1 Introduction
DempsterShafer theory (DS theory for short) (Dempster, 1967; Shafer, 1976) has been widely used because it allows to handle uncertain data (Durbach and Stewart, 2012; Yang et al., 2013; Yang and Xu, 2013). In DS theory, various belief structures are employed to represent the uncertain data. Recently, the study of parameter estimation based on belief structures has attracted many attentions (Come et al., 2009; Denoeux, 2010, 2013; Su et al., 2013). Typically, Denoeux (Denoeux, 2013) proposed an evidential EM algorithm for parameter estimation in the case of crisp belief structures, and Su et al. (Su et al., 2013) developed a parameter estimation approach for fuzzy belief structures. In this paper, the parameter estimation based on intervalvalued belief structures (Yager, 2001; Wang et al., 2006) has been considered. A novel parameter estimation method is proposed for the case of intervalvalued belief structures. Within the proposed method, two criteria, the maximization of observation data’s likelihood and the minimization of estimated parameter’s uncertainty, are both considered simultaneously. The proposed method is effective for both crisp (deterministic) and intervalvalued (uncertain) belief structures, and promising for various applications.
2 DS theory and belief structures
DS theory (Dempster, 1967; Shafer, 1976) is often regarded as an extension of the Bayesian theory. Please refer to (Shafer, 1976; Yang and Xu, 2013) for more knowledge about DS theory. In DS theory, various belief structures, such as crisp, intervalvalued and fuzzy belief structures, are employed as basic data structures. They are used to express various uncertain information. A crisp belief structure is defined as follows. Let a finite nonempty set be a frame of discernment, and denote the power set of . A crisp belief structure is a mapping , satisfying
(1) 
The crisp belief structure is deterministic because its belief degree is expressed by real numbers. By contrast, the intervalvalued belief structure (IBS) is a kind of uncertain belief structures, which is an extension of the crisp belief structure. It is more capable to represent the uncertain information. Some basic concepts about IBS are given as below (Yager, 2001; Wang et al., 2006). Let be a frame of discernment, , , , be the focal elements on . An IBS satisfies such conditions

, where and ;

and ;

, .
An IBS is valid if it satisfies and . In the rest of this paper, all the IBSs are valid.
3 Proposed parameter estimation method
In previous literatures (Denoeux, 2013; Su et al., 2013)
, parameter estimation based on crisp and fuzzy belief structures has been studied. However, the parameter estimation based on intervalvalued belief structures is still an unsettled problem. In this paper, a novel parameter estimation method based on IBSs is proposed to fill that gap. Without loss of generality, some concepts about interval probabilities are introduced first.
3.1 Interval probabilities
(Guo and Tanaka, 2010)Let be a finite set , a set of intervals satisfying is an interval probabilities of if there are for such that .
Interval probabilities are the extension of pointvalued probability mass functions, which can be degenerated to the classical probability distribution.
(Guo and Tanaka, 2010) Let be an interval probabilities, the th ignorance of , denoted as , is(2) 
Obviously, . for and for the pointvalued probabilities. can be seen as an effective index to measure the uncertainty/imprecision of interval probabilities.
3.2 Likelihood function model for IBS
To do the parameter estimation under IBS environment, the likelihood function model for IBS should be developed first. Let
be a discrete random variable taking values in
, with interval probabilities which depends on unknown parameter . There are several types of observational data.If the observational data is completely certain, for example happened, the likelihood function given a singleton can be represented as
(3) 
If an event , , is observed, the likelihood function given a subset is now
(4) 
where ,
If the observational data is described by a piece of uncertain belief structure — an IBS , the likelihood function given such uncertain data is
(5) 
where
(6) 
Now assuming there are observational data, expressed by IBSs, . The likelihood of is represented as
(7) 
Observation  1  2  3  4  5  6 

1.0  1.0  1.0  0.3  0.0  0.0  
0.0  0.0  0.0  0.3  1.0  1.0  
0.0  0.0  0.0  0.4  0.0  0.0 
3.3 Solution for parameter estimation
The likelihood function model developed above is the foundation for the parameter estimation based on IBSs. Depending on that, an optimization model is proposed to make an estimation for parameter .
(8) 
where is the th ignorance of , and is a distance measure for two intervals and presented in (Tran and Duckstein, 2002). For and ,
(9) 
The model is formulated based on two criteria, namely maximization of the likelihood of observational data indicated by and minimization of the uncertainty/imprecision of estimated parameter indicated by , respectively. Within model , is a control parameter to adjust the impact of these two criteria. A pointvalued probability distribution will be obtained if , and a set of interval probabilities will be obtained if . A global optimization algorithm called OQNLP (Ugray et al., 2007) is employed to solve the optimization model . Generally, the proposed method is superior to Denoeux’s method because not only can this method handle the case of crisp belief structures, but it can also deal with the case of IBSs, as shown in the next section.
Probability  

Denoeux’s method (Denoeux, 2013)  0.6  0.4 
Proposed method ()  0.6  0.4 
Observation  1  2  3  4 

[0.30, 0.40]  [0.35, 0.45]  [0.10, 0.25]  [0.30, 0.45]  
[0.10, 0.25]  [0.10, 0.20]  [0.30, 0.45]  [0.30, 0.50]  
[0.25, 0.35]  [0.20, 0.30]  [0.35, 0.50]  [0.15, 0.40]  
[0.10, 0.20]  [0.05, 0.15]  [0.10, 0.25]  [0.00, 0.20] 
’s value  

[0.9823, 0.9823]  [0.0000, 0.0000]  [0.0177, 0.0177]  0.0000  
[0.8397, 0.9433]  [0.0057, 0.1093]  [0.0510, 0.1547]  0.1036  
[0.5821, 0.9331]  [0.0122, 0.3632]  [0.0547, 0.4058]  0.3510  
[0.4614, 0.9569]  [0.0085, 0.5040]  [0.0346, 0.5301]  0.4955  
[0.2963, 0.8751]  [0.0324, 0.6112]  [0.0925, 0.6713]  0.5788  
[0.2580, 0.8907]  [0.0288, 0.6615]  [0.0805, 0.7132]  0.6327  
[0.3228, 0.9988]  [0.0002, 0.6763]  [0.0010, 0.6770]  0.6760  
[0.2687, 0.9744]  [0.0055, 0.7112]  [0.0201, 0.7259]  0.7057  
[0.1876, 0.9136]  [0.0235, 0.7496]  [0.0629, 0.7889]  0.7260  
[0.2272, 0.9768]  [0.0050, 0.7546]  [0.0182, 0.7678]  0.7496  
[0.1339, 0.9815]  [0.0042, 0.8518]  [0.0143, 0.8619]  0.8476 
4 Numerical Examples
4.1 Example for the case of crisp belief structures
In this example, a set of observational data is composed of 6 crisp belief structures, as shown in Table 1. The estimated parameter is the probability distribution of random variable taking values in . Two methods, namely Denoeux’s (Denoeux, 2013) and proposed in this paper, are employed. As shown in Table 2, the results obtained by these two methods are identical, which demonstrates the proposed method is effective for crisp belief structures.
4.2 Example for the case of intervalbased belief structures
While, Denoeux’s method is incapable when the observational data are represented by IBSs, as shown in Table 3. In this situation, a set of interval probabilities can be estimated based on different by using the proposed method. As seen in Table 4, a pointvalued probability distribution is obtained that , , when . The estimated probability distribution becomes a set of interval probabilities when . The uncertainty of obtained interval probabilities rises with the increase of .
Criteria  HIS trustworthiness evaluations 

Reliability  {(,[0.0393,0.2159]), (,[0.3305,0.6476]), (,[0.2266,0.5128]), (,[0,0.1026]), (,[0,0.1449])} 
Safety  {(,[0.0728,0.3119]), (,[0.4817,0.8246]), (,[0.068,0.1832]), (,[0,0.1814])} 
Realtime  {(,[0.229,0.7]), (,[0.2727,0.75]), (,[0,0.1778])} 
Maintainability  {(,[0.1515,0.2849]), (,[0.4545,0.6648]), (,[0.048,0.2424]), (,[0,0.186])} 
Availability  {(,[0.0867,0.2537]), (,[0.5034,0.7258]), (,[0.1438,0.2722])(,[0,0.0899])} 
Security  {(,[0.0513,0.1967]), (,[0.3213,0.473]), (,[0.4017,0.5676]), (,[0,0.0939])} 
4.3 Trustworthiness assessment of hospital information system (HIS)
In this example, the trustworthiness assessment of HIS is studied. Generally, the rating of HIS trustworthiness can be . Table 5 shows the assessment criteria and the evaluation for each criterion in a HIS, derived from literature (Fu and Yang, 2012). Here, the evaluations on various criteria are treated as a set of observational data composed of IBSs. Based on our proposed method, various sets of interval probabilities are obtained when takes different values, as shown in Table 6. According to these results, the rating is appropriate for the given HIS.
’s value  

[0.0000, 0.0000]  [0.0000, 0.0000]  [0.0000, 0.0000]  [1.0000, 1.0000]  [0.0000, 0.0000]  
[0.0007, 0.0291]  [0.0007, 0.0558]  [0.0015, 0.1789]  [0.8187, 0.9961]  [0.0010, 0.1784]  
[0.0004, 0.0874]  [0.0004, 0.2284]  [0.0008, 0.4799]  [0.5187, 0.9978]  [0.0006, 0.4796]  
[0.0001, 0.0994]  [0.0001, 0.3615]  [0.0001, 0.6076]  [0.3919, 0.9996]  [0.0001, 0.6075]  
[0.0003, 0.1422]  [0.0003, 0.4503]  [0.0005, 0.6729]  [0.3262, 0.9986]  [0.0004, 0.6727] 
5 Conclusion
In this paper, the problem of parameter estimation based on belief structures has been studied. The proposed method provides a unified framework for this problem. Not only crisp belief structures but also uncertain belief structures — IBSs, are both can be handled. As an important technique in DS theory, it has the ability to handle various types of uncertain data and knowledge represented as belief structures.
Acknowledgements
The work is supported by National Natural Science Foundation of China, Grant No. 61174022.
References

Come et al. (2009)
Come, E., Oukhellou, L., Denoeux, T., Aknin, P., 2009. Learning from partially supervised data using mixture models and belief functions. Pattern Recognition 42 (3), 334–348.
 Dempster (1967) Dempster, A. P., 1967. Upper and lower probabilities induced by a multivalued mapping. Annals of Mathematics and Statistics 38 (2), 325–339.
 Denoeux (2010) Denoeux, T., 2010. Maximum likelihood from evidential data: an extension of the EM algorithm. In: Proc. Int’l Conf. Soft Methods in Probability and Statistics (SMPS’10). pp. 181–188.
 Denoeux (2013) Denoeux, T., 2013. Maximum likelihood estimation from uncertain data in the belief function framework. IEEE Transactions on knowledge and data engineering 25 (1), 119–130.
 Durbach and Stewart (2012) Durbach, I. N., Stewart, T. J., 2012. Modeling uncertainty in multicriteria decision analysis. European Journal of Operational Research 223 (1), 1–14.
 Fu and Yang (2012) Fu, C., Yang, S., 2012. The conjunctive combination of intervalvalued belief structures from dependent sources. International Journal of Approximate Reasoning 53 (5), 769–785.
 Guo and Tanaka (2010) Guo, P., Tanaka, H., 2010. Decision making with interval probabilities. European Journal of Operational Research 203 (2), 444–454.
 Shafer (1976) Shafer, G., 1976. A Mathematical Theory of Evidence. Princeton University Press, Princeton.

Su et al. (2013)
Su, Z.G., Wang, Y.F., Wang, P.H., 2013. Parametric regression analysis of imprecise and uncertain data in the fuzzy belief function framework. International Journal of Approximate Reasoning 54 (8), 1217–1242.
 Tran and Duckstein (2002) Tran, L., Duckstein, L., 2002. Comparison of fuzzy numbers using a fuzzy distance measure. Fuzzy sets and Systems 130 (3), 331–341.
 Ugray et al. (2007) Ugray, Z., Lasdon, L., Plummer, J., Glover, F., Kelly, J., Martí, R., 2007. Scatter search and local NLP solvers: A multistart framework for global optimization. INFORMS Journal on Computing 19 (3), 328–340.
 Wang et al. (2006) Wang, Y.M., Yang, J.B., Xu, D.L., Chin, K.S., 2006. The evidential reasoning approach for multiple attribute decision analysis using interval belief degrees. European Journal of Operational Research 175 (1), 35–66.
 Yager (2001) Yager, R. R., 2001. Dempstershafer belief structures with interval valued focal weights. International Journal of Intelligent Systems 16 (4), 497–512.
 Yang et al. (2013) Yang, G.L., Yang, J.B., Liu, W.B., Li, X.X., 2013. Crossefficiency aggregation in DEA models using the evidentialreasoning approach. European Journal of Operational Research 231 (2), 393–404.

Yang and Xu (2013)
Yang, J.B., Xu, D.L., 2013. Evidential reasoning rule for evidence combination. Artificial Intelligence 205, 1–29.
Comments
There are no comments yet.