Related concepts of entropy play a very important role in dealing with uncertainty in terms of Shannon's information theory. However, for uncertain information involving epistemic uncertainty, which is usually modelled by using Dempster-Shafer theory, the concepts of cross entropy and relative entropy are still not well defined currently. Facing this issue, by reviewing and importing existing related work, this study gives new definitions of cross entropy and relative entropy of mass functions, which are respectively named as cross plausibility entropy and relative plausibility entropy since they are both based on an uncertainty measure called plausibility entropy. The properties of cross and relative plausibility entropies are also given, which shows a strong connection with classical cross entropy and relative entropy in Shannon's information theory. An example of application regarding parameter estimation is provided to show the effectiveness and reasonability of the presented entropies, which has implemented the parameter estimation for a generalized Bernoulli distribution with plausibility distribution observations.
How to represent and measure the uncertainty is one of the central issues in information sciences. In general, the uncertainty can be briefly classified into random uncertainty and epistemic uncertainty [33]. Attributed to Shannon's innovative contributions [1], quantifying the randomness of uncertain information is solved through the mathematical framework of information theory. Dempster-Shafer theory [2, 3], also known as belief function theory, is a widely used framework to represent information with epistemic uncertainty, where a mathematical structure called mass functions is provided to simultaneously describe discord and non-specificity involved in the given information [4]. However, measuring the uncertainty of mass functions is not yet well solved currently, especially it still has not a consensus with respect to the definitions of entropy, cross entropy, and relative entropy of mass functions.
With respect to the entropy of mass functions, many researchers have paid attention on the problem [5, 6, 7]. Klir [8] has proposed generalized information theory (GIT) which aims to generalize Shannon's information theory for probabilities to various uncertainty theories including imprecise probabilities, fuzzy sets, belief functions, and so on. However, the proposed aggregated uncertainty (AU) in GIT for mass functions is of some shortcomings [9], especially some of the underlying axiomatic requirements of AU have been challenged [10]. Recent years, with the proposal of Deng's entropy [11], a new entropy measure for calculating the uncertainty of a mass function, the research of uncertainty measures in Dempster-Shafer theory has welcomed a "strong resurgence" [12]. Many novel entropy definitions of mass functions have been put forward [13, 14, 15]. For example, Jirousek and Shenoy [10] designed an entropy measure for mass functions by combining plausibility transformation and weighted Hartley entropy. Zhou and Deng [16] proposed a fractal-based belief entropy on the basis of Deng's entropy. In terms of belief intervals of single elements, Moral-Garcia and Abellan [17] developed an uncertainty measure of mass functions which is analogous to AU. Besides, Cui and Deng [18] presented a total uncertainty measure of mass functions based on plausibility function, which is called plausibility entropy. Facing existing uncertainty measures, Dezert and Tchamova [19] have raised the effectiveness problem of uncertainty measures, and provided four desiderata to check if an uncertainty measure is effective, and a new effective measure of uncertainty for mass functions has been proposed in [20].
Cross entropy and relative entropy are another two important concepts according to Shannon's information theory. Although the research of entropy of mass functions is flourishing, cross entropy and relative entropy of mass functions, however, are relatively rare. This matter of fact is easy to understand, because cross and relative entropies are strongly connected with the concept of entropy, their forms are usually on the basis of the definition of entropy. Many existing entropy measures of mass functions are hard to yield corresponding cross and relative entropies. Very recently, Dezert and Dambreville [21] have provided definitions of cross entropy and relative entropy of mass functions in terms of Dezert's effective measure of uncertainty presented in [20]. In addition, Gao et al. [22] proposed a definition of cross entropy of mass functions based on Deng's entropy [11]. However, as will be analyzed in this paper, the existing definitions of cross entropy of mass functions are of some defects in underlying properties, new cross entropy, as well as corresponding relative entropy, of mass functions are required, which is the purpose of the study.
Specifically, inspired by the plausibility entropy, a total uncertainty measure of mass functions, presented in [18], new cross entropy and relative entropy of mass functions are given in this paper, which are named as cross plausibility entropy and relative plausibility entropy, respectively. The properties of cross and relative plausibility entropies are given, which shows a strong connection with classical cross entropy and relative entropy in Shannon's information theory. In addition, an illustrative example in parameter estimation is given to show the potential application of the presented entropies.
The remainder of the paper is organized as follows. Section 2 briefly introduces the basic knowledge of Dempster-Shafer theory. Related work regarding existing definitions of cross entropy of mass functions are reviewed in Section 3. Then, new cross and relative entropies of mass functions are given in Section 4. An example of application is provided in Section 5. Finally, Section 6 concludes the study.
Basics of Dempster-Shafer theory
Dempster-Shafer theory [2, 3] has provided a well-defined framework to represent and deal with uncertainty information with epistemic uncertainty. In this theory, the set of possible answers to a given question of interest is called as a frame of discernment (FOD), denoted as Ω={θ1,θ2,⋯,θn}, in which the set is collectively exhaustive and all elements in a FOD are mutually exclusive. The power set of FOD Ω is represented by 2Ω.
In order to represent the uncertain information involving epistemic uncertainty, mass functions, also known as basic probability assignments (BPAs), are defined in Dempster-Shafer theory. A mass function is a mapping from the power set of a FOD to interval [0,1], denoted as m:2Ω→[0,1], satisfying the following conditions
m(∅)=0and∑A∈2Ωm(A)=1
where A is called as a focal element of m if m(A)>0, and m(A) measures the belief assigned exactly to A. In general, a probability distribution can be treated as a Bayesian mass function in Dempster-Shafer theory, where ∀A with |A|≥2 there is m(A)=0.
Belief function Bel and plausibility function Pl are two equivalent forms of mass functions, which respectively express the lower bound and upper bound of the support degree of a set A based on a given mass function m, A⊆Ω. Given a mass function m, Bel and Pl are defined as follows
Bel(A)=∑B⊆Am(B)
Pl(A)=1−Bel(A¯)=∑B∩A≠∅m(B)
where A¯=Ω−A. For each A⊆Ω, there is Bel(A)≤Pl(A), and [Bel(A),Pl(A)] is called the belief interval of A. In general, the wider the belief interval of a set A, the larger the uncertainty it contributes to the whole mass function. Therefore, in Dempster-Shafer theory, given a FOD Ω, the vacuous mass function mγ, in which mγ(Ω)=1, has the largest uncertainty.
Related work
In this section, two existing definitions of cross entropy of mass functions, proposed very recently, are reviewed.
Dezert and Dambreville's cross entropy
Dezert [20] has given an entropy definition to measure the uncertainty of a mass function m on a FOD Ω
U(m)=∑A∈2Ωs(A)
with
s(A)=−m(A)(1−u(A))ln(m(A))+u(A)(1−m(A))
where ln(⋅) represents the natural logarithm, and u(A)=Pl(A)−Bel(A) for any A∈2Ω.
On the basis of U(m), Dezert and Dambreville [21] have further proposed a cross entropy of mass functions
It is proved in [21] that U(m), U(m1,m2), and U(m1||m2) can respectively degenerate into classical Shannon's entropy, cross entropy, and relative entropy (also known as Kullback-Leibler (KL) divergence), and they own a relation U(m1,m2)=U(m1||m2)+U(m1).
Gao et al.'s cross entropy
Gao et al. [22] presented another cross entropy definition of mass functions as below
HD(m1,m2)=−∑A⊆Ωm1(A)log2m2(A)2|A|−1
in which the underlying entropy definition for mass functions is based on Deng's entropy [11] as follows
HD(m)=−∑A⊆Ωm(A)log2m(A)2|A|−1
Since the relative entropy inspired by Deng's entropy is not defined at present, quantitative relation between HD(m) and HD(m1,m2) is not established yet.
New cross and relative entropies of mass functionsAnalysis of existing cross entropy definitions
In this subsection, these two cross entropy definitions of mass functions mentioned in the above section are analyzed simply.
At first, Gao et al.'s cross entropy HD(m1,m2) has been questioned in [21] because the underlying Deng's entropy HD(m) is non-effective. According to the four desiderata proposed in [19], "unicity of max value of MoU" (D4), i.e., MoU(mγ)>MoU(m) for any m≠mγ in which mγ is the vacuous mass function, is not satisfied by HD(m). In terms of Deng's entropy, given a FOD Ω, the vacuous mass function, i.e., m(Ω)=1, does not have the maximum uncertainty. Please refer to literature [21, 31, 32] for more details.
Secondly, based on a similar consideration, Dezert and Dambreville's cross entropy U(m1,m2) may also not be recommended since its underlying entropy measure U(m) violates the monotonicity [17] of a rational uncertainty measure in belief function theory. Specifically, the monotonicity means that Uncertainty(m1)≤Uncertainty(m2) holds if ∀A⊆Ω:[Bel(A)m1,Pl(A)m1]⊆[Bel(A)m2,Pl(A)m2] for arbitrary BPAs m1, m2 defined on a same FOD Ω. Literature [23] has first revealed the problem of U(m), however the given counterexample in [23] is a little problem in the calculation of U(m) (specifically, log2 is misused). In this paper, a real counterexample of U(m) on the monotonicity is given as below.
Given a FOD Ω={a,b,c}, m1 and m2 are two mass functions defined on Ω, in which
m1(a)=0.05,m1(ac)=0.05,m1(bc)=0.9;
m2(ab)=0.05,m2(ac)=0.05,m2(bc)=0.9.
Obviously, there are Bel(A)m1≥Bel(A)m2 and Pl(A)m1≤Pl(A)m2 for any A⊆Ω. Therefore, it clearly should be Uncertainty(m1)≤Uncertainty(m2) in terms of the monotonicity. However, by means of the uncertainty measure U(m) expressed in nats, it obtains U(m1)=3.9549 and U(m2)=3.9153. Namely, there is U(m1)>U(m2). Therefore, the monotonicity is violated by U(m).
Based on the above analysis, new cross entropy definition of mass functions is required and it is exactly the purpose of the study.
Cross and relative plausibility entropies
In this paper, new cross and relative entropies of mass functions are presented, which is on the basis of an uncertainty measure called plausibility entropy.
The plausibility entropy was recently proposed in [18], which is defined as
where m is a mass function defined on FOD Ω={θ1,θ2,⋯,θn}. Alternatively, the plausibility entropy HPl(m) can also be expressed in the form of Shannon's entropy
HPl(m)=∑θi∈ΩPl(θi)×HS(Pl_Pm)
where Pl_Pm is the plausibility transformation [24] of m, satisfying Pl_Pm(θi)=Pl(θi)∑θj∈ΩPl(θj), θi∈Ω, and HS(Pl_Pm)=−∑θi∈ΩPl_Pm(θi)log2Pl_Pm(θi) is Shannon's entropy of probability distribution Pl_Pm on Ω.
It can be proved that the plausibility entropy HPl(m) has satisfied four desiderata given in [19] for an effective measure of uncertainty (MoU) including "zero min value of MoU" (D1), "increasing of MoU of vacuous BPA" (D2), "compatibility with Shannon's entropy" (D3), and "unicity of max value of MoU" (D4). In addition, many other desirable properties are also satisfied by HPl(m), please refer to [18, 25, 26] for more information. Based on the well-defined plausibility entropy, new cross and relative entropies of mass functions, named as cross plausibility entropy and relative plausibility entropy respectively, are given.
Cross plausibility entropy. Given two mass functions m1 and m2 on a same FOD Ω={θ1,θ2,⋯,θn}, a cross plausibility entropy, denoted as HPl(m1,m2), is defined as
where Pl_Pm1 and Pl_Pm2 are the plausibility transformations of m1 and m2 respectively, and HS(Pl_Pm1,Pl_Pm2)=−∑θi∈ΩPl_Pm1(θi)log2Pl_Pm2(θi) is the classical cross entropy between probability distributions Pl_Pm1 and Pl_Pm2.
In terms of Eq. (4), a series of properties satisfied by the cross plausibility entropy HPl(m1,m2) can be obtained easily.
<bold>.</bold>
HPl(m1,m2)≥HPl(m1), the equality holds if and only if Pl_Pm1=Pl_Pm2.
<bold>.</bold>
HPl(m1,m2)≠HPl(m2,m1), i.e., the cross plausibility entropy is not symmetric in general.
<bold>.</bold>
If m1 and m2 are Bayesian mass functions, the cross plausibility entropy coincides with the classical cross entropy for probabilities, i.e., HPl(m1,m2)=−∑θi∈Ωm1(θi)log2m2(θi).
Having the above definition of cross plausibility entropy, the relative entropy of mass functions can be defined immediately in a similar means. We have noted that reference [27] provided a KL divergence as a straightforward derivative of the plausibility entropy, it is exactly the expected form of relative entropy, which is introduced as follows.
Relative plausibility entropy. Let m1 and m2 be two mass functions defined on a FOD Ω={θ1,θ2,⋯,θn}, a relative plausibility entropy, denoted as HPl(m1||m2), is defined as follows
Similarly, the relative plausibility entropy HPl(m1||m2) can also be simply represented as
HPl(m1||m2)=∑θi∈ΩPlm1(θi)×HS(Pl_Pm1||Pl_Pm2)
where HS(Pl_Pm1||Pl_Pm2) is the classical cross entropy between probability distributions Pl_Pm1 and Pl_Pm2, i.e., HS(Pl_Pm1||Pl_Pm2)=∑θi∈ΩPl_Pm1(θi)log2Pl_Pm1(θi)Pl_Pm2(θi).
From Eq. (6), some properties of the relative plausibility entropy HPl(m1||m2) are derived as below.
<bold>.</bold>
HPl(m1||m2)≥0, the equality holds if and only if Pl_Pm1=Pl_Pm2.
<bold>.</bold>
HPl(m1||m2)≠HPl(m2||m1) in general, i.e., the cross plausibility entropy is not symmetric.
<bold>.</bold>
For two Bayesian mass functions m1 and m2, the cross plausibility entropy degenerates into classical relative entropy (or KL divergence) for probabilities, i.e., HPl(m1||m2)=∑θi∈Ωm1(θi)log2m1(θi)m2(θi).
What's more, as same as the equality relation for probability distributions P1 and P2 in terms of Shannon's entropy, classical cross entropy and relative entropy, i.e., HS(P1,P2)=HS(P1||P2)+HS(P1), the presented HPl(m1,m2), HPl(m1||m2), as well as plausibility entropy HPl(m1), of mass functions also meet the following equality relation
HPl(m1,m2)=HPl(m1||m2)+HPl(m1)
An example of application
In this section, an illustrative example regarding parameter estimation is provided to show the potential application of proposed cross plausibility entropy HPl(m1,m2). The example is originally from literature [28].
Assuming there are n patients randomly taken from a population which has a proportion θ to suffer from a disease, and each of them is represented by Xi to show if he/she has the disease (i.e., Xi=1) or not (i.e., Xi=0). Then, these random samples 𝐗=(X1,X2,⋯,Xn), which are independent and identically distributed (iid), can be viewed as the outcome of a Bernoulli variable. For realizations 𝐱=(x1,x2,⋯,xn)∈Ω𝐗={0,1}n, the probability can be calculated by
pX(𝐱;θ)=∏i=1nθxi(1−θ)1−xi
The task is to estimate the unknown parameter θ according to state descriptions 𝐱=(x1,x2,⋯,xn). However, due to the uncertainty, these states are only partially known, and let mi be the mass function concerning the state xi associated with patient i. Table 1 gives a data set composed of n=6 observations, in which the fourth one m4 is uncertain and represented by m4({1})=α, m4({0})=β, and m4({1,0})=1−α−β, where α,β,1−α−β∈[0,1].
Data set for the example of parameter estimation.
Observation i
1
2
3
4
5
6
mi({1})
0.0
0.0
0.0
α
1.0
1.0
mi({0})
1.0
1.0
1.0
β
0.0
0.0
mi({1,0})
0.0
0.0
0.0
1−α−β
0.0
0.0
Literature [28] proposed an evidential expectation-maximization (E2M) algorithm to estimate the parameter θ, in terms of a maximum likelihood principle. Figure 1 gives the results of using E2M algorithm with respect to different α and β. It is found that, by using the E2M algorithm, the result of estimated θ is unchanged if the plausibility transformations of different m4 caused by changed α and β are the same. For example, let m41 and m42 be m41({1})=0.6, m41({0})=0.4, m41({1,0})=0, and m42({1})=0.4, m42({0})=0.1, m42({1,0})=0.5, respectively. There are Pl_Pm41({1})=Pl_Pm42({1})=0.6 and Pl_Pm41({0})=Pl_Pm42({0})=0.4, i.e., Pl_Pm41=Pl_Pm42. Then, by using the E2M algorithm, parameter θ is estimated as θ1,∗=0.4201 associated with m41 and θ2,∗=0.4201 associated with m42. Two observations with different uncertainty degrees, m41 and m42, lead to the same estimation of θ.
Estimation of θ with the use of E2M algorithm by considering different values of α and β.
Now, let us use a cross entropy-based method to solve the issue of estimating parameter θ, where θ is derived by minimizing a total cross entropy loss, which coincides with the maximum likelihood principle widely used in machine learning.
For the data set shown in Table 1, since there is uncertain observation m4 involving epistemic uncertainty, the proposed cross plausibility entropy is used to calculate the total cross entropy loss LPl. Let P be a distribution relying on parameter θ with P(1)=θ and P(0)=1−θ, then
Figure 2 shows the results of using the proposed cross plausibility entropy with the consideration of different α and β. Compared with the results of E2M algorithm, the estimated θ is changing with m4 having different uncertainty degrees. For example, for the cases of m4=m41 and m4=m42, it is obtained that θPl1,∗=0.4333 and θPl2,∗=0.4462.
Estimation of θ with the use of proposed cross plausibility entropy by considering different values of α and β.
For the comparison, Dezert and Dambreville's cross entropy U(m1,m2) is also used in the example to obtain the estimation of parameter θ. Let mθ be the estimation of θ, in which mθ({1})=θ, mθ({0})=1−θ−ε, and mθ({1,0})=ε, where ε→0. Then, in terms of Dezert and Dambreville's cross entropy, the total loss is
Then, the estimation of θ is derived via ∂LD∂θ=0 as below
θD∗=2+α5+α+β(1−ε)
in which ε→0. The results with the use of cross entropy HD(m1,m2) are shown in Figure 4.
Estimation of θ with the use of cross entropy by considering different values of α and β.
In addition, cross entropies inspired by two widely used entropies of mass functions, pignistic entropy [29, 4] and Yager entropy [30], are also considered for comparison. In terms of the formula of pignistic entropy Hpignistic(m)=−∑θi∈ΩBetPm(θi)log2BetPm(θi), where BetPm(θi)=∑θi∈Am(A)|A|, cross pignistic entropy is naturally defined as Hpignistic(m1,m2)=−∑θi∈ΩBetPm1(θi)log2BetPm2(θi). Then, a total cross entropy loss can be obtained by
where P(1)=θ and P(0)=1−θ. By letting ∂Lpignistic∂θ=0, we have
θpignistic∗=5+α−β12
whose graphical results with different values of α and β are given in Figure 5.
Estimation of θ with the use of cross pignistic entropy by considering different values of α and β.
Similarly, according to the expression of Yager entropy HYager(m)=−∑A∈Ωm(A)log2Plm(A), cross Yager entropy is defined as HYager(m1,m2)=−∑A∈Ωm1(A)log2Plm2(A). Then, the corresponding total cross entropy loss is
which is same with θD∗ that is obtained by using cross entropy HD(m1,m2). Figure 6 shows the results of using cross Yager entropy by considering different α and β.
Estimation of θ with the use of cross Yager entropy by considering different values of α and β.
For the sake of further comparison of these methods, by letting α=β, the uncertain observation m4 becomes m4({1})=m4({0})=α and m4({1,0})=1−2α, where α∈[0,0.5]. If α=0, m4 has the maximum uncertainty, and the uncertainty of m4 is the least while α=0.5. And it is noted that there is not any preference in m4 between states {1} and {0} because Belm4({1})=Belm4({0})=α and Plm4({1})=Plm4({0})=1−α. Figure 7 shows the estimation results of θ by using different methods for the case of new m4 in which α=β.
Estimation results of θ generated by using different methods where observation 4 is set as m4({1})=m4({0})=α, m4({1,0})=1−2α.
Generalized Bernoulli distribution with plausibility distribution observations.
Observation i
1
2
3
4
5
6
Pli({1})
0.0
0.0
0.0
1−β
1.0
1.0
Pli({0})
1.0
1.0
1.0
1−α
0.0
0.0
In theory, if we only have observations 1,2,3,5,6, in terms of the maximum likelihood principle, the estimation of θ should be θ∗=2/5=0.4. With the consideration of observation 4, i.e., unbiased m4: (i)When α=0, the estimation of θ should lie in the interval [2/6,3/6] (whose midpoint is 2.5/6) since the existence of epistemic uncertainty in observation 4 with m4({1,0})=1; (ii) When α=0.5, the estimation of θ should be 2.5/6 since in this case the example becomes a mixture model for a two-dimensional Bernoulli distribution with m4({1})=m4({0})=0.5 in which there is only random uncertainty; (iii)In the process of increasing α's value from 0 to 0.5, the value of θ should be changing monotonically, because for the unbiased m4 the only change is its uncertainty degree which is decreasing monotonically; (iv)Therefore, there is a path for the estimation of θ from start point θ|α=0∈[26,36] to end point θ|α=0.5=2.56.
From Figure 7, the result of E2M algorithm is not reasonable since the estimated θ is 0.4 when α=0.5. Cross pignistic entropy Hpignistic(m1,m2) produces insensitive result for the change of observation m4. The proposed cross plausibility entropy HPl(m1,m2) gives that the value of θ is declining monotonically with the rise of α, while Dezert and Dambreville's cross entropy U(m1,m2), cross entropy HD(m1,m2), and cross Yager entropy HYager(m1,m2) present opposite trend of change. The difference between result of HPl(m1,m2) and those of U(m1,m2), HD(m1,m2), and HYager(m1,m2), is caused by the underlying logic of different entropy measures. The plausibility entropy is based on plausibility function and it tends to get the maximum uncertainty degree that a mass function could possibly have, therefore the cross plausibility entropy HPl(m1,m2) gives a relatively big estimation of θ. In contrast, it seems that U(m1,m2), HD(m1,m2) and HYager(m1,m2) are to obtain relatively small estimation of θ.
Compared with U(m1,m2), HD(m1,m2) and HYager(m1,m2), the proposed HPl(m1,m2) is more recommended in theory because, at first, the underlying entropy definitions of the formers are of some defects as analyzed in Section 4.1 and related references [19, 5], and the underlying mechanism of using cross plausibility entropy HPl(m1,m2) to obtain the estimation of θ is more clear. In this example, with the use of cross plausibility entropy HPl(m1,m2), the classical Bernoulli distribution based on probabilities is generalized to a new Bernoulli distribution with plausibility distribution observations as shown in Table 2. According to Table 2, the estimation of parameter θ can be obtained immediately as θPl∗=3−β7−α−β. Moreover, this Bernoulli distribution with plausibility distribution observations can be easily extended to the case of multi-dimensional Bernoulli distribution.
Conclusion
In this paper, new definitions of cross entropy and relative entropy of mass functions have been given on the basis of a recently presented total uncertainty measure of mass functions called plausibility entropy. And, properties of the cross plausibility entropy and relative plausibility entropy have been presented in the study. In addition, an illustrative example of application has been provided to show the effectiveness of the presented entropies compared with other methods and entropy definitions of mass functions. The presented cross plausibility entropy and relative plausibility entropy of mass functions can be used in the scenarios of multi-source information fusion based on Dempster-Shafer evidence theory for target recognition, fault diagnosis, and so on.
In the future study, on one hand, more theoretical analysis about the presented cross and relative plausibility entropies will be conducted; on the other hand, practical applications with the use of cross and relative plausibility entropies for multi-sensor information fusion, decision support systems, intelligent diagnosis, and so forth, will be further considered.
Acknowledgments
The work was partially supported by the National Natural Science Foundation of China under Grant 62173272.
Conflict of interest
The authors declare no conflicts of interest.
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