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Probability Elicitation in Influence Diagram Modeling by Using Interval Probability

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Probability Elicitation in Influence Diagram Modeling by Using Interval Probability International Journal of Intelligence Science, 2012, 2, 89-95 http://dx.doi.org/10.4236/ijis.2012.24012 Published Online October 2012 (http://www.SciRP.org/journal/ijis) Probability Elicitation in Influence Diagram Modeling by Using Interval Probability Xiaoxuan Hu1,2, He Luo1,2, Chao Fu1,2 1School of Management, Hefei University of Technology, Hefei, China 2Key Laboratory of Process Optimization and Intelligent Decision-Making, Ministry of Education, Hefei, China Email: [email protected] Received May 29, 2012; revised July 31, 2012; accepted August 10, 2012 ABSTRACT In decision modeling with influence diagrams, the most challenging task is probability elicitation from domain experts. It is usually very difficult for experts to directly assign precise probabilities to chance nodes. In this paper, we propose an approach to elicit probability effectively by using the concept of interval probability (IP). During the elicitation process, a group of experts assign intervals to probabilities instead of assigning exact values. Then the intervals are combined and converted into the point valued probabilities. The detailed steps of the elicitation process are given and illustrated by constructing the influence diagram for employee recruitment decision for a China’s IT Company. The proposed approach provides a convenient and comfortable way for experts to assess probabilities. It is useful in influ- ence diagrams modeling as well as in other subjective probability elicitation situations. Keywords: Influence Diagram; Probability Elicitation; Interval Probability; Decision-Making 1. Introduction To deal with the imprecision and inconsistence of sub- jective probabilities, quite a few methods are developed An influence diagram (ID) [1] is a directed acyclic graph to guide experts correctly giving probabilities, such as for modeling and solving decision problems under un- using visual tools like probability scale [8], probability certainty. It provides a more compact way to represent wheel [9] and scaled probability bar [10], adapting Ana- complex decision situations than a decision tree does. In lytic Hierarchy Process (AHP) [11]. Wiegmann gives an recent years, influence diagrams have been used as ef- overview of the popular elicitation methods [12]. fective modeling tools for Bayesian decision analysis. In this paper, we focus on probability elicitation for The work of constructing an influence diagram can be influence diagrams. We present a new approach to elicit divided into two sub-works: The first one is to build the probabilities from experts. Our approach does not require influence diagram structure; the other one is to assign experts to give point-valued probabilities but to give in- parameters to all kinds of nodes, including assigning terval-valued probabilities instead. Because in daily life, conditional probabilities for chance nodes, acquiring the it is unrealistic to expect experts to provide exact values utilities for value nodes and generating decision al- of many probabilities. They are used to describing prob- terna-tives for decision nodes. The whole work is chal- abilities by verbal or other inexact expressions, such as lenging and time consuming, and a number of difficulties “possible”, “rare” or “likely”. Each expression is an im- may be faced. Bielza et al. have discussed the important precise or fuzzy description of probability that actually issues in modeling with influence diagrams [2-4]. means an interval of probability. Cano and Moral pointed During the entire construction process, the assignment out that imprecise probability model such as interval of conditional probabilities for chance nodes is consid- probability is more useful than exact probability model in ered the most difficult problem. There are two common many situations [13]. The experts would be more con- used solutions to get probability values: learning from fident and feel more comfortable to deal with intervals data or acquiring from experts. In the machine learning probabilities. So, we apply the concept of interval proba- community, many algorithms have been presented to bility [14-18] to elicit probabilities, and we combine learn probabilities from data [5-7]. However, in many multiple experts’ judgments to increase the accuracy of real-world applications, we do not have available data set the final results. and have to elicit probabilities from experts. The prob- The paper is organized as follows: In Section 2, we abilities are thus called subjective probabilities. briefly introduce influence diagrams. Section 3 shows the Copyright © 2012 SciRes. IJIS 90 X. X. HU ET AL. basic concept of interval probabilities. In Section 4, we mation arc from the node “weather forecast” to the deci- describe our approach of probabilities elicitation from sion node, and a relevance arc from the node “weather experts. First we introduce the entire process of the ap- forecast” to the node “weather”. Bob’s utility varies with proach, and then give the details of each step. In Section various instances of decisions and weather conditions. 5, we illustrate our approach by a real application: estab- The evaluation of an influence diagram is to find the lishing an influence diagram model for employee re- best alternative by comparing the expected utility (EU) cruitment decision for a China’s IT company. Finally, we among every decision alternatives. Suppose Dj is a deci- give a conclusion in Section 6. sion node with a set of decision alternativesdLd1,,n . First we calculate the EU of each decision alternative di: 2. Influence Diagram EUdd,ii U cj P cj e (1) An influence diagram can be defined as a four-tuple [19] j in which e represents the evidences. IDGXPU ,,,r Then we select the best alternative d* in dLd,, , such that, 1 n which satisfies 1) G = (V, E) is a directed acyclic graph (DAG), with * nodes V and edges E. V are partitioned into three sets, V dE arg max Udi (2) =VVCDUV. VC, VD and VU are the set of chance nodes, decision nodes and value nodes, respectively. The The original evaluation approach is to unfold an in- dependence relations and information precedence among fluence diagram into a decision tree. Obviously it is inef- all the nodes are encoded in E. ficient. Shachter presents a way to evaluate influence diagrams with two operations: node-removal and arc- 2) X is a set of variables. X = X CD X , XC is a set of reversal [20]. By recursively using the two operations, an random variables, and each variable in XC is represented influence diagram is transformed into a diagram with by a chance node of G. XD is a set of decision variables, only a utility node. The utilities for individual decision and each variable in XD is represented by a decision node of G . alternatives are computed during the process. F. Jensen et 3) Pr is a set of conditional probability distributions. al. describe a way to convert an influence diagram into a junction tree [21], and then the message passing algo- Each random variable CXiC associates with a dis- rithm is operated on the junction tree for calculating the tribution PiiCParC. 4) U is a set of utility functions. Each value node expected utility. Zhang describes a method to reduce VViiU1, 2 contains one utility function u(Xpa(vi)). influence diagrams evaluation into Bayesian network In influence diagrams, the chance nodes are repre- inference problems [22], so that many Bayesian network sented by circulars, and each chance node associates with inference algorithms can be used to evaluate influence a probability distribution. The decision nodes are rectan- diagrams. gles. Each decision node has a set of alternatives. The value nodes are diamonds. The parameters of each value 3. Interval Probability node show the utility of various outcomes according to The theory of imprecise probability has received much the decision makers. Directed arcs in influence diagrams attention [13-17,23-26]. Interval probability is a major have different meanings. Arcs pointing to decision nodes expression of imprecise probability that has been used in are called information arcs, which indicate information uncertain reasoning [14,18], decision making [15,16], precedence. An information arc from a chance node A to a decision node B denotes that variable A will be ob- and some other applications. It is an extension of classi- servable before the decision is made. Arcs pointing to cal probability so that can be adapted in more complex chance nodes are called relevance arcs, which represent uncertain situations. the dependency between the variables and their parents. Definition 1. (Interval probability) [14,17,27]: Let Ω be The missing arc between two chance nodes means condi- tional independence. Weather Watch prain( ) 0.15 {,}yes no An example of an influence diagram is shown in Fig- forecast the game? ure 1. Bob is going to decide whether to go to watch a football game tomorrow. The only factor he considering p(|)0.90 rain rain is the weather. If there is no rain, he will go; otherwise he p(| rain no rain )0.05 u(, yes rain )0 prefers to stay at home. He has a weather forecast sensor, u(, yes no rain )10 0 Weather Utility from which he can know whether it will rain or not to- unorain(, )60 morrow. But the sensor has a small probability to make unonorain(, )20 wrong forecast. As shown in Figure 1, there is an infor- Figure 1. An influence diagram. Copyright © 2012 SciRes. IJIS X. X. HU ET AL. 91 a sample space, A aii 1, 2,..., n be a σ-field of ， Setp 1. Selection of domain experts random events in Ω a set of intervals Laii, Ua is called interval probabilities if it satisfies the following Domain conditions： experts iLaUa,0 ii 1 (3) Four criteria n Appropriate experts ipa,,ii LaUa i, such that pai 1; (4) i1 the software tool ipaLa,inf ii pP , Setp 2. Elicitation interval probabilities from the experts ipaUa,sup ii (5) pP Semi interval probabilities wherePp | LapaUaiii i, The linear The intervals satisfy conditions (3)-(5) are also called programming model reachable probability intervals [14] or F-probability [17].
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