
- Lindley
- 1996
(Show Context)
Citation Context ...oners to start using the new concepts and methods. It is amusing that quite many mathematicians and statisticians have expressed fairly strong views against the use of imprecise probability, see e.g. =-=[23]-=-, whereas the onus of justification is clearly on the side of using the far more restrictive precise probabilities, which are a special case of imprecise probability. There are several possible interp... |

2 |
Computing system reliability given interval-valued characteristics
- Utkin, Kozine
- 2002
(Show Context)
Citation Context ...ction we briefly mentioned reliability results for systems corresponding to the distribution classes by Utkin and Gurov [30]. Utkin and Gurov [28, 29], Kozine and Filimonov [20], and Utkin and Kozine =-=[31]-=- have presented a variety of useful results on system reliability with imprecise probability, mostly by 0.8 0.6 0.4 0.2 Upper bound Lower bound .... BH ......... KM 0 250 500 750 1000 1250 1500 Figure... |