## Using probability trees to compute marginals with imprecise probabilities (2002)

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Venue: | INTERNATIONAL JOURNAL OF APPROXIMATE REASONING |

Citations: | 26 - 2 self |

### BibTeX

@ARTICLE{Cano02usingprobability,

author = {Andres Cano and Serafín Moral},

title = {Using probability trees to compute marginals with imprecise probabilities},

journal = {INTERNATIONAL JOURNAL OF APPROXIMATE REASONING},

year = {2002},

volume = {29},

pages = {2002}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper presents an approximate algorithm to obtain a posteriori intervals of probability, when available information is also given with intervals. The algorithm uses probability trees as a means of representing and computing with the convex sets of

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Citation Context ...i cult to solve. The Cano and Moral methodology [8] to build a probability tree is based on the methods for inducing classi®cation rules from examples. One of these methods is Quinlan's ID3 algorithm =-=[38]-=-, that builds a decision tree from a set of examples. The process of constructing a tree consists of choosing, given a tree T with structure T , the branch to be expanded and the variable to be placed... |

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Citation Context ...ilable information is not speci®c enough. For example, when we take out balls from an urn with 10 balls, where ®ve are red and ®ve are white or black �but we do not know the exact number of each one) =-=[21,31,42]-=-. · In robust Bayesian inference, to model uncertainty about a prior distribution [3,22]. · To model the con¯ict between several sources of information [35,51]. There are various mathematical models f... |

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Citation Context ...hematical models for imprecise probability [52,56,57]: comparative probability orderings [25,26], possibility measures [23,61], fuzzy measures [28,47,58], belief functions [42,46], Choquet capacities =-=[16,30]-=-, interval probabilities [6,59,60], coherent lower previsions [52,57], convex sets of probabilities [7,15,52,57], sets of desirable gambles [52,57]. Out of all these models, we think that convex sets ... |

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Citation Context ...bilities. In general, the use of imprecise probability models is useful in many situations. We can highlight the following situations [56]: · When we have little information to evaluate probabilities =-=[52,53,55]-=-. · When available information is not speci®c enough. For example, when we take out balls from an urn with 10 balls, where ®ve are red and ®ve are white or black �but we do not know the exact number o... |

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Citation Context ...ilable information is not speci®c enough. For example, when we take out balls from an urn with 10 balls, where ®ve are red and ®ve are white or black �but we do not know the exact number of each one) =-=[21,31,42]-=-. · In robust Bayesian inference, to model uncertainty about a prior distribution [3,22]. · To model the con¯ict between several sources of information [35,51]. There are various mathematical models f... |

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Citation Context ...the con¯ict between several sources of information [35,51]. There are various mathematical models for imprecise probability [52,56,57]: comparative probability orderings [25,26], possibility measures =-=[23,61]-=-, fuzzy measures [28,47,58], belief functions [42,46], Choquet capacities [16,30], interval probabilities [6,59,60], coherent lower previsions [52,57], convex sets of probabilities [7,15,52,57], sets ... |

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Citation Context ...ponential in the number of parameters �number of variables in the distribution).s6 A. Cano, S. Moral / Internat. J. Approx. Reason. 29 �2002) 1±46 De®nition 1 �Probability tree). A probability tree T =-=[5,8,13,27,32,37,41,45,62]-=- for a set of variables XI is a directed labeled tree, where each internal node represents a variable Xi 2 XI and each leaf node represents a real number r 2 R. Each internal node will have as many ou... |

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Citation Context ...s paper is divided into six sections. In Section 2we describe the problem of the propagation of probabilities in Bayesian networks and how it can be solved by using the variable elimination algorithm =-=[20,34,62]-=-. Section 3 studies the use of probability trees to represent potentials compactly and how they can represent context speci®c independences; we also study how to build and operate with probability tre... |

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Citation Context ...phs: Boerlage92[4], Boblo [39,40], Car Starts �a somewhat large network contributed by Sreekanth Nagarajan based on the automobile belief network that D. Heckerman et al. presented in [29]) and Alarm =-=[2]-=-. These graphs can be found in the literature for the probabilistic case, i.e. at each node we have a conditional probability distribution. Propagation on these networks is not very di cult from a pro... |

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Citation Context ...ponential in the number of parameters �number of variables in the distribution).s6 A. Cano, S. Moral / Internat. J. Approx. Reason. 29 �2002) 1±46 De®nition 1 �Probability tree). A probability tree T =-=[5,8,13,27,32,37,41,45,62]-=- for a set of variables XI is a directed labeled tree, where each internal node represents a variable Xi 2 XI and each leaf node represents a real number r 2 R. Each internal node will have as many ou... |

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Citation Context ...the con¯ict between several sources of information [35,51]. There are various mathematical models for imprecise probability [52,56,57]: comparative probability orderings [25,26], possibility measures =-=[23,61]-=-, fuzzy measures [28,47,58], belief functions [42,46], Choquet capacities [16,30], interval probabilities [6,59,60], coherent lower previsions [52,57], convex sets of probabilities [7,15,52,57], sets ... |

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Citation Context ...ral sources of information [35,51]. There are various mathematical models for imprecise probability [52,56,57]: comparative probability orderings [25,26], possibility measures [23,61], fuzzy measures =-=[28,47,58]-=-, belief functions [42,46], Choquet capacities [16,30], interval probabilities [6,59,60], coherent lower previsions [52,57], convex sets of probabilities [7,15,52,57], sets of desirable gambles [52,57... |

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Citation Context ...ral sources of information [35,51]. There are various mathematical models for imprecise probability [52,56,57]: comparative probability orderings [25,26], possibility measures [23,61], fuzzy measures =-=[28,47,58]-=-, belief functions [42,46], Choquet capacities [16,30], interval probabilities [6,59,60], coherent lower previsions [52,57], convex sets of probabilities [7,15,52,57], sets of desirable gambles [52,57... |

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Citation Context ...s paper is divided into six sections. In Section 2we describe the problem of the propagation of probabilities in Bayesian networks and how it can be solved by using the variable elimination algorithm =-=[20,34,62]-=-. Section 3 studies the use of probability trees to represent potentials compactly and how they can represent context speci®c independences; we also study how to build and operate with probability tre... |

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Inferences from multinomial data: learning about a bag of marbles
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Citation Context ...bilities. In general, the use of imprecise probability models is useful in many situations. We can highlight the following situations [56]: · When we have little information to evaluate probabilities =-=[52,53,55]-=-. · When available information is not speci®c enough. For example, when we take out balls from an urn with 10 balls, where ®ve are red and ®ve are white or black �but we do not know the exact number o... |

112 |
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Citation Context ... think that convex sets of probabilities are the most suitable for calculating with and representing imprecise probabilities. We think that because there is a speci®c interpretation of numeric values =-=[52,54]-=-, they are powerful enough to represent the result of basic operations within the model without having to make approximations that cause loss of information, as in interval probabilities [50]. Convex ... |

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Citation Context ...hematical models for imprecise probability [52,56,57]: comparative probability orderings [25,26], possibility measures [23,61], fuzzy measures [28,47,58], belief functions [42,46], Choquet capacities =-=[16,30]-=-, interval probabilities [6,59,60], coherent lower previsions [52,57], convex sets of probabilities [7,15,52,57], sets of desirable gambles [52,57]. Out of all these models, we think that convex sets ... |

99 |
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Citation Context ...ral sources of information [35,51]. There are various mathematical models for imprecise probability [52,56,57]: comparative probability orderings [25,26], possibility measures [23,61], fuzzy measures =-=[28,47,58]-=-, belief functions [42,46], Choquet capacities [16,30], interval probabilities [6,59,60], coherent lower previsions [52,57], convex sets of probabilities [7,15,52,57], sets of desirable gambles [52,57... |

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Citation Context ...stribution [3,22]. · To model the con¯ict between several sources of information [35,51]. There are various mathematical models for imprecise probability [52,56,57]: comparative probability orderings =-=[25,26]-=-, possibility measures [23,61], fuzzy measures [28,47,58], belief functions [42,46], Choquet capacities [16,30], interval probabilities [6,59,60], coherent lower previsions [52,57], convex sets of pro... |

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Citation Context ...ponential in the number of parameters �number of variables in the distribution).s6 A. Cano, S. Moral / Internat. J. Approx. Reason. 29 �2002) 1±46 De®nition 1 �Probability tree). A probability tree T =-=[5,8,13,27,32,37,41,45,62]-=- for a set of variables XI is a directed labeled tree, where each internal node represents a variable Xi 2 XI and each leaf node represents a real number r 2 R. Each internal node will have as many ou... |

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Citation Context ...robability [52,56,57]: comparative probability orderings [25,26], possibility measures [23,61], fuzzy measures [28,47,58], belief functions [42,46], Choquet capacities [16,30], interval probabilities =-=[6,59,60]-=-, coherent lower previsions [52,57], convex sets of probabilities [7,15,52,57], sets of desirable gambles [52,57]. Out of all these models, we think that convex sets of probabilities are the most suit... |

56 |
Probability intervals : a tool for uncertain reasoning
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Citation Context ...robability [52,56,57]: comparative probability orderings [25,26], possibility measures [23,61], fuzzy measures [28,47,58], belief functions [42,46], Choquet capacities [16,30], interval probabilities =-=[6,59,60]-=-, coherent lower previsions [52,57], convex sets of probabilities [7,15,52,57], sets of desirable gambles [52,57]. Out of all these models, we think that convex sets of probabilities are the most suit... |

55 |
A valuation-based language for expert systems
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Citation Context ...The Cartesian product Q i2I Ui will be denoted by UI. Given x 2 UI and J I, xJ will denote the element of UJ obtained from x dropping the coordinates which are not in J. Following Shenoy and Shafer's =-=[43,44]-=- general terminology, a mapping from a set UI on [0,1] will be called a valuation de®ned on UI. In the probabilistic case, valuations are known as potentials. Suppose V is the set of all our initial v... |

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Citation Context ...independence graphs: Boerlage92[4], Boblo [39,40], Car Starts �a somewhat large network contributed by Sreekanth Nagarajan based on the automobile belief network that D. Heckerman et al. presented in =-=[29]-=-) and Alarm [2]. These graphs can be found in the literature for the probabilistic case, i.e. at each node we have a conditional probability distribution. Propagation on these networks is not very di ... |

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An overview of robust Bayesian analysis (with Discussion
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Citation Context ...th 10 balls, where ®ve are red and ®ve are white or black �but we do not know the exact number of each one) [21,31,42]. · In robust Bayesian inference, to model uncertainty about a prior distribution =-=[3,22]-=-. · To model the con¯ict between several sources of information [35,51]. There are various mathematical models for imprecise probability [52,56,57]: comparative probability orderings [25,26], possibil... |

41 |
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Citation Context ...s paper is divided into six sections. In Section 2we describe the problem of the propagation of probabilities in Bayesian networks and how it can be solved by using the variable elimination algorithm =-=[20,34,62]-=-. Section 3 studies the use of probability trees to represent potentials compactly and how they can represent context speci®c independences; we also study how to build and operate with probability tre... |

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Citation Context |

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Citation Context ...h interval probabilities, but computations are carried out by considering their associated convex sets. Some authors have considered the propagation of probabilistic intervals in graphical structures =-=[1,24,48,49]-=-. However in the procedures proposed, there is no guarantee that the calculated intervals are always the same as those obtained by using a global computation. In general, it can be said that calculate... |

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Citation Context ...ty measures [23,61], fuzzy measures [28,47,58], belief functions [42,46], Choquet capacities [16,30], interval probabilities [6,59,60], coherent lower previsions [52,57], convex sets of probabilities =-=[7,15,52,57]-=-, sets of desirable gambles [52,57]. Out of all these models, we think that convex sets of probabilities are the most suitable for calculating with and representing imprecise probabilities. We think t... |

28 |
Examples of independence for imprecise probabilities
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Citation Context ...onditioning is HjA ˆ CHfp… jA† : p 2 H; p…A† 6ˆ 0g. For convex sets we will use the so-called strong conditional independence as a notion of independence. See De Campos and Moral [7] and Couso et al. =-=[17]-=- for a detailed study of de®nitions of independence in convex sets. If H X ;Y ;Z is a global convex set of probabilities for …X ; Y ; Z†, we say that X and Y are conditionally strong independent given... |

25 | The ALARM monitoring system - Beinlich, Suermondt, et al. - 1989 |

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Citation Context ...nother solution to the problem of propagating the convex sets associated to the intervals, is by using an approximate algorithm using combinatorial optimization techniques such as simulated annealing =-=[9,10]-=-, genetic algorithms [11], and gradient techniques [18,19]. Probability trees [13,41] can be used to represent probability potentials. In [13], the authors have used probability trees to propagate pro... |

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Citation Context ...th 10 balls, where ®ve are red and ®ve are white or black �but we do not know the exact number of each one) [21,31,42]. · In robust Bayesian inference, to model uncertainty about a prior distribution =-=[3,22]-=-. · To model the con¯ict between several sources of information [35,51]. There are various mathematical models for imprecise probability [52,56,57]: comparative probability orderings [25,26], possibil... |

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Citation Context ...lem of propagating the convex sets associated to the intervals, is by using an approximate algorithm using combinatorial optimization techniques such as simulated annealing [9,10], genetic algorithms =-=[11]-=-, and gradient techniques [18,19]. Probability trees [13,41] can be used to represent probability potentials. In [13], the authors have used probability trees to propagate probabilities e - ciently in... |

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Citation Context ...sets associated to the intervals, is by using an approximate algorithm using combinatorial optimization techniques such as simulated annealing [9,10], genetic algorithms [11], and gradient techniques =-=[18,19]-=-. Probability trees [13,41] can be used to represent probability potentials. In [13], the authors have used probability trees to propagate probabilities e - ciently in Bayesian networks using a join t... |

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Citation Context ...rvals, is by using an approximate algorithm using combinatorial optimization techniques such as simulated annealing [9,10], genetic algorithms [11], and gradient techniques [18,19]. Probability trees =-=[13,41]-=- can be used to represent probability potentials. In [13], the authors have used probability trees to propagate probabilities e - ciently in Bayesian networks using a join tree when resources �memory ... |

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Citation Context ...h interval probabilities, but computations are carried out by considering their associated convex sets. Some authors have considered the propagation of probabilistic intervals in graphical structures =-=[1,24,48,49]-=-. However in the procedures proposed, there is no guarantee that the calculated intervals are always the same as those obtained by using a global computation. In general, it can be said that calculate... |

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Citation Context ...rvals, is by using an approximate algorithm using combinatorial optimization techniques such as simulated annealing [9,10], genetic algorithms [11], and gradient techniques [18,19]. Probability trees =-=[13,41]-=- can be used to represent probability potentials. In [13], the authors have used probability trees to propagate probabilities e - ciently in Bayesian networks using a join tree when resources �memory ... |

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Citation Context ...cted acyclic graph where each node represents a random variable, and the topology of the graph shows the independence relations between variables �see Fig. 1), according to the d-separation criterion =-=[36]-=-.s4 A. Cano, S. Moral / Internat. J. Approx. Reason. 29 �2002) 1±46 Let X ˆfX1; ...; Xng be the set of variables in the network. Let us assume that each variable Xi takes values on a ®nite set Ui. For... |

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Citation Context ...bilities. In general, the use of imprecise probability models is useful in many situations. We can highlight the following situations [56]: · When we have little information to evaluate probabilities =-=[52,53,55]-=-. · When available information is not speci®c enough. For example, when we take out balls from an urn with 10 balls, where ®ve are red and ®ve are white or black �but we do not know the exact number o... |

13 |
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Citation Context ...ot know the exact number of each one) [21,31,42]. · In robust Bayesian inference, to model uncertainty about a prior distribution [3,22]. · To model the con¯ict between several sources of information =-=[35,51]-=-. There are various mathematical models for imprecise probability [52,56,57]: comparative probability orderings [25,26], possibility measures [23,61], fuzzy measures [28,47,58], belief functions [42,4... |

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Citation Context ...ty measures [23,61], fuzzy measures [28,47,58], belief functions [42,46], Choquet capacities [16,30], interval probabilities [6,59,60], coherent lower previsions [52,57], convex sets of probabilities =-=[7,15,52,57]-=-, sets of desirable gambles [52,57]. Out of all these models, we think that convex sets of probabilities are the most suitable for calculating with and representing imprecise probabilities. We think t... |