Technology and Health Care 23 (2015) S161–S167 DOI 10.3233/THC-150950 IOS Press
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Evidence reasoning method for constructing conditional probability tables in a Bayesian network of multimorbidity Yuanwei Du∗ and Yubin Guo Faculty of Management and Economics, Kunming University of Science and Technology, Kunming, Yunnan, China Abstract. BACKGROUND: The intrinsic mechanism of multimorbidity is difficult to recognize and prediction and diagnosis are difficult to carry out accordingly. Bayesian networks can help to diagnose multimorbidity in health care, but it is difficult to obtain the conditional probability table (CPT) because of the lack of clinically statistical data. OBJECTIVE: Today, expert knowledge and experience are increasingly used in training Bayesian networks in order to help predict or diagnose diseases, but the CPT in Bayesian networks is usually irrational or ineffective for ignoring realistic constraints especially in multimorbidity. METHODS: In order to solve these problems, an evidence reasoning (ER) approach is employed to extract and fuse inference data from experts using a belief distribution and recursive ER algorithm, based on which evidence reasoning method for constructing conditional probability tables in Bayesian network of multimorbidity is presented step by step. RESULTS: A multimorbidity numerical example is used to demonstrate the method and prove its feasibility and application. Bayesian network can be determined as long as the inference assessment is inferred by each expert according to his/her knowledge or experience. CONCLUSIONS: Our method is more effective than existing methods for extracting expert inference data accurately and is fused effectively for constructing CPTs in a Bayesian network of multimorbidity. Keywords: Evidence reasoning, Bayesian network, multimorbidity, health care, construction method
1. Introduction Multimorbidity is an epidemic in which a patient has a variety of diseases at the same time, which can produce diverse effects on the status of the patient’s health because of the possible interactions between various diseases [1]. At present the existing research can be divided into two categories: One focuses on considering multimorbidity as an aggregation of special diseases and establishing an index system in order to find which category it belongs to through the sampling investigation method or monitoring experiment method [2,3]; the other focuses on predicting the trend of multimorbidity development by integrating basic discriminant information in all kinds of diseases with the hierarchical regression analysis method [4,5]. Obviously, these two types of research have important contributions to the study of multimorbidity. However, it has to be recognized that the former focuses on the overall performance ∗ Corresponding author: Yuanwei Du, Faculty of Management and Economics, Kunming University of Science and Technology, Kunming, Yunnan, China. Tel.: +86 871 65154194; Fax: +86 871 65154194; E-maill: [email protected].
c 2015 – IOS Press and the authors. All rights reserved 0928-7329/15/$35.00
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of multimorbidity, but ignores the interaction between different diseases, and the latter recognizes the interactions among diseases but it can predict results from the cause but not diagnose the cause from the result. As a result, the intrinsic mechanism of multimorbidity is difficult to be recognized roundly and the prediction and the diagnosis are hardly carried out accordingly. Bayesian networks are an important method in the field of artificial intelligence and have been successfully used to solve many prediction and diagnosis problems in health care [6,7]. It is not difficult to see how Bayesian networks can address predicting and diagnosing a single disease by considering the causal relationship between interrelated variables, but they are rarely useful when considering multimorbidity. One reason is that there is numerous clinically statistical data for single diseases, which can be used to construct a Bayesian network for predicting or diagnosing. The other reason is that there is little clinically statistical data about multimorbidity, and it is difficult to get the conditional probability table (CPT) by relevant methods such as machine learning. Experts in the health care field, engaged in preventing, diagnosing, or treating diseases, have accumulated rich professional knowledge and clinical experience and have the ability to infer to some extent the mechanism of multimorbidity, and the integrated results will be the result of inference data of more than one expert. In this paper, we assume that the Bayesian multimorbidity network structure has been completed and some CPTs are empty and need to be inferred by experts.
2. Problem description and inference information expression A Bayesian network can be described as a tuple G = ((N, V ), D), where (N, V ) is a directed acyclic graph that is used to represent the characteristics of qualitative knowledge in the model, N = (N1 , N2 , · · · , NR ), which represents a set of nodes in the described field, and V is a set of arcs among nodes. If there is no connection between nodes Nr and Nr (r = r ), they are independent of each other and have no direct causal relationship; if there is a connection Nr → Nr between double nodes, then there is a direct causal relationship between Nr and Nr , and we call Nr a parent node and Nr a child node. The direction of the arc determines the causal relationship between the double nodes, and it can be used in causal reasoning. D = (D1 , D2 , · · · , DT ) represents the set of parameters of conditional probability (CP) in the network, giving the quantitative knowledge of the model. Assume that node Nr has two states {Nr = True, Nr = False} in general, r = 1, · · · , R. The (s) CPT Dt attached to Nr can be built by P (Nr = xr |Nr = xr ). Assume that {Nt |s= 1, 2, · · · ,S} is the set of parent nodes of Nr , then the number of combination of states in the CPT attached to Nr is I = 2s . The CPT is usually determined by a machine-learning algorithm, but these algorithms can lose efficacy in complex situations such as multimorbidity. When traditional machine-learning algorithms all (i) fail, determining the CPT attached to Nr denoted by pt (i = 1, 2, · · · , I) is an important issue in the Bayesian network. Today, expert knowledge and experience is increasingly introduced into the training process of Bayesian networks in order to enhance the learning effect [8]. However, most existing studies assume that all experts have the ability to estimate the CPs in the CPT using a specific information demand structure (such as a pairwise comparison judgment matrix). This research does not take into account the fact that there are many differences in expert inference abilities and inference perspectives. These realistic constraints can easily cause problems since the individually inferred data lacks rationality and integrated data lacks effectiveness. Therefore, the realistic constraint that the ability of experts is always limited has to be considered in the inference process. Therefore, how to accurately extract the inference
Y. Du and Y. Guo / Evidence reasoning method for constructing CPT
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data of experts and how to effectively fuse this information are pivotal problems for constructing the CPT in a multimorbidity Bayesian network. The ER approach can model various kinds of uncertainties such as ignorance, fuzziness, interval data, and interval belief degrees in a unified format. It has been used in recent years to uniformly solve uncertain multiple attribute decision making (MADM) problems [9,10]. The above analysis shows that the extension of the ER approach to model and solve CPT construction problems under uncertainties is a new and significant research area and different from most existing relevant methods. The ER approach not only allows an expert to give a certain inference degree about familiar CPs, but also allows the expert to surrender judgment and give no inference information on about unfamiliar CPs. Assume that H = {H1 , H2 , · · · , HN } denotes a set of assessment grades and the number of grades (i) is determined according to the requirement for accuracy. If pt is assessed to a grade Hn by expert ek (i) with a belief degree of βn,k (pt ), the assessment can be expressed by the following belief distribution: (i)
(i)
(i)
Bk (pt ) = {(Hn , βn,k (pt )), n = 1, · · · , N ; (H, βH,k (pt ))} (1) (i) (i) (i) (i) (i) In Eq. (1), βn,k (pt ) 0, n βn,k (pt ) 1, and βH,k (pt ) = 1 − n βn,k (pt ). Note that βn,k (pt ) (i) is a degree of belief that pt is assessed as grade Hn by expert ek , H = {H1 , H2 , · · · , Hn } is the frame (i) of discernment, and βH,k (pt ) denotes the degree of global ignorance with the meaning of the degree of unknown of this inference problem by expert ek . According to the above information expression (i) method, each expert can give their inference data to all of the CPs for Nt , which is described as Bk (pt ), k = 1, 2, · · · , K , i = 1, 2, · · · , I .
3. CPT construction method In a Bayesian network multimorbidity structure, assume that the CPT of node Nt needs to be inferred (s) by experts. Also assume {Nt |s= 1, 2, · · · ,S} is the set of parent nodes of Nt , each with two states, True or False states. The ER method for constructing the CPT in a Bayesian network of multimorbidity is constructed as follows: Step 1: Definition and representation of the CPT inference problem. Define a set of K experts as E = {e k |k = 1, 2, · · · , K}, and estimate the relative weights of the experts as W = {wk |0 wk 1; k wk = 1; k = 1, 2, · · · , K}. Define distinctive evaluation grades Hn as a complete set of standards for assessing each CP in CPT by all experts, or H = {H1 , H2 , · · · , HN }. The number of grades should be determined in advance according to the requirements for accuracy. Then the CPT inference problem can be stated as how to determine a CP from the inference information as (i) (1) given by each expert, i.e., Bk (pt ), k = 1, 2, · · · , K ,i = 1, 2, · · · , I . (i) Step 2: Let i = 1. Define the current inference problem as pt , i = 1. Step 3: Determine the basic probability assignments for each expert. Let mn,k be a basic probability (i) mass representing the degree to which ek supports a hypothesis and pt is assessed to Hn . Let mH,k be a remaining probability mass unassigned to any individual grade after ek has been assessed. mn,k and mH,k are calculated as (2). Let mH,k = m ¯ H,k + m ˜ H,k , where m ¯ H,k = 1 − wk and (i) (i) m ˜ H,k = wk (1 − n βn,k (pt )) = wk βΩ,k (pt ). (i) mn,k = wk βn,k (pt ), n = 1, 2, · · · , N (2) (i) mH,k = 1 − n mn,k = 1 − wk n βn,k (pt ), k = 1, 2, · · · , K
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Step 4: Combine the probability assignments for all of experts. Let mn,e(1) = mn,1 (n = 1, 2, · · · , N ), m ¯ H,e(1) = m ¯ H,1 , m ˜ H,e(1) = m ˜ H,1 and mH,e(1) = mH,1 . The combined probability assignments mn,e(K)(n = 1, 2, · · · , N ), m ¯ n,e(K), m ˜ n,e(K), and m ˜ H,e(K) can be generated by aggregating all the basic probability assignments using the following recursive ER algorithm:
{Hn } : mn,e(k+1) = Ke(k+1) [mn,e(k) mn,k+1 + mH,e(k) mn,k+1 + mn,e(k)mH,k+1 ] ˜ H,e(k) + m ¯ H,e(k) {H} : mH,e(k) = m
(3)
˜ H,e(k+1) = Ke(k+1) [m ˜ H,e(k) m ˜ H,k+1 + m ¯ H,e(k) m ˜ H,k+1 + m ˜ H,e(k) m ¯ H,k+1 ], m ¯ H,e(k+1) = where m N N −1 m ¯ H,e(k) m ¯ H,k+1 ], Ke(k+1) = [1 − t=1 m ] . Ke(k+1) [m j = 1 t,e(k) j,k+1 j = t Step 5: Calculate the combined degrees of belief for all of experts. Let βn denote a degree of belief that (i) all experts at pt are assessed to grade Hn , which is generated by combining the assessments for (i) all the associated experts ek at pt . Then βn is calculated by
{Hn } : βn = mn,e(K) (1 − m ¯ H,e(K)), n = 1, 2, · · · , N ¯ H,e(K)) {H} : βH = mH,e(K) (1 − m
(4)
Step 6: Determine the pignistic probability for the current CP. Let belief function and plausibility function of Hn as Bel(Hn ) = m(B) and P l(H ) = n B⊆Hn Hn ∩B=∅ m(B), set factorε = (i) [1 − Hn ⊆H Bel(Hn )] / Hn ⊆H P l(Hn ), and the pignistic probability (9) for pt corresponding toHn is (i)
αn (pt ) = Bel(Hn ) + ε · P l(Hn ),
n = 1, 2, · · · , N.
(5)
Step 7: Let i = i + 1. If i I then turn to Step 3, otherwise go to Step 8. (i) Step 8: Construct the CPT. The distributed overall assessment of pt is given by the following distribution: (i)
(i)
B(pt ) = {(Hn , αn (pt )), n = 1, 2, · · · , N },
i = 1, 2, · · · , I.
(6) (i)
If the overall assessment is complete, then the expected inference probability of pt is calculated by (i)
p˜t =
n
(i)
Hn αn (pt ), (i)
i = 1, 2, · · · , I.
(7) (i)
Note from Eq. (7) that p˜t is the inference CP corresponding to pt by integration with inference in(i) (i) formation generated by all of experts. p˜t and 1 − p˜t can be seen as the CPs that the probability of Nt = T rue and Nt = F alse under the condition of the ith combination of states in the CPT attached (i) (i) to Nt . When the CPs are all inferred, i.e., p˜t and 1 − p˜t , i = 1, 2, · · · , I , the CPT attached to node Nt has been determined.
Y. Du and Y. Guo / Evidence reasoning method for constructing CPT
State of parent nodes
CH=T EN=T GE=T
DB=T DB=F
pt (1) 1-pt
(1)
Table 1 CPT to be inferred and attached to node DB CH=T CH=T CH=T CH=F EN=T EN=F EN=F EN=T GE=F GE=T GE=F GE=T (2)
pt (2) 1-pt
(3)
pt (3) 1-pt
(4)
pt (4) 1-pt
(5)
pt (5) 1-pt
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CH=F EN=T GE=F (6)
pt (6) 1-pt
CH=T EN=F GE=T (7)
pt (7) 1-pt
CH=T EN=T GE=T (8)
pt (8) 1-pt
Table 2 Belief distributions inferred by three experts e1 e2 e3
H1 =0.1 0.4 0.0 0.0
H2 =0.3 0.0 0.5 0.0
H3 =0.5 0.3 0.4 0.3
H4 =0.7 0.0 0.0 0.5
H5 =0.9 0.0 0.0 0.0
H 0.3 0.1 0.2
Fig. 1. Bayesian network for multimorbidity.
4. Numerical example Suppose the multimorbidity Bayesian network has been recognized as Fig. 1 [1], and the CPTs existing in this network are almost determined by machine-learning methods or other methods, but the CPT attached to node “disease B” (DB for short) is empty because of a lack of prior data. Now this CPT attached to node DB has to be inferred by utilizing the construction method presented in this paper. Suppose a set of three experts, E = {ek |k = 1, 2, 3}, is invited to participate in this CPT inference problem, and their corresponding relative weights are W = {w1 = 0.5, w2 = 0.3, w3 = 0.2}. The evaluation grades are defined as H = {H1 =0.1, H2 =0.3, · · · , H5 =0.9}. The node has three parent nodes (characteristics, environment, and genetics, respectively, CH, EN, and GE for short), and each has two states, True or False (T and F for short), so that there are I = 23 CPs to be inferred in this CPT, i.e., (i) pt ,i = 1, 2, · · · , 8. The CPT is described as Table 1.
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mn,e(3) βn (i) αn (pt )
H1 =0.1 0.1419 0.2100 0.2563
H2 =0.3 0.0948 0.1402 0.1787
H3 =0.5 0.2470 0.3655 0.4292
H4 =0.7 0.0549 0.0813 0.1131
H5 =0.9 0.0000 0.0000 0.0228
H 0.1372 0.2031 −
m ¯ H,e(K) 0.3241 − −
(1)
Due to space limitations, we select pt to demonstrate the inference method, and other CPs could be inferred similarly. Suppose the belief distributions assessed by three experts are shown in Table 2. Taking the belief distributions in Table 2 into the construction method proposed in Section 3, we can derive the combined probability assignments, the pignistic probability, and the expected inference probability (see Table 3). We integrate the expected inference probability with its corresponding assessment grade by (i) (i) (2) (8) Eq. (7) and obtain p˜t as well as 1 − p˜t . Similarly, p˜t − p˜t in Table 1 can be inferred. From the above procedure, it is easy to see that the CPT in the multimorbidity Bayesian network can be determined as long as the inference assessment (e.g., belief distribution) can be inferred by each expert according to his/her knowledge or experience. 5. Conclusions The intrinsic mechanism of multimorbidity is difficult to recognize roundly and prediction and diagnosis are difficult to carry out accordingly. Bayesian networks can help predict or diagnose multimorbidity in health care, but it is difficult to get its CPT since there is little clinically statistical data. Today, expert knowledge and experience have been increasingly used as part of the Bayesian network training process in order to enhance the learning effect, but they do not take into account the fact that there are many differences in expert inference ability and perspective. In order to solve these problems, the ER approach was employed to extract and integrate the information from experts using belief distribution and a recursive ER algorithm, based on which an evidence reasoning method for constructing a CPT in the multimorbidity Bayesian network. Finally, a multimorbidity numerical example was utilized to demonstrate the proposed method and prove its feasibility and applicability. As long as the Bayesian network is established by the method proposed in this paper, the work of preventing and diagnosing multimorbidity, for example, could be carried out using Bayes, conditional probability, and total probability formulas. Acknowledgements This work is supported by National Natural Science Funds (71261011, 71462022), Applied Basic Research Project in Yunnan Province (2011FZ021, 2013FB030), and Innovation Team Building Project for Philosophy and Social Sciences in Yunnan Province (2014cx05). References [1] [2]
Lappenschaar M, Hommersom A, Lucas PJF, Lagro J, Visscher S. Multilevel Bayesian networks for the analysis of hierarchical health care data. Artif Intell Med. 2013; 57(3): 171-83. Diederichs C, Berger K, Bartels DB. The measurement of multiple chronic diseases – a systematic review on existing multimorbidity indices. The journals of gerontology. Series A, Biological Sciences and Medical Sciences. 2011; 66(3): 301-11.
Y. Du and Y. Guo / Evidence reasoning method for constructing CPT [3] [4] [5] [6] [7] [8] [9] [10]
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Fortin M, Hudon C, Haggerty J, Akker MVD, Almirall J. Prevalence estimates of multimorbidity: A comparative study of two sources. Bmc Health Serv Res. 2010; 10(1): 111. Nielen MMJ, Schellevis FOG, Verheij RA. Inter-practice variation in diagnosing hypertension and diabetes mellitus: A cross-sectional study in general practice. BMC Family Practice. 2009; 10(1): 6. Henriksson GR, Weitoft GRC, Allebeck P. Associations between income inequality at municipality level and health depend on context – a multilevel analysis on myocardial infarction in Sweden. Social Science & Medicine (1982). 2010; 71(6): 1141-9. van Gerven MAJ, Taal BG, Lucas PJF. Dynamic Bayesian networks as prognostic models for clinical patient management. J Biomed Inform. 2008; 41(4): 515-29. Velikova M, van Scheltinga JT, Lucas PJF, Spaanderman M. Exploiting causal functional relationships in Bayesian network modelling for personalised healthcare. Int J Approx Reason. 2014 2014-01-01; 55(1, Part 1): 59-73. Masegosa AR, Moral S. An interactive approach for Bayesian network learning using domain/expert knowledge. Int J Approx Reason. 2013; 54(8): 1168-81. Fu C, Chin K. Robust evidential reasoning approach with unknown attribute weights. Knowl-Based Syst. 2014; 59: 9-20. Xu D. An introduction and survey of the evidential reasoning approach for multiple criteria decision analysis. Ann Oper Res. 2012; 195(1): 163-87.
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Evidence reasoning method for constructing conditional probability tables in a Bayesian network of multimorbidity Yuanwei Du∗ and Yubin Guo Faculty of Management and Economics, Kunming University of Science and Technology, Kunming, Yunnan, China Abstract. BACKGROUND: The intrinsic mechanism of multimorbidity is difficult to recognize and prediction and diagnosis are difficult to carry out accordingly. Bayesian networks can help to diagnose multimorbidity in health care, but it is difficult to obtain the conditional probability table (CPT) because of the lack of clinically statistical data. OBJECTIVE: Today, expert knowledge and experience are increasingly used in training Bayesian networks in order to help predict or diagnose diseases, but the CPT in Bayesian networks is usually irrational or ineffective for ignoring realistic constraints especially in multimorbidity. METHODS: In order to solve these problems, an evidence reasoning (ER) approach is employed to extract and fuse inference data from experts using a belief distribution and recursive ER algorithm, based on which evidence reasoning method for constructing conditional probability tables in Bayesian network of multimorbidity is presented step by step. RESULTS: A multimorbidity numerical example is used to demonstrate the method and prove its feasibility and application. Bayesian network can be determined as long as the inference assessment is inferred by each expert according to his/her knowledge or experience. CONCLUSIONS: Our method is more effective than existing methods for extracting expert inference data accurately and is fused effectively for constructing CPTs in a Bayesian network of multimorbidity. Keywords: Evidence reasoning, Bayesian network, multimorbidity, health care, construction method
1. Introduction Multimorbidity is an epidemic in which a patient has a variety of diseases at the same time, which can produce diverse effects on the status of the patient’s health because of the possible interactions between various diseases [1]. At present the existing research can be divided into two categories: One focuses on considering multimorbidity as an aggregation of special diseases and establishing an index system in order to find which category it belongs to through the sampling investigation method or monitoring experiment method [2,3]; the other focuses on predicting the trend of multimorbidity development by integrating basic discriminant information in all kinds of diseases with the hierarchical regression analysis method [4,5]. Obviously, these two types of research have important contributions to the study of multimorbidity. However, it has to be recognized that the former focuses on the overall performance ∗ Corresponding author: Yuanwei Du, Faculty of Management and Economics, Kunming University of Science and Technology, Kunming, Yunnan, China. Tel.: +86 871 65154194; Fax: +86 871 65154194; E-maill: [email protected].
c 2015 – IOS Press and the authors. All rights reserved 0928-7329/15/$35.00
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of multimorbidity, but ignores the interaction between different diseases, and the latter recognizes the interactions among diseases but it can predict results from the cause but not diagnose the cause from the result. As a result, the intrinsic mechanism of multimorbidity is difficult to be recognized roundly and the prediction and the diagnosis are hardly carried out accordingly. Bayesian networks are an important method in the field of artificial intelligence and have been successfully used to solve many prediction and diagnosis problems in health care [6,7]. It is not difficult to see how Bayesian networks can address predicting and diagnosing a single disease by considering the causal relationship between interrelated variables, but they are rarely useful when considering multimorbidity. One reason is that there is numerous clinically statistical data for single diseases, which can be used to construct a Bayesian network for predicting or diagnosing. The other reason is that there is little clinically statistical data about multimorbidity, and it is difficult to get the conditional probability table (CPT) by relevant methods such as machine learning. Experts in the health care field, engaged in preventing, diagnosing, or treating diseases, have accumulated rich professional knowledge and clinical experience and have the ability to infer to some extent the mechanism of multimorbidity, and the integrated results will be the result of inference data of more than one expert. In this paper, we assume that the Bayesian multimorbidity network structure has been completed and some CPTs are empty and need to be inferred by experts.
2. Problem description and inference information expression A Bayesian network can be described as a tuple G = ((N, V ), D), where (N, V ) is a directed acyclic graph that is used to represent the characteristics of qualitative knowledge in the model, N = (N1 , N2 , · · · , NR ), which represents a set of nodes in the described field, and V is a set of arcs among nodes. If there is no connection between nodes Nr and Nr (r = r ), they are independent of each other and have no direct causal relationship; if there is a connection Nr → Nr between double nodes, then there is a direct causal relationship between Nr and Nr , and we call Nr a parent node and Nr a child node. The direction of the arc determines the causal relationship between the double nodes, and it can be used in causal reasoning. D = (D1 , D2 , · · · , DT ) represents the set of parameters of conditional probability (CP) in the network, giving the quantitative knowledge of the model. Assume that node Nr has two states {Nr = True, Nr = False} in general, r = 1, · · · , R. The (s) CPT Dt attached to Nr can be built by P (Nr = xr |Nr = xr ). Assume that {Nt |s= 1, 2, · · · ,S} is the set of parent nodes of Nr , then the number of combination of states in the CPT attached to Nr is I = 2s . The CPT is usually determined by a machine-learning algorithm, but these algorithms can lose efficacy in complex situations such as multimorbidity. When traditional machine-learning algorithms all (i) fail, determining the CPT attached to Nr denoted by pt (i = 1, 2, · · · , I) is an important issue in the Bayesian network. Today, expert knowledge and experience is increasingly introduced into the training process of Bayesian networks in order to enhance the learning effect [8]. However, most existing studies assume that all experts have the ability to estimate the CPs in the CPT using a specific information demand structure (such as a pairwise comparison judgment matrix). This research does not take into account the fact that there are many differences in expert inference abilities and inference perspectives. These realistic constraints can easily cause problems since the individually inferred data lacks rationality and integrated data lacks effectiveness. Therefore, the realistic constraint that the ability of experts is always limited has to be considered in the inference process. Therefore, how to accurately extract the inference
Y. Du and Y. Guo / Evidence reasoning method for constructing CPT
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data of experts and how to effectively fuse this information are pivotal problems for constructing the CPT in a multimorbidity Bayesian network. The ER approach can model various kinds of uncertainties such as ignorance, fuzziness, interval data, and interval belief degrees in a unified format. It has been used in recent years to uniformly solve uncertain multiple attribute decision making (MADM) problems [9,10]. The above analysis shows that the extension of the ER approach to model and solve CPT construction problems under uncertainties is a new and significant research area and different from most existing relevant methods. The ER approach not only allows an expert to give a certain inference degree about familiar CPs, but also allows the expert to surrender judgment and give no inference information on about unfamiliar CPs. Assume that H = {H1 , H2 , · · · , HN } denotes a set of assessment grades and the number of grades (i) is determined according to the requirement for accuracy. If pt is assessed to a grade Hn by expert ek (i) with a belief degree of βn,k (pt ), the assessment can be expressed by the following belief distribution: (i)
(i)
(i)
Bk (pt ) = {(Hn , βn,k (pt )), n = 1, · · · , N ; (H, βH,k (pt ))} (1) (i) (i) (i) (i) (i) In Eq. (1), βn,k (pt ) 0, n βn,k (pt ) 1, and βH,k (pt ) = 1 − n βn,k (pt ). Note that βn,k (pt ) (i) is a degree of belief that pt is assessed as grade Hn by expert ek , H = {H1 , H2 , · · · , Hn } is the frame (i) of discernment, and βH,k (pt ) denotes the degree of global ignorance with the meaning of the degree of unknown of this inference problem by expert ek . According to the above information expression (i) method, each expert can give their inference data to all of the CPs for Nt , which is described as Bk (pt ), k = 1, 2, · · · , K , i = 1, 2, · · · , I .
3. CPT construction method In a Bayesian network multimorbidity structure, assume that the CPT of node Nt needs to be inferred (s) by experts. Also assume {Nt |s= 1, 2, · · · ,S} is the set of parent nodes of Nt , each with two states, True or False states. The ER method for constructing the CPT in a Bayesian network of multimorbidity is constructed as follows: Step 1: Definition and representation of the CPT inference problem. Define a set of K experts as E = {e k |k = 1, 2, · · · , K}, and estimate the relative weights of the experts as W = {wk |0 wk 1; k wk = 1; k = 1, 2, · · · , K}. Define distinctive evaluation grades Hn as a complete set of standards for assessing each CP in CPT by all experts, or H = {H1 , H2 , · · · , HN }. The number of grades should be determined in advance according to the requirements for accuracy. Then the CPT inference problem can be stated as how to determine a CP from the inference information as (i) (1) given by each expert, i.e., Bk (pt ), k = 1, 2, · · · , K ,i = 1, 2, · · · , I . (i) Step 2: Let i = 1. Define the current inference problem as pt , i = 1. Step 3: Determine the basic probability assignments for each expert. Let mn,k be a basic probability (i) mass representing the degree to which ek supports a hypothesis and pt is assessed to Hn . Let mH,k be a remaining probability mass unassigned to any individual grade after ek has been assessed. mn,k and mH,k are calculated as (2). Let mH,k = m ¯ H,k + m ˜ H,k , where m ¯ H,k = 1 − wk and (i) (i) m ˜ H,k = wk (1 − n βn,k (pt )) = wk βΩ,k (pt ). (i) mn,k = wk βn,k (pt ), n = 1, 2, · · · , N (2) (i) mH,k = 1 − n mn,k = 1 − wk n βn,k (pt ), k = 1, 2, · · · , K
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Step 4: Combine the probability assignments for all of experts. Let mn,e(1) = mn,1 (n = 1, 2, · · · , N ), m ¯ H,e(1) = m ¯ H,1 , m ˜ H,e(1) = m ˜ H,1 and mH,e(1) = mH,1 . The combined probability assignments mn,e(K)(n = 1, 2, · · · , N ), m ¯ n,e(K), m ˜ n,e(K), and m ˜ H,e(K) can be generated by aggregating all the basic probability assignments using the following recursive ER algorithm:
{Hn } : mn,e(k+1) = Ke(k+1) [mn,e(k) mn,k+1 + mH,e(k) mn,k+1 + mn,e(k)mH,k+1 ] ˜ H,e(k) + m ¯ H,e(k) {H} : mH,e(k) = m
(3)
˜ H,e(k+1) = Ke(k+1) [m ˜ H,e(k) m ˜ H,k+1 + m ¯ H,e(k) m ˜ H,k+1 + m ˜ H,e(k) m ¯ H,k+1 ], m ¯ H,e(k+1) = where m N N −1 m ¯ H,e(k) m ¯ H,k+1 ], Ke(k+1) = [1 − t=1 m ] . Ke(k+1) [m j = 1 t,e(k) j,k+1 j = t Step 5: Calculate the combined degrees of belief for all of experts. Let βn denote a degree of belief that (i) all experts at pt are assessed to grade Hn , which is generated by combining the assessments for (i) all the associated experts ek at pt . Then βn is calculated by
{Hn } : βn = mn,e(K) (1 − m ¯ H,e(K)), n = 1, 2, · · · , N ¯ H,e(K)) {H} : βH = mH,e(K) (1 − m
(4)
Step 6: Determine the pignistic probability for the current CP. Let belief function and plausibility function of Hn as Bel(Hn ) = m(B) and P l(H ) = n B⊆Hn Hn ∩B=∅ m(B), set factorε = (i) [1 − Hn ⊆H Bel(Hn )] / Hn ⊆H P l(Hn ), and the pignistic probability (9) for pt corresponding toHn is (i)
αn (pt ) = Bel(Hn ) + ε · P l(Hn ),
n = 1, 2, · · · , N.
(5)
Step 7: Let i = i + 1. If i I then turn to Step 3, otherwise go to Step 8. (i) Step 8: Construct the CPT. The distributed overall assessment of pt is given by the following distribution: (i)
(i)
B(pt ) = {(Hn , αn (pt )), n = 1, 2, · · · , N },
i = 1, 2, · · · , I.
(6) (i)
If the overall assessment is complete, then the expected inference probability of pt is calculated by (i)
p˜t =
n
(i)
Hn αn (pt ), (i)
i = 1, 2, · · · , I.
(7) (i)
Note from Eq. (7) that p˜t is the inference CP corresponding to pt by integration with inference in(i) (i) formation generated by all of experts. p˜t and 1 − p˜t can be seen as the CPs that the probability of Nt = T rue and Nt = F alse under the condition of the ith combination of states in the CPT attached (i) (i) to Nt . When the CPs are all inferred, i.e., p˜t and 1 − p˜t , i = 1, 2, · · · , I , the CPT attached to node Nt has been determined.
Y. Du and Y. Guo / Evidence reasoning method for constructing CPT
State of parent nodes
CH=T EN=T GE=T
DB=T DB=F
pt (1) 1-pt
(1)
Table 1 CPT to be inferred and attached to node DB CH=T CH=T CH=T CH=F EN=T EN=F EN=F EN=T GE=F GE=T GE=F GE=T (2)
pt (2) 1-pt
(3)
pt (3) 1-pt
(4)
pt (4) 1-pt
(5)
pt (5) 1-pt
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CH=F EN=T GE=F (6)
pt (6) 1-pt
CH=T EN=F GE=T (7)
pt (7) 1-pt
CH=T EN=T GE=T (8)
pt (8) 1-pt
Table 2 Belief distributions inferred by three experts e1 e2 e3
H1 =0.1 0.4 0.0 0.0
H2 =0.3 0.0 0.5 0.0
H3 =0.5 0.3 0.4 0.3
H4 =0.7 0.0 0.0 0.5
H5 =0.9 0.0 0.0 0.0
H 0.3 0.1 0.2
Fig. 1. Bayesian network for multimorbidity.
4. Numerical example Suppose the multimorbidity Bayesian network has been recognized as Fig. 1 [1], and the CPTs existing in this network are almost determined by machine-learning methods or other methods, but the CPT attached to node “disease B” (DB for short) is empty because of a lack of prior data. Now this CPT attached to node DB has to be inferred by utilizing the construction method presented in this paper. Suppose a set of three experts, E = {ek |k = 1, 2, 3}, is invited to participate in this CPT inference problem, and their corresponding relative weights are W = {w1 = 0.5, w2 = 0.3, w3 = 0.2}. The evaluation grades are defined as H = {H1 =0.1, H2 =0.3, · · · , H5 =0.9}. The node has three parent nodes (characteristics, environment, and genetics, respectively, CH, EN, and GE for short), and each has two states, True or False (T and F for short), so that there are I = 23 CPs to be inferred in this CPT, i.e., (i) pt ,i = 1, 2, · · · , 8. The CPT is described as Table 1.
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Y. Du and Y. Guo / Evidence reasoning method for constructing CPT Table 3 Progress data in calculation
mn,e(3) βn (i) αn (pt )
H1 =0.1 0.1419 0.2100 0.2563
H2 =0.3 0.0948 0.1402 0.1787
H3 =0.5 0.2470 0.3655 0.4292
H4 =0.7 0.0549 0.0813 0.1131
H5 =0.9 0.0000 0.0000 0.0228
H 0.1372 0.2031 −
m ¯ H,e(K) 0.3241 − −
(1)
Due to space limitations, we select pt to demonstrate the inference method, and other CPs could be inferred similarly. Suppose the belief distributions assessed by three experts are shown in Table 2. Taking the belief distributions in Table 2 into the construction method proposed in Section 3, we can derive the combined probability assignments, the pignistic probability, and the expected inference probability (see Table 3). We integrate the expected inference probability with its corresponding assessment grade by (i) (i) (2) (8) Eq. (7) and obtain p˜t as well as 1 − p˜t . Similarly, p˜t − p˜t in Table 1 can be inferred. From the above procedure, it is easy to see that the CPT in the multimorbidity Bayesian network can be determined as long as the inference assessment (e.g., belief distribution) can be inferred by each expert according to his/her knowledge or experience. 5. Conclusions The intrinsic mechanism of multimorbidity is difficult to recognize roundly and prediction and diagnosis are difficult to carry out accordingly. Bayesian networks can help predict or diagnose multimorbidity in health care, but it is difficult to get its CPT since there is little clinically statistical data. Today, expert knowledge and experience have been increasingly used as part of the Bayesian network training process in order to enhance the learning effect, but they do not take into account the fact that there are many differences in expert inference ability and perspective. In order to solve these problems, the ER approach was employed to extract and integrate the information from experts using belief distribution and a recursive ER algorithm, based on which an evidence reasoning method for constructing a CPT in the multimorbidity Bayesian network. Finally, a multimorbidity numerical example was utilized to demonstrate the proposed method and prove its feasibility and applicability. As long as the Bayesian network is established by the method proposed in this paper, the work of preventing and diagnosing multimorbidity, for example, could be carried out using Bayes, conditional probability, and total probability formulas. Acknowledgements This work is supported by National Natural Science Funds (71261011, 71462022), Applied Basic Research Project in Yunnan Province (2011FZ021, 2013FB030), and Innovation Team Building Project for Philosophy and Social Sciences in Yunnan Province (2014cx05). References [1] [2]
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