Contradiction measures and specificity degrees of basic belief assignments
aa r X i v : . [ c s . A I] S e p Contradiction measures and specificity degrees ofbasic belief assignments
Florentin Smarandache
Math. & Sciences Dept.University of New Mexico,200 College Road,Gallup, NM 87301, U.S.A.Email: [email protected]
Arnaud Martin
IRISAUniversity of Rennes 1Rue ´Edouard Branly BP 3021922302 Lannion, FranceEmail: [email protected]
Christophe Osswald
E3I2ENSTA Bretagne2, rue Franc¸ois Verny29806 Brest, Cedex 9, FranceEmail: [email protected]
Abstract —In the theory of belief functions, many measuresof uncertainty have been introduced. However, it is not alwayseasy to understand what these measures really try to represent.In this paper, we re-interpret some measures of uncertainty inthe theory of belief functions. We present some interests anddrawbacks of the existing measures. On these observations, weintroduce a measure of contradiction. Therefore, we present somedegrees of non-specificity and Bayesianity of a mass. We proposea degree of specificity based on the distance between a mass andits most specific associated mass. We also show how to use thedegree of specificity to measure the specificity of a fusion rule.Illustrations on simple examples are given.
Keywords: Belief function, uncertainty measures, speci-ficity, conflict.
I. I
NTRODUCTION
The theory of belief functions was first introduced by [1]in order to represent some imprecise probabilities with upper and lower probabilities . Then [15] proposed a mathematicaltheory of evidence.Let Θ be a frame of discernment. A basic belief assignment (bba) m is the mapping from elements of the powerset Θ onto [0 , such that: X X ∈ Θ m ( X ) = 1 . (1)The axiom m ( ∅ ) = 0 is often used, but not mandatory. A focal element X is an element of Θ such that m ( X ) = 0 ..The difference of a bba with a probability is the domain ofdefinition. A bba is defined on the powerset Θ and not onlyon Θ . In the powerset, each element is not equivalent in termsof precision. Indeed, θ ∈ Θ is more precise than θ ∪ θ ∈ Θ .In the case of the DSmT introduced in [17], the bba aredefined on an extension of the powerset: the hyper powersetnoted D Θ , formed by the closure of Θ by union and inter-section. The problem of signification of each focal element isthe same as in Θ . For instance, θ ∈ Θ is less precise than θ ∩ θ ∈ D Θ . In the rest of the paper, we will note G Θ foreither Θ or D Θ .In order to try to quantify the measure of uncertainty suchas in the set theory [5] or in the theory of probabilities[16], some measures have been proposed and discussed in the theory of belief functions [2], [7], [8], [21]. However,the domain of definition of the bba does not allow an idealdefinition of measure of uncertainty. Moreover, behind theterm of uncertainty, different notions are hidden.In the section II, we present different kinds of measuresof uncertainty given in the state of art, we discuss them andgive our definitions of some terms concerning the uncertainty.In section III, we introduce a measure of contradiction anddiscuss it. We introduce simple degrees of uncertainty in thesection IV, and propose a degree of specificity in the sectionV. We show how this degree of specificity can be used tomeasure the specificity of a combination rule.II. M EASURES OF UNCERTAINTY ON BELIEF FUNCTIONS
In the framework of the belief functions, several functions(we call them belief functions ) are in one to one correspon-dence with the bba: bel , pl and q . From these belief functions,we can define several measures of uncertainty. Klir in [8]distinguishes two kinds of uncertainty: the non-specificityand the discord. Hence, we recall hereafter the main belieffunctions, and some non-specificity and discord measures. A. Belief functions
Hence, the credibility and plausibility functions representrespectively a minimal and maximal belief. The credibility function is given from a bba for all X ∈ G Θ by: bel( X ) = X Y ⊆ X,Y m ( Y ) . (2)The plausibility is given from a bba for all X ∈ G Θ by: pl( X ) = X Y ∈ G Θ ,Y ∩ X m ( Y ) . (3)The commonality function is also another belief function givenby: q( X ) = X Y ∈ G Θ ,Y ⊇ X m ( Y ) . (4)These functions allow an implicit model of imprecise anduncertain data. However, these functions are monotonic byinclusion: bel and pl are increasing, and q is decreasing. Thiss the reason why the most of time we use a probability to takea decision. The most used projection into probability subspaceis the pignistic probability transformation introduced by [18]and given by: betP( X ) = X Y ∈ G Θ ,Y | X ∩ Y || Y | m ( Y ) , (5)where | X | is the cardinality of X , in the case of the DSmTthat is the number of disjoint elements corresponding in theVenn diagram.From this probability, we can use the measure of uncertaintygiven in the theory of probabilities such as the Shannonentropy [16], but we loose the interest of the belief functionsand the information given on the subsets of the discernmentspace Θ . B. Non-specificity
The non-specificity in the classical set theory is the impre-cision of the sets. Such as in [14], we define in the theory ofbelief functions, the non-specificity related to vagueness andnon-specificity.
Definition
The non-specificity in the theory of belieffunctions quantifies how a bba m is imprecise. The non-specificity of a subset X is defined by Hartley[5] by log ( | X | ) . This measure was generalized by [2] in thetheory of belief functions by: NS( m ) = X X ∈ G Θ , X m ( X ) log ( | X | ) . (6)That is a weighted sum of the non-specificity, and the weightsare given by the basic belief in X . Ramer in [13] has shownthat it is the unique possible measure of non-specificity in thetheory of belief functions under some assumptions such assymmetry, additivity, sub-additivity, continuity, branching andnormalization.If the measure of the non-specificity on a bba is low, we canconsider the bba is specific. Yager in [21] defined a specificitymeasure such as: S ( m ) = X X ∈ G Θ , X m ( X ) | X | . (7)Both definitions corresponded to an accumulation of afunction of the basic belief assignment on the focal elements.Unlike the classical set theory, we must take into account thebba in order to quantify (to weight) the belief of the imprecisefocal elements. The imprecision of a focal element can ofcourse be given by the cardinality of the element.First of all, we must be able to compare the non-specificity(or specificity) between several bba’s, event if these bba’s arenot defined on the same discernment space. That is not thecase with the equations (6) and (7). The non-specificity of theequation (6) takes its values in [0 , log ( | Θ | )] . The specificityof the equation (7) can have values in [ | Θ | , . We will showhow we can easily define a degree of non-specificity in [0 , .We could also define a degree of specificity from the equation (7), but that is more complicated and we will later show howwe can define a specificity degree.The most non-specific bba’s for both equations (6) and (7)are the total ignorance bba given by the categorical bba m Θ : m (Θ) = 1 . We have NS( m ) = log ( | Θ | ) and S ( m ) = | Θ | .This categorical bba is clearly the most non-specific for us.However, the most specific bba’s are the Bayesian bba’s. Theonly focal elements of a Bayesian bba are the simple elementsof Θ . On these kinds of bba m we have NS( m ) = 0 and S ( m ) = 1 . For example, we take the three Bayesian bba’sdefined on Θ = { θ , θ , θ } by: m ( θ ) = m ( θ ) = m ( θ ) = 1 / , (8) m ( θ ) = m ( θ ) = 1 / , m ( θ ) = 0 , (9) m ( θ ) = 1 , m ( θ ) = m ( θ ) = 0 . (10)We obtain the same non-specificity and specificity for thesethree bba’s.That hurts our intuition; indeed, we intuitively expect thatthe bba m is the most specific and the m is the less specific.We will define a degree of specificity according to a mostspecific bba that we will introduce. C. Discord
Different kinds of discord have been defined as extensionsfor belief functions of the Shannon entropy, given for theprobabilities. Some discord measures have been proposed fromplausibility, credibility or pignistic probability: E ( m ) = − X X ∈ G Θ m ( X ) log (pl( X )) , (11) C ( m ) = − X X ∈ G Θ m ( X ) log (bel( X )) , (12) D ( m ) = − X X ∈ G Θ m ( X ) log (betP( X )) , (13)with E ( m ) ≤ D ( m ) ≤ C ( m ) . We can also give the Shanonentropy on the pignistic probability: − X X ∈ G Θ betP( X ) log (betP( X )) . (14)Other measures have been proposed, [8] has shown that thesemeasures can be resumed by: − X X ∈ G Θ m ( X ) log (1 − Con m ( X )) , (15)where Con is called a conflict measure of the bba m on X . However, in our point of view that is not a conflictsuch presented in [20], but a contradiction. We give the bothfollowing definitions: Definition A contradiction in the theory of belief functionsquantifies how a bba m contradicts itself. Definition (C1)
The conflict in the theory of belief functionscan be defined by the contradiction between 2 or more bba’s.
In order to measure the conflict in the theory of belieffunctions, it was usual to use the mass k given by theonjunctive combination rule on the empty set. This rule isgiven by two basic belief assignments m and m and for all X ∈ G Θ by: m c ( X ) = X A ∩ B = X m ( A ) m ( B ) := ( m ⊕ m )( X ) . (16) k = m c ( ∅ ) can also be interpreted as a non-expected solution.In [21], Yager proposed another conflict measure from thevalue of k given by − log (1 − k ) .However, as observed in [9], the weight of conflict givenby k (and all the functions of k ) is not a conflict measurebetween the basic belief assignments. Indeed this value iscompletely dependant of the conjunctive rule and this ruleis non-idempotent - the combination of identical basic beliefassignments leads generally to a positive value of k . Tohighlight this behavior, we defined in [12] the auto-conflict which quantifies the intrinsic conflict of a bba. The auto-conflict of order n for one expert is given by: a n = (cid:18) n ⊕ i =1 m (cid:19) ( ∅ ) . (17)The auto-conflict is a kind of measure of the contradiction,but depends on the order. We studied its behavior in [11].Therefore we need to define a measure of contradictionindependent on the order. This measure is presented in thenext section III.III. A CONTRADICTION MEASURE
The definition of the conflict (C1) involves firstly to measureit on the bba’s space and secondly that if the opinions of twoexperts are far from each other, we consider that they are inconflict. That suggests a notion of distance. That is the reasonwhy in [11], we give a definition of the measure of conflictbetween experts assertions through a distance between theirrespective bba’s. The conflict measure between experts isdefined by: Conf(1 ,
2) = d ( m , m ) . (18)We defined the conflict measure between one expert i and theother M − experts by: Conf( i, E ) = 1 M − M X j =1 ,i = j Conf( i, j ) , (19)where E = { , . . . , M } is the set of experts in conflict with i .Another definition is given by: Conf( i, M ) = d ( m i , m M ) , (20)where m M is the bba of the artificial expert representing thecombined opinions of all the experts in E except i .We use the distance defined in [6], which is for us the mostappropriate, but other distances are possible. See [4] for acomparison of distances in the theory of belief functions. Thisdistance is defined for two basic belief assignments m and m on G Θ by: d ( m , m ) = r
12 ( m − m ) T D ( m − m ) , (21) where D is an G | Θ | × G | Θ | matrix based on Jaccard distancewhose elements are: D ( A, B ) = , if A = B = ∅ , | A ∩ B || A ∪ B | , ∀ A, B ∈ G Θ . (22)However, this measure is a total conflict measure. In orderto define a contradiction measure we keep the same spirit.First, the contradiction of an element X with respect to a bba m is defined as the distance between the bba’s m and m X ,where m X ( X ) = 1 , X ∈ G Θ , is the categorical bba: Contr m ( X ) = d ( m, m X ) , (23)where the distance can also be the Jousselme distance on thebba’s. The contradiction of a bba is then defined as a weightedcontradiction of all the elements X of the considered space G Θ : Contr m = c X X ∈ G Θ m ( X ) d ( m, m X ) , (24)where c is a normalized constant which depends on thetype of distance used and on the cardinality of the frame ofdiscernment in order to obtain values in [0 , as shown in thefollowing illustration. A. Illustration
Here the value c in the equation (24) is equal to 2. First wenote that for all categorical bbas m Y , the contradiction givenby the equation (23) gives Contr m Y ( Y ) = 0 and the contra-diction given by the equation (24) brings also Contr m Y = 0 .Considering the bba m ( θ ) = 0 . and m ( θ ) = 0 . , wehave Contr m = 1 . That is the maximum of the contradiction,hence the contraction of a bba takes its values in [0 , . Figure 1. Bayesian bba’s θ . θ . θ m : θ . θ . θ . m :Taking the Bayesian bba given by: m ( θ ) = 0 . , m ( θ ) =0 . , and m ( θ ) = 0 . . We obtain: Contr m ( θ ) ≃ . , Contr m ( θ ) ≃ . , Contr m ( θ ) ≃ . The contradiction for m is Contr m = 0 . . igure 2. Non-dogmatic bba θ θ . θ . . m :Take m ( { θ , θ , θ } ) = 0 . , m ( θ ) = 0 . , and m ( θ ) =0 . ; the masses are the same than m , but the highest one ison Θ = { θ , θ , θ } instead of θ . We obtain: Contr m ( { θ , θ , θ } ) ≃ . , Contr m ( θ ) ≃ . , Contr m ( θ ) ≃ . The contradiction for m is Contr m = 0 . . We can seethat the contradiction is lowest thanks to the distance takinginto account the imprecision of Θ . Figure 3. Focal elements of cardinality 2 θ θ θ . . . m :If we consider now the same mass values but onfocal elements of cardinality 2: m ( { θ , θ } ) = 0 . , m ( θ , θ ) = 0 . , and m ( θ , θ ) = 0 . . We obtain: Contr m ( { θ , θ } ) ≃ . , Contr m ( { θ , θ } ) ≃ . , Contr m ( { θ , θ } ) ≃ . The contradiction for m is Contr m = 0 . .Fewer of focal elements there are, smaller the contradictionof the bba will be, and more the focal elements are precise,higher the contradiction of the bba will be.IV. D EGREES OF UNCERTAINTY
We have seen in the section II that the measure non-specificity given by the equation (6) take its values in a spacedependent on the size of the discernment space Θ . Indeed, themeasure of non-specificity takes its values in [0 , log ( | Θ | )] .In order to compare some non-specificity measures in anabsolute space, we define a degree of non-specificity from the equation (6) by: δ NS ( m ) = X X ∈ G Θ , X m ( X ) log ( | X | )log ( | Θ | )= X X ∈ G Θ , X m ( X ) log | Θ | ( | X | ) . (25)Therefore, this degree takes its values into [0 , for all bba’s m , independently of the size of discernment. We still have δ NS ( m Θ ) = 1 , where m Θ is the categorical bba giving thetotal ignorance. Moreover, we obtain δ NS ( m ) = 0 for allBayesian bba’s.From the definition of the degree of non-specificity, we canpropose a degree of specificity such as: δ B ( m ) = 1 − X X ∈ G Θ , X m ( X ) log ( | X | )log ( | Θ | )= 1 − X X ∈ G Θ , X m ( X ) log | Θ | ( | X | ) . (26)As we observe already the degree of non-specificity givenby the equation (26) does not really measure the specificitybut the Bayesianity of the considered bba. This degree is equalto 1 for Bayesian bba’s and not one for other bba’s. Definition
The
Bayesianity in the theory of belief functionsquantify how far a bba m is from a probability. We illustrate these degrees in the next subsection.
A. Illustration
In order to illustrate and discuss the previous introduceddegrees we take some examples given in the table I. Thebba’s are defined on Θ where Θ = { θ , θ , θ } . We onlyconsider here non-Bayesian bba’s, else the values are stillgiven hereinbefore.We can observe for a given sum of basic belief on thesingletons of Θ the Bayesianity degree can change accordingto the basic belief on the other focal elements. For example,the degrees are the same for m and m , but different for m .For the bba m , note that the sum of the basic beliefs on thesingletons is equal to the basic belief on the ignorance. In thiscase the Bayesianity degree is exactly 0.5. That is conform tothe intuitive signification of the Bayesianity. If we look m and m , we first note that there is no basic belief on the singletons.As a consequence, the Bayesianity is weaker. Moreover, thebba m is naturally more Bayesian than m because of thebasic belief on Θ .We must add that these degrees are dependent on thecardinality of the frame of discernment for non Bayesian bba’s.Indeed, if we consider the given bba m with Θ = { θ , θ , θ } ,the degree δ B = 0 . . Now if we consider the same bbawith Θ = { θ , θ , θ , θ } (no beliefs are given on θ ), theBayesianity degree is δ B = 0 . . The larger is the frame, thelarger becomes the Bayesianity degree.V. D EGREE OF SPECIFICITY
In the previous section, we showed the importance to con-sider a degree instead of a measure. Moreover, the measures able IE
VALUATION OF B AYESIANITY ON EXAMPLES m m m m m m m Θ θ θ θ θ ∪ θ θ ∪ θ θ ∪ θ Θ δ B δ NS of specificity and non-specificity given by the equations (7)and (6) give the same values for every Bayesian bba’s as seenon the examples of the section II.We introduce here a degree of specificity based on compar-ison with the bba the most specific. This degree of specificityis given by: δ S ( m ) = 1 − d ( m, m s ) , (27)where m s is the bba the most specific associated to m and d is a distance defined onto [0 , . Here we also choose theJousselme distance, the most appropriated on the bba’s space,with values onto [0 , . This distance is dependent on the sizeof the space G Θ , we have to compare the degrees of specificityfor bba’s defined from the same space. Accordingly, the mainproblem is to define the bba the most specific associated to m . A. The most specific bba
In the theory of belief functions, several partial ordershave been proposed in order to compare the bba’s [3]. Thesepartial ordering are generally based on the comparisons oftheir plausibilities or their communalities, specially in orderto find the least-committed bba. However, comparing bba’swith plausibilities or communality can be complex and withoutunique solution.The problem to find the most specific bba associated to a bba m does not need to use a partial ordering. We limit the specificbba’s to the categorical bba’s: m X ( X ) = 1 where X ∈ G Θ and we will use the following axiom and proposition: Axiom
For two categorical bba’s m X and m Y , m X is morespecific than m Y if and only if | X | < | Y | . In case of equality, the both bba’s are isospecific . Proposition
If we consider two isospecific bba’s m X and m Y , the Jousselme distance between every bba m and m X isequal to the Jousselme distance between m and m Y : ∀ m, d ( m, m X ) = d ( m, m Y ) (28) if and only if m ( X ) = m ( Y ) . Proof
The proof is obvious considering the equations (21) and (22) . As the both bba’s m X and m Y are categoric there isonly one non null term in the difference of vectors m − m X and m − m Y . These both terms a X and a Y are equal, because m X and m Y are isospecific and so according to the equation (22) D ( Z, X ) = D ( Z, Y ) ∀ Z ∈ G Θ . Therefore m ( X ) = m ( Y ) ,that proves the proposition (cid:3) We define the most specific bba m s associated to a bba m as a categorical bba as follows: m s ( X max ) = 1 where X max ∈ G Θ .Therefore, the matter is now how to find X max . We proposetwo approaches: First approach, Bayesian case If m is a Bayesian bba we define X max such as: X max = arg max( m ( X ) , X ∈ Θ) . (29)If there exist many X max ( i.e. having the samemaximal bba and giving many isospecific bba’s),we can take any of them. Indeed, according to theprevious proposition, the degree of specificity of m will be the same with m s given by either X max satisfying (29). First approach, non-Bayesian case If m is a non-Bayesian bba, we can define X max ina similar way such as: X max = arg max (cid:18) m ( X ) | X | , X ∈ G Θ , X
6≡ ∅ (cid:19) . (30)In fact, this equation generalizes the equation (29).However, in this case we can also have several X max not giving isospecific bba’s. Therefore, we chooseone of the more specific bba, i.e. believing in theelement with the smallest cardinality. Note also thatwe keep the terms of Yager from the equation (7). Second approach
Another way in the case of non-Bayesian bba m isto transform m into a Bayesian bba, thanks to one ofthe probability transformation such as the pignisticprobability. Afterwards, we can apply the previousBayesian case. With this approach, the most specificbba associated to a bba m is always a categoricalbba such as: m s ( X max ) = 1 where X max ∈ Θ andnot in G Θ . B. Illustration
In order to illustrate this degree of specificity we give someexamples. The table II gives the degree of specificity forsome Bayesian bba’s. The smallest degree of specificity ofa Bayesian bba is obtained for the uniform distribution ( m ),and the largest degree of specificity is of course obtain forcategorical bba ( m ).The degree of specificity increases when the differencesbetween the mass of the largest singleton and the massesof other singletons are getting bigger: δ S ( m ) < δ S ( m ) <δ S ( m ) < δ S ( m ) . In the case when one has three disjointsingletons and the largest mass of one of them is 0.45 (on θ ),the minimum degree of specificity is reached when the massesof θ and θ are getting further from the mass of θ ( m ). If able III LLUSTRATION OF THE DEGREE OF SPECIFICITY ON B AYESIAN BBA . θ θ θ δ S m m m m m m m m two singletons have the same maximal mass, bigger this massis and bigger is the degree of specificity: δ S ( m ) < δ S ( m ) .In the case of non-Bayesian bba, we first take a simpleexample: m ( θ ) = 0 . , m ( θ ∪ θ ) = 0 . (31) m ( θ ) = 0 . , m ( θ ∪ θ ) = 0 . . (32)For these two bba’s m and m , both proposed approachesgive the same most specific bba: m θ . We obtain δ S ( m ) =0 . and δ S ( m ) = 0 . . Therefore, m is more specificthan m . Remark that these degrees are the same if weconsider the bba’s defined on Θ and Θ , with Θ = { θ , θ } and Θ = { θ , θ , θ } . If we now consider Bayesian bba m ( θ ) = m ( θ ) = 0 . , the associated degree of specificityis δ S ( m ) = 0 . . As expected by intuition, m is more specificthan m .We consider now the following bba: m ( θ ) = 0 . , m ( θ ∪ θ ∪ θ ) = 0 . . (33)The most specific bba is also m θ , and we have δ S ( m ) =0 . . This degree of specificity is naturally smaller than δ S ( m ) because of the mass 0.4 on a more imprecise focalelement.Let’s now consider the following example: m ( θ ∪ θ ) = 0 . , m ( θ ∪ θ ) = 0 . . (34)We do not obtain the same most specific bba with bothproposed approaches: the first one will give the categoricalbba m θ ∪ θ and the second one m θ . Hence, the first degreeof specificity is δ S ( m ) = 0 . and the second one is δ S ( m ) = 0 . . We note that the second approach producesnaturally some smaller degrees of specificity. C. Application to measure the specificity of a combination rule
We propose in this section to use the proposed degree ofspecificity in order to measure the quality of the result ofa combination rule in the theory of belief functions. Indeed,many combination rules have been developed to merge thebba’s [10], [19]. The choice of one of them is not alwaysobvious. For a special application, we can compare the pro-duced results of several rules, or try to choose according to the special proprieties wanted for an application. We also proposedto study the comportment of the rules on generated bba’s[12]. However, no real measures have been used to evaluatethe combination rules. Hereafter, we only show how we canuse the degree of specificity to evaluate and compare thecombination rules in the theory of belief functions. A completestudy could then be done for example on generated bba’s.We recall here the used combination rules, see [10] for theirdescription.The
Dempster rule is the normalized conjunctive combi-nation rule of the equation (16) given for two basic beliefassignments m and m and for all X ∈ G Θ , X
6≡ ∅ by: m DS ( X ) = 11 − k X A ∩ B = X m ( A ) m ( B ) . (35)where k is either m c ( ∅ ) or the sum of the masses of theelements of ∅ equivalence class for D Θ .The Yager rule transfers the global conflict on the totalignorance Θ : m Y ( X ) = m c ( X ) if X ∈ Θ \ {∅ , Θ } m c (Θ) + m c ( ∅ ) if X = Θ0 if X = ∅ (36)The disjunctive combination rule is given for two basicbelief assignments m and m and for all X ∈ G Θ by: m Dis ( X ) = X A ∪ B = X m ( A ) m ( B ) . (37)The Dubois and Prade rule is given for two basic beliefassignments m and m and for all X ∈ G Θ , X
6≡ ∅ by: m DP ( X ) = X A ∩ B = X m ( A ) m ( B )+ X A ∪ B = XA ∩ B ≡∅ m ( A ) m ( B ) . (38)The PCR rule is given for two basic belief assignments m and m and for all X ∈ G Θ , X
6≡ ∅ by: m PCR ( X ) = m c ( X ) + X Y ∈ G Θ ,X ∩ Y ≡∅ (cid:18) m ( X ) m ( Y ) m ( X )+ m ( Y ) + m ( X ) m ( Y ) m ( X )+ m ( Y ) (cid:19) , (39)The principle is very simple: compute the degree of speci-ficity of the bba’s you want combine, then calculate the degreeof specificity obtained on the bba after the chosen combinationrule. The degree of specificity can be compared to the degreesof specificity of the combined bba’s.In the following example given in the table III we com-bine two Bayesian bba’s with the discernment frame Θ = { θ , θ , θ } . Both bba’s are very contradictory. The valuesare rounded up. The first approach gives the same degree ofspecificity than the second one except for the rules m Dis , m DP and m Y . We observe that the smallest degree of specificity isobtained for the conjunctive rule because of the accumulatedmass on the empty set not considered in the calculus of thedegree. The highest degree of specificity is reached for the able IIID EGREES OF SPECIFICITY FOR COMBINATION RULES ON B AYESIAN BBA ’ S . m m m c m DS m Y m Dis m DP m PCR ∅ θ θ θ θ ∪ θ θ ∪ θ θ ∪ θ Θ m s m θ m θ m θ m θ m Θ m θ ∪ θ m θ ∪ θ m θ m s m θ m θ m θ m θ m θ m θ m θ m θ δ S
1- 0.639 0.655 0.176 0.567 0.857 0.619 0.619 0.497 δ S
2- 0.639 0.655 0.176 0.567 0.457 0.478 0.478 0.497
Yager rule, for the same reason. That is the only rule given adegree of specificity superior to δ S ( m ) and to δ S ( m ) . Thesecond approach shows well the loss of specificity with therules m Dis , m Y and m DP having a cautious comportment.With the example, the degree of specificity obtained by thecombination rules are almost all inferior to δ S ( m ) and to δ S ( m ) , because the bba’s are very conflicting. If the degreesof specificity of the rule such as m DS and m PCR are superiorto δ S ( m ) and to δ S ( m ) , that means that the bba’s are notin conflict.Let’s consider now a simple non-Bayesian example intable IV. Figure 4. Two non-Bayesian bba’s θ . θ . θ . . m : θ . θ . θ . . . . m : VI. C ONCLUSION
First, we propose in this article a reflection on the mea-sures on uncertainty in the theory of belief functions. A lotof measures have been proposed to quantify different kindof uncertainty such as the specificity - very linked to theimprecision - and the discord. The discord, we do not haveto confuse with the conflict, is for us a contradiction of asource (giving information with a bba in the theory of belief
Table IVD
EGREES OF SPECIFICITY FOR COMBINATION RULES ON NON -B AYESIANBBA ’ S . m m m c m DS m Y m Dis m DP m PCR ∅ θ θ θ θ ∪ θ θ ∪ θ θ ∪ θ Θ m s m θ m θ m θ m θ m θ m θ ∪ θ m θ m θ m s m θ m θ m θ m θ m θ m θ m θ m θ δ S
1- 0.553 0.522 0.336 0.488 0.389 0.609 0.428 0.497 δ S
2- 0.553 0.522 0.336 0.488 0.389 0.456 0.428 0.497 functions) with oneself. We distinguish the contradiction andthe conflict that is the contradiction between 2 or more bba’s.We introduce a measure of contradiction for a bba based onthe weighted average of the conflict between the bba and thecategorical bba’s of the focal elements.The previous proposed specificity or non-specificity mea-sures are not defined on the same space. Therefore that isdifficult to compare them. That is the reason why we proposethe use of degree of uncertainty. Moreover these measures givesome counter-intuitive results on Bayesian bba’s. We proposea degree of specificity based on the distance between a massand its most specific associated mass that we introduce. Thismost specific associated mass can be obtained by two ways andgive the nearest categorical bba for a given bba. We proposealso to use the degree of specificity in order to measure thespecificity of a fusion rule. That is a tool to compare andevaluate the several combination rules given in the theory ofbelief functions.
Acknowledgments
The authors want to thanks B
REST M ETROPOLE O C ´ EANE and ENSTA B
RETAGNE for funding this collaboration andproviding them an excellent research environment duringspring 2010. R
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