## Propagating Imprecise Probabilities In Bayesian Networks (1996)

Venue: | Artificial Intelligence |

Citations: | 16 - 5 self |

### BibTeX

@ARTICLE{Kleiter96propagatingimprecise,

author = {Gernot D. Kleiter},

title = {Propagating Imprecise Probabilities In Bayesian Networks},

journal = {Artificial Intelligence},

year = {1996},

volume = {88},

pages = {143--161}

}

### OpenURL

### Abstract

Often experts are incapable of providing `exact' probabilities; likewise, samples on which the probabilities in networks are based must often be small and preliminary. In such cases the probabilities in the networks are imprecise. The imprecision can be handled by second-order probability distributions. It is convenient to use beta or Dirichlet distributions to express the uncertainty about probabilities. The problem of how to propagate point probabilities in a Bayesian network now is transformed into the problem of how to propagate Dirichlet distributions in Bayesian networks. It is shown that the propagation of Dirichlet distributions in Bayesian networks with incomplete data results in a system of probability mixtures of beta-binomial and Dirichlet distributions. Approximate first order probabilities and their second order probability density functions are be obtained by stochastic simulation. A number of properties of the propagation of imprecise probabilities are discuss...

### Citations

7698 |
Probabilistic reasoning in intelligent systems: networks of plausible inference
- Pearl
- 1988
(Show Context)
Citation Context ...ability distributions are organized in tables and attached to the nodes of the graph. The tables are not `visible' in the graphical representation. Consider the example shown in Figure 1 and Table 1. =-=[7, 37, 34]-=-. The network in Figure 1 represents the dependencies in a graphical model. The nodes A to E represent clinical absent/present variables like diseases, test results, or symptoms. Table 1 contains the ... |

1776 |
Statistical Analysis With Missing Data
- RJA, DB
- 2002
(Show Context)
Citation Context ...ithin the same family of probability distributions. What shall we do when the cell counts and the marginals do not add up? In inferential statistics this case occurs when some of the data are missing =-=[29]-=-. The treatment of the missing data can be related to the complete case by calculating the weighted averages of complete solutions, where averaging is performed over the space of the missing data. Thi... |

1227 |
Bayesian theory
- Bernardo, Smith
- 2000
(Show Context)
Citation Context ...nd second-order distributions [16, 21, 25, 26, 39, 40, 33]. A tutorial is provided in [17]. We treat probabilities that are not known precisely in the same way they are treated in Bayesian statistics =-=[3]-=-, as uncertain quantities to which a (second order) probability density function is attached. The distributions expresses the imprecision. If little is known about the uncertain quantity, the distribu... |

983 | Learning bayesian networks: the combination of knowledge and statistical data
- Heckerman, Geiger, et al.
- 1995
(Show Context)
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940 |
Sampling-based approaches to calculating marginal densities
- Gelfand, Smith
- 1990
(Show Context)
Citation Context ... hX 1 ; X 2 ; : : : ; Xn i is an n- dimensional weight table, we denote the marginal along the subset fZ 1 ; : : : ; Zm g ` fX 1 ; : : : ; Xn g by hX 1 ; X 2 ; : : : ; Xn i #fZ1 ;:::;Z mg . We follow =-=[14]-=- and denote a probability density function (pdf) by brackets. Joint, conditional, and marginal distributions are written as [X; Y ], [XjY ], and [X], respectively. The product of densities is denoted ... |

863 |
Statistical Reasoning with Imprecise Probabilities
- Walley
- 1991
(Show Context)
Citation Context ...ayesian network cannot be considered to be precise point values. In the literature, several proposals have been made how to handle imprecision in dependency structures, such as lower and upper bounds =-=[9, 12, 44]-=-, propagation of variances [8, 31, 38], and second-order distributions [16, 21, 25, 26, 39, 40, 33]. A tutorial is provided in [17]. We treat probabilities that are not known precisely in the same way... |

573 | Discrete Multivariate Analysis: Theory and Practice, Cambridge and
- Fienberg, Holland
- 1977
(Show Context)
Citation Context ...functions g of a random variable X we have E[g(X)] = g(E[X]). This is not true if g is not linear. In many cases, though, the mean and the variance of g(X) can be approximated by the ffi method [32], =-=[4]-=-: Definition 6 (ffi rule) Let (X 1 ; : : : ; Xn ) be independent random variables with means (E 1 ; : : : ; En ) and variances (V 1 ; : : : ; Vn ). If f(X 1 ; : : : ; Xn ) is a function of the variabl... |

481 |
Graphical Models in Applied Multivariate Statistics
- Whittaker
- 1990
(Show Context)
Citation Context ...ike medical diagnosis, prediction, or explanation are special cases of propagating probabilities in a Bayesian network. Bayesian belief networks belong to the class of graphical probabilistic models (=-=[6, 11, 15, 30, 42]-=-; for tutorials and related work on uncertainty in artificial intelligence see the http://www.auai.org page and the references given there). Usually, the probabilities in Bayesian networks are treated... |

470 |
Maximum Likelihood Estimation from Incomplete Data via the EM Algorithm
- Dempster, Laird, et al.
- 1977
(Show Context)
Citation Context ...missing data. This results in probability mixtures [26]. For computational purposes the solutions are too complicated. Incomplete data are usually analyzed by expectation maximization (EM) algorithms =-=[10]-=-. EM is an iterative procedure providing maximum likelihood estimates in the presence of missing data. The precision (variance) of the estimates can be approximated [29]. EM has several disadvantages.... |

447 |
A Treatise on Probability
- Keynes
- 1963
(Show Context)
Citation Context ...e about the point probabilitys1 =( 1 +s2 ).s1 is the weight in favor of an event, a proposition, or a hypothesis, ands2 the weight against it. Weights of evidence were extensively discussed by Keynes =-=[20]-=-. The beta or Dirichlet distributions implement a system of second order pdf s on the probability parameters underlying the network. If a node X has no parents, then the pdf is a marginal distribution... |

336 | Learning Bayesian Networks: The
- Heckerman, Geiger, et al.
- 1995
(Show Context)
Citation Context ...sion in dependency structures, such as lower and upper bounds [9, 12, 44], propagation of variances [8, 31, 38], and second-order distributions [16, 21, 25, 26, 39, 40, 33]. A tutorial is provided in =-=[17]-=-. We treat probabilities that are not known precisely in the same way they are treated in Bayesian statistics [3], as uncertain quantities to which a (second order) probability density function is att... |

284 |
The statistical analysis of compositional data
- Aitchison
- 1982
(Show Context)
Citation Context ...t 8 and in the second case the missing data can be predicted probabilistically. In statistics the probability distribution of a future sample given an observed one is called a predictive distribution =-=[2, 1]-=-. It may be shown that the predictive distribution in both our cases is a Polya-Eggenberger distribution [19]. The member distribution turns out to be a probability mixture of beta distributions where... |

254 | Operations for Learning with Graphical Models
- Buntine
- 1994
(Show Context)
Citation Context ... a parent together with its natural child in a plate. A plate is a rectangle drawn around a set of nodes with a repetition number N written in its left lower corner. Plates were introduced by Buntine =-=[5]-=- (who gives credit to Spiegelhalter). Plates indicate a data set of the same kind. The set of nodes is instantiated simultaneously or repeatedly by N observations. A plate shows a complete set of N da... |

206 | Learning Bayesian belief networks: An approach based on the MDL principle. Computational Intelligence 10:269–293
- Lam, Bacchus
- 1994
(Show Context)
Citation Context ...hould therefore strive to keep the inferential systems simple. The trade-off between complexity and accuracy has recently been studied in Bayesian networks by the minimum description length criterion =-=[27]-=-. From our viewpoint the trade-off may be illustrated by an example that at first looks terribly counter-intuitive. Can more data make us more uncertain about our inferences? Consider the following pr... |

202 |
Bayesian Analysis in Expert Systems
- Spiegelhalter, Dawid, et al.
- 1993
(Show Context)
Citation Context ..., several proposals have been made how to handle imprecision in dependency structures, such as lower and upper bounds [9, 12, 44], propagation of variances [8, 31, 38], and second-order distributions =-=[16, 21, 25, 26, 39, 40, 33]-=-. A tutorial is provided in [17]. We treat probabilities that are not known precisely in the same way they are treated in Bayesian statistics [3], as uncertain quantities to which a (second order) pro... |

180 | A Guide to the Literature on Learning Probabilistic Networks from Data
- Buntine
- 1996
(Show Context)
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178 | N.: Graphical Models for Association Between Variables, Some of Which are Qualitative and Some Quantitative - Lauritzen, Wermuth - 1989 |

175 |
Introduction to Graphical Modelling
- Edwards
- 2000
(Show Context)
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172 |
Urn Models and Their Application
- Johnson, Kotz
- 1977
(Show Context)
Citation Context ...istribution of a future sample given an observed one is called a predictive distribution [2, 1]. It may be shown that the predictive distribution in both our cases is a Polya-Eggenberger distribution =-=[19]-=-. The member distribution turns out to be a probability mixture of beta distributions where the mixing weights are Polya-Eggenberger probabilities [26]: Theorem 3 (Non-Natural neighbors) If 1. ff, fi ... |

155 |
Probabilistic Reasoning in Expert Systems
- Neapolitan
- 1990
(Show Context)
Citation Context |

71 |
BUGS: A program to perform Bayesian inference using Gibbs sampling
- Thomas, Spiegelhalter, et al.
- 1992
(Show Context)
Citation Context ...works was proposed by Pearl [34]. Hrycej [18] has shown that the stochastic simulation in a Bayesian network is a special case of Gibbs sampling. It has extensively been employed to Bayesian networks =-=[13, 36, 41, 40]-=-. At the start each instantiated node is clamped to its constant value and each noninstantiated node is set to an arbitrary value. Then, iteratively, the following steps are performed: 1. Select a non... |

39 |
Probabilistic reasoning in predictive expert systems
- Spiegelhalter
- 1986
(Show Context)
Citation Context ...ability distributions are organized in tables and attached to the nodes of the graph. The tables are not `visible' in the graphical representation. Consider the example shown in Figure 1 and Table 1. =-=[7, 37, 34]-=-. The network in Figure 1 represents the dependencies in a graphical model. The nodes A to E represent clinical absent/present variables like diseases, test results, or symptoms. Table 1 contains the ... |

26 |
A randomized approximation algorithm for probabilistic inference on Bayesian belief networks
- Chavez, Cooper
- 1990
(Show Context)
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23 |
Gibbs sampling in Bayesian networks
- Hrycej
- 1990
(Show Context)
Citation Context ...xtures becomes cumbersom. Below, we employ an approximation based on the ffi method. 9 5 Stochastic simulation The use of stochastic simulation in Bayesian networks was proposed by Pearl [34]. Hrycej =-=[18]-=- has shown that the stochastic simulation in a Bayesian network is a special case of Gibbs sampling. It has extensively been employed to Bayesian networks [13, 36, 41, 40]. At the start each instantia... |

23 |
Assessment, criticism and improvement of imprecise subjective probabilities for a medical expert system
- Spiegelhalter, Franklin, et al.
- 1989
(Show Context)
Citation Context ..., several proposals have been made how to handle imprecision in dependency structures, such as lower and upper bounds [9, 12, 44], propagation of variances [8, 31, 38], and second-order distributions =-=[16, 21, 25, 26, 39, 40, 33]-=-. A tutorial is provided in [17]. We treat probabilities that are not known precisely in the same way they are treated in Bayesian statistics [3], as uncertain quantities to which a (second order) pro... |

17 |
Interval influence diagrams
- Fertig, Breese
- 1990
(Show Context)
Citation Context ...ayesian network cannot be considered to be precise point values. In the literature, several proposals have been made how to handle imprecision in dependency structures, such as lower and upper bounds =-=[9, 12, 44]-=-, propagation of variances [8, 31, 38], and second-order distributions [16, 21, 25, 26, 39, 40, 33]. A tutorial is provided in [17]. We treat probabilities that are not known precisely in the same way... |

17 |
Natural sampling: Rationality without base rates
- Kleiter
- 1994
(Show Context)
Citation Context ... for Bayes' formula. It tells us how precise our posterior probability is. 4.2 Beta and mixed beta member distributions Let A and B be two binary random variables. We have shown the following theorem =-=[23]-=-, [26]: Theorem 1 (Natural child) If 1. ff; fi 1 ; fi 2 , andsare the probability parameters underlying the propositional variables A, BjA, Bj:A, and Ajb, respectively, and if the second order pdfs of... |

15 |
A note on the delta method. American Statistician 46:27–29
- Oehlert
- 1992
(Show Context)
Citation Context ...inear functions g of a random variable X we have E[g(X)] = g(E[X]). This is not true if g is not linear. In many cases, though, the mean and the variance of g(X) can be approximated by the ffi method =-=[32]-=-, [4]: Definition 6 (ffi rule) Let (X 1 ; : : : ; Xn ) be independent random variables with means (E 1 ; : : : ; En ) and variances (V 1 ; : : : ; Vn ). If f(X 1 ; : : : ; Xn ) is a function of the va... |

8 |
Second order probabilities for uncertain and con icting evidence
- Paa
- 1991
(Show Context)
Citation Context |

7 |
A unified approach to imprecision and sensitivity of beliefs in expert systems
- Spiegelhalter
- 1989
(Show Context)
Citation Context ...o be precise point values. In the literature, several proposals have been made how to handle imprecision in dependency structures, such as lower and upper bounds [9, 12, 44], propagation of variances =-=[8, 31, 38]-=-, and second-order distributions [16, 21, 25, 26, 39, 40, 33]. A tutorial is provided in [17]. We treat probabilities that are not known precisely in the same way they are treated in Bayesian statisti... |

6 |
Statistical Prediction Analysis Cambridge
- Aitchison, Dunsmore
- 1975
(Show Context)
Citation Context ...t 8 and in the second case the missing data can be predicted probabilistically. In statistics the probability distribution of a future sample given an observed one is called a predictive distribution =-=[2, 1]-=-. It may be shown that the predictive distribution in both our cases is a Polya-Eggenberger distribution [19]. The member distribution turns out to be a probability mixture of beta distributions where... |

5 |
An implementation of a method for computing the uncertainty in inferred probabilities in belief networks
- Che, Neapolitan, et al.
- 1993
(Show Context)
Citation Context ...o be precise point values. In the literature, several proposals have been made how to handle imprecision in dependency structures, such as lower and upper bounds [9, 12, 44], propagation of variances =-=[8, 31, 38]-=-, and second-order distributions [16, 21, 25, 26, 39, 40, 33]. A tutorial is provided in [17]. We treat probabilities that are not known precisely in the same way they are treated in Bayesian statisti... |

4 |
Conditional events with vague information in expert systems
- Coletti, Gilio, et al.
- 1991
(Show Context)
Citation Context ...ayesian network cannot be considered to be precise point values. In the literature, several proposals have been made how to handle imprecision in dependency structures, such as lower and upper bounds =-=[9, 12, 44]-=-, propagation of variances [8, 31, 38], and second-order distributions [16, 21, 25, 26, 39, 40, 33]. A tutorial is provided in [17]. We treat probabilities that are not known precisely in the same way... |

4 |
A Bayesian approach to imprecision in belief nets
- Kleiter, Kardinal
- 1995
(Show Context)
Citation Context |

2 |
Exact and approximate algorithms and their implementations in mixed graphical models
- Gammerman, Luo, et al.
- 1995
(Show Context)
Citation Context ...works was proposed by Pearl [34]. Hrycej [18] has shown that the stochastic simulation in a Bayesian network is a special case of Gibbs sampling. It has extensively been employed to Bayesian networks =-=[13, 36, 41, 40]-=-. At the start each instantiated node is clamped to its constant value and each noninstantiated node is set to an arbitrary value. Then, iteratively, the following steps are performed: 1. Select a non... |

2 |
Expressing imprecision in probabilistic knowledge
- Kleiter
- 1994
(Show Context)
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1 |
Bayesian Diagnosis in
- Kleiter
- 1992
(Show Context)
Citation Context |

1 |
Properties of probabilistic imprecision
- Kleiter
- 1993
(Show Context)
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1 |
The precision of Bayesian classification: the multivariate normal case
- Kleiter
- 1994
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1 |
Investigations of variances in belief networks
- Neapolitan, Kenevan
- 1991
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1 |
A survey of sampling methods for inference on directed graphs
- Runnalls
- 1994
(Show Context)
Citation Context ...works was proposed by Pearl [34]. Hrycej [18] has shown that the stochastic simulation in a Bayesian network is a special case of Gibbs sampling. It has extensively been employed to Bayesian networks =-=[13, 36, 41, 40]-=-. At the start each instantiated node is clamped to its constant value and each noninstantiated node is set to an arbitrary value. Then, iteratively, the following steps are performed: 1. Select a non... |