Contradiction measures and specificity degrees of basic belief assignments

Contradiction measures and s pecificity de grees of basic belief assignments Florentin Smarandac he Math. & Sciences Dept. University of New Mexico, 200 College Road , Gallup, NM 8 7301 , U.S.A. Email: smarand@unm .edu Arnaud Martin IRISA University of Rennes 1 Rue ´ Edouar d Branly BP 302 19 22302 Lannio n, Fran ce Email: Arnaud.Ma rtin@univ-rennes1.fr Christophe Osswald E3I2 ENST A Bretagne 2, rue Franc ¸ ois V erny 29806 Brest, Ced ex 9, France Email: Christophe .Osswald@ensta-bretagne.f r Abstract —In the theory of belief functions, many measur es of uncertainty hav e been in troduced. Howev er , it is not always easy to understand what th ese measures really try to represent. In this paper , we re-interpret some measures of uncertainty i n the theory of belief functions. W e p resent some interests and drawbacks of the existing measures. On these observations, we introduce a measure of contradiction. Theref ore, we present some degrees of non-specificity and Bay esianity of a mass. W e pro pose a degr ee of specificity based on the distance between a mass and its most specific associated mass. W e also sh ow how to use th e degree of sp ecificity to measure the specificity of a f usion rule. Illustrations on simple examples are given. Keywords: Belief function, uncertainty measures, speci- ficity , conflict. I . I N T RO D U C T I O N The theory of belief f unctions was first introduced b y [1] in o rder to rep resent som e im precise pr obabilities with upp er and lo wer pr obab ilities . T hen [ 15] pro posed a mathematical theory of e viden ce. Let Θ be a frame of discernment. A basic belief assignment (bba) m is the mapp ing from elements of the po werset 2 Θ onto [0 , 1] su ch that: X X ∈ 2 Θ m ( X ) = 1 . (1) The ax iom m ( ∅ ) = 0 is often used, but not man datory . A focal elem ent X is an element of 2 Θ such that m ( X ) 6 = 0 .. The d ifference o f a bba with a probability is the d omain of definition. A bba is defined on the p owerset 2 Θ and not only on Θ . In the powerset, each element is not equ iv alent in terms of precision. Ind eed, θ 1 ∈ Θ is mor e precise than θ 1 ∪ θ 2 ∈ 2 Θ . In the case o f the DSmT introduced in [1 7], th e bba are defined on an extension of the powerset: the h yper powerset noted D Θ , formed by the closure of Θ by un ion and inter- section. The p roblem of signification o f each focal element is the same as in 2 Θ . For instance, θ 1 ∈ Θ is less p recise than θ 1 ∩ θ 2 ∈ D Θ . In th e rest of the pa per, we w ill n ote G Θ for either 2 Θ or D Θ . In ord er to try to quan tify the m easure of uncertainty such as in the set th eory [5] o r in the th eory of probab ilities [16], som e m easures h av e been p ropo sed and d iscussed in the theor y of belief fu nctions [2], [7], [8], [21]. Howe ver , the do main of definition of the bba does not allow an ideal definition of measure of u ncertainty . M oreover , beh ind th e term of uncertainty , different notion s are hidd en. In th e section II, we pr esent different kinds of m easures of uncer tainty given in the state of ar t, we discuss them a nd giv e our defin itions of some terms concerning th e uncertainty . In section III, we introduce a measure of contradiction and discuss it. W e intro duce simple degrees of uncertain ty in th e section IV, an d p ropo se a degree o f specificity in the section V. W e show how this degree o f specificity can be u sed to measure th e specificity of a combina tion ru le. I I . M E A S U R E S O F U N C E RTA I N T Y O N BE L I E F F U N C T I O N S In th e fra mew ork of the belief fun ctions, se veral fu nctions (we call them b elief func tions ) ar e in on e to one cor respon- dence with the bba: b el , pl and q . From th ese belief fu nctions, we can d efine se veral measur es o f unce rtainty . Klir in [8] distinguishes two kind s of un certainty: the no n-specificity and the disco rd. Hence, we recall he reafter the main be lief function s, and some non- specificity and discord measur es. A. Belief functions Hence, the cre dibility and plausibility fun ctions represent respectively a minim al a nd maximal belief. The cr edibility function is giv en f rom a bba for all X ∈ G Θ by: bel( X ) = X Y ⊆ X,Y 6≡∅ m ( Y ) . (2) The plausibility is giv en f rom a bba for all X ∈ G Θ by: pl( X ) = X Y ∈ G Θ ,Y ∩ X 6≡∅ m ( Y ) . (3) The commona lity function is also ano ther belief fu nction giv en by: q( X ) = X Y ∈ G Θ ,Y ⊇ X m ( Y ) . (4) These fun ctions allow an implicit mod el of imp recise and uncertain d ata. Howev er , these functio ns are monoto nic b y inclusion: bel an d pl are increasing, and q is decre asing. This is the reason wh y the mo st of time we use a pro bability to take a decision . Th e mo st used projection into pr obability subsp ace is the pign istic pro bability tran sformation intro duced by [ 18] and given by: betP ( X ) = X Y ∈ G Θ ,Y 6≡∅ | X ∩ Y | | Y | m ( Y ) , (5) where | X | is the card inality of X , in the case of the DSmT that is the num ber o f d isjoint elements corr espondin g in th e V enn diag ram. From this probability , we can use the measure of uncertain ty giv en in the theor y of probab ilities such as the Shann on entropy [ 16], but we loose the interest of the belief functions and th e info rmation given on the sub sets of the discernm ent space Θ . B. Non-specificity The non-spec ificity in th e classical set theory is the im pre- cision of th e sets. Suc h as in [14], we d efine in th e th eory of belief f unctions, the non-sp ecificity related to vagueness and non-spe cificity . Definition The no n-specificity in the theo ry of belief functions q uantifi es how a bba m is imprecis e. The n on-specificity of a sub set X is d efined b y Hartley [5] b y log 2 ( | X | ) . This mea sure was g eneralized by [ 2] in the theory of belief functions by: NS( m ) = X X ∈ G Θ , X 6≡ ∅ m ( X ) log 2 ( | X | ) . (6) That is a weighted sum of the non-specificity , and the weights are given by the basic belief in X . Ram er in [13] h as shown that it is the unique possible me asure of non-specificity in the theory of belief fun ctions u nder some assum ptions su ch as symmetry , ad ditivity , sub-ad ditivity , continuity , b ranchin g and normalizatio n. If th e measure of th e no n-specificity on a b ba is low , we ca n consider the bba is specific. Y ager in [21] defined a specificity measure such as: S ( m ) = X X ∈ G Θ , X 6≡∅ m ( X ) | X | . (7) Both definitions corresp onded to an a ccumulatio n of a function of the basic belief assignme nt on the fo cal elements. Unlike the classical set theory , we must take into a ccount the bba in order to quantify (to weight) the b elief of the imp recise focal elem ents. Th e impr ecision of a focal element can of course be giv en b y the cardinality of the element. First of all, we must be able to compare the non-specificity (or specificity) between several bba’ s, ev ent if these bba’ s are not defined on the same discernment space . T hat is n ot the case with th e equations ( 6) and (7) . The non-specificity o f the equation (6) takes its values in [0 , log 2 ( | Θ | )] . The specificity of the equ ation (7) can have values in [ 1 | Θ | , 1] . W e will show how we can easily define a d egree of non-sp ecificity in [0 , 1] . W e could also define a degree of specificity fro m the equation (7), but that is mor e comp licated and we will later show how we can define a specificity degree. The most n on-spec ific bba’ s fo r bo th eq uations ( 6) and (7) are the total ignoranc e b ba given by the categorical bba m Θ : m (Θ) = 1 . W e h av e NS( m ) = log 2 ( | Θ | ) and S ( m ) = 1 | Θ | . This cate gorical bba is clearly the most non-specific for us. Howe ver, the most specific bb a’ s are the Baye sian bba’ s. The only foca l elemen ts of a Bay esian bb a are the simple elements of Θ . On the se kinds o f bba m we have NS( m ) = 0 and S ( m ) = 1 . For example, we take the three Bayesian bba’ s defined on Θ = { θ 1 , θ 2 , θ 3 } b y: m 1 ( θ 1 ) = m 1 ( θ 2 ) = m 1 ( θ 3 ) = 1 / 3 , (8) m 2 ( θ 1 ) = m 2 ( θ 2 ) = 1 / 2 , m 2 ( θ 3 ) = 0 , (9) m 3 ( θ 1 ) = 1 , m 3 ( θ 2 ) = m 3 ( θ 3 ) = 0 . (10) W e obtain the same non -specificity and specificity for these three bba’ s. That hurts our intu ition; indeed , we intuitively expect that the b ba m 3 is th e most s pecific an d the m 1 is th e less s pecific. W e will define a degree o f specificity a ccording to a most specific b ba that we will introdu ce. C. Discord Different kin ds of discor d h ave been d efined as extensions for be lief fun ctions of the Shannon en tropy , g iv en for th e probab ilities. Some discord measures hav e been prop osed from plausibility , c redibility or p ignistic pro bability: E ( m ) = − X X ∈ G Θ m ( X ) log 2 (pl( X )) , (11) C ( m ) = − X X ∈ G Θ m ( X ) log 2 (bel ( X )) , (12) D ( m ) = − X X ∈ G Θ m ( X ) log 2 (betP ( X )) , (13) with E ( m ) ≤ D ( m ) ≤ C ( m ) . W e can also give the Sh anon entropy o n the p ignistic probab ility: − X X ∈ G Θ betP ( X ) log 2 (betP ( X )) . (14) Other measures have b een prop osed, [ 8] has shown that these measures can be r esumed by: − X X ∈ G Θ m ( X ) log 2 (1 − Con m ( X )) , (15) where Co n is ca lled a conflict measure of the bb a m on X . Howe ver, in our poin t of view th at is no t a conflict such presen ted in [20], but a c ontradictio n. W e g i ve the both following definition s: Definition A contradiction in the theory of belief functions quantifi es how a bba m contradicts itself. Definition (C1 ) The conflict in th e the ory o f belief fu nctions can be defin ed by the contradiction between 2 or mo r e bba’s. In order to measure the conflict in the th eory of belief function s, it was usual to use the mass k g iv en by the conjunc ti ve combina tion ru le on th e empty set. This rule is giv en by two basic belief assign ments m 1 and m 2 and for all X ∈ G Θ by: m c ( X ) = X A ∩ B = X m 1 ( A ) m 2 ( B ) := ( m 1 ⊕ m 2 )( X ) . (16) k = m c ( ∅ ) can also be inte rpreted as a n on-expected solution . In [ 21], Y ager pr oposed anoth er conflict measure f rom the value of k given by − log 2 (1 − k ) . Howe ver, as observed in [9], the weight of conflict given by k (and all the functio ns of k ) is not a conflict measure between th e ba sic belief assignments. I ndeed this value is completely d ependan t of the conjun ctiv e rule a nd th is ru le is non- idempo tent - the co mbination of identica l basic belief assignments leads ge nerally to a positiv e value of k . T o highligh t th is behavior, we defined in [ 12] the auto-con flict which quantifies the intrinsic conflict o f a bb a. Th e auto- conflict of order n for one expert is g iv en b y: a n =  n ⊕ i =1 m  ( ∅ ) . (17) The auto-con flict is a kind of m easure of th e contradiction, but depen ds on the or der . W e stu died its b ehavior in [1 1]. Therefo re we need to define a measure of contrad iction indepen dent on the order . T his measur e is pr esented in the next section III. I I I . A C O N T R A D I C T I O N M E A S U R E The definition of the conflict (C1) in volves firstly to measure it on th e bba’ s space and secondly th at if the opinion s of two experts are far fr om each other, we con sider th at they are in conflict. That sugg ests a notio n of distance. That is the reason why in [11], we give a definition o f the mea sure o f conflict between experts assertion s thr ough a d istance between the ir respective bba’ s. The conflict measure between 2 experts is defined by: Conf (1 , 2) = d ( m 1 , m 2 ) . (18) W e defined the conflict measure between on e expert i and the other M − 1 exper ts by : Conf ( i, E ) = 1 M − 1 M X j =1 ,i 6 = j Conf ( i, j ) , (19) where E = { 1 , . . . , M } is the set of expe rts in conflict wit h i . Another definition is given by: Conf ( i, M ) = d ( m i , m M ) , (20) where m M is the bb a of th e artificial exper t re presenting the combined o pinion s of all the experts in E except i . W e use th e d istance defined in [6], which is for us th e most approp riate, but oth er distan ces are possible. See [4] for a compariso n o f distances in th e th eory of belief fu nctions. This distance is defined fo r two b asic belief assignments m 1 and m 2 on G Θ by: d ( m 1 , m 2 ) = r 1 2 ( m 1 − m 2 ) T D ( m 1 − m 2 ) , (21) where D is an G | Θ | × G | Θ | matrix based o n Jaccard distance whose elements are: D ( A, B ) =        1 , if A = B = ∅ , | A ∩ B | | A ∪ B | , ∀ A, B ∈ G Θ . (22) Howe ver, this m easure is a total confl ict measu re. In order to define a contr adiction mea sure we keep the same spir it. First, the contradiction of an element X with respect to a bba m is defin ed as the d istance between the b ba’ s m and m X , where m X ( X ) = 1 , X ∈ G Θ , is the c ategorical bb a: Contr m ( X ) = d ( m, m X ) , (23) where the distance can a lso be the Jou sselme distance on th e bba’ s. The contra diction of a b ba is then defined as a weighted contradictio n o f all the elemen ts X of th e considered space G Θ : Contr m = c X X ∈ G Θ m ( X ) d ( m, m X ) , (24) where c is a normalized con stant which depen ds on the type o f distan ce used and on the ca rdinality of the fr ame o f discernmen t in orde r to obtain values in [0 , 1] as shown in the following illustration. A. Illustration Here the v alu e c in the equation (24 ) is equal to 2. First we note tha t for all categor ical bbas m Y , th e co ntradiction g iv en by the equation (2 3) gives Contr m Y ( Y ) = 0 and th e con tra- diction given by th e equation (24) brin gs also Contr m Y = 0 . Considering the bb a m 1 ( θ 1 ) = 0 . 5 an d m 1 ( θ 2 ) = 0 . 5 , we have Cont r m 1 = 1 . Th at is the maximum of the co ntradiction , hence th e contractio n of a bb a takes its values in [0 , 1] . Figure 1. Bayesian bba ’ s θ 1 0 . 5 θ 2 0 . 5 θ 3 0 m 1 : θ 1 0 . 6 θ 2 0 . 3 θ 3 0 . 1 m 2 : T aking the Bayesian bba given by: m 2 ( θ 1 ) = 0 . 6 , m 2 ( θ 2 ) = 0 . 3 , and m 2 ( θ 3 ) = 0 . 1 . W e obtain : Contr m 2 ( θ 1 ) ≃ 0 . 36 , Contr m 2 ( θ 2 ) ≃ 0 . 66 , Contr m 2 ( θ 3 ) ≃ 0 . 79 The contradictio n for m 2 is Contr m 2 = 0 . 9 8 49 . Figure 2. Non-dogmatic bb a θ 1 0 θ 2 0 . 3 θ 3 0 . 1 0 . 6 m 3 : T ake m 3 ( { θ 1 , θ 2 , θ 3 } ) = 0 . 6 , m 3 ( θ 2 ) = 0 . 3 , and m 3 ( θ 3 ) = 0 . 1 ; the masses are th e same than m 2 , but the high est one is on Θ = { θ 1 , θ 2 , θ 3 } instead of θ 1 . W e obtain : Contr m 3 ( { θ 1 , θ 2 , θ 3 } ) ≃ 0 . 28 , Contr m 3 ( θ 2 ) ≃ 0 . 56 , Contr m 3 ( θ 3 ) ≃ 0 . 71 The contr adiction fo r m 3 is Contr m 3 = 0 . 80 92 . W e can see that the co ntradiction is lowest th anks to the distan ce takin g into account the imprecision of Θ . Figure 3. Focal elements of card inality 2 θ 1 θ 2 θ 3 0 . 6 0 . 1 0 . 3 m 4 : If we consider now the sam e m ass values but o n focal elements of card inality 2: m 4 ( { θ 1 , θ 2 } ) = 0 . 6 , m 4 ( θ 1 , θ 3 ) = 0 . 3 , and m 4 ( θ 2 , θ 3 ) = 0 . 1 . W e obtain : Contr m 4 ( { θ 1 , θ 2 } ) ≃ 0 . 29 , Contr m 4 ( { θ 1 , θ 3 } ) ≃ 0 . 53 , Contr m 4 ( { θ 2 , θ 3 } ) ≃ 0 . 65 The contradictio n for m 4 is Contr m 4 = 0 . 8 0 . Fewer of focal elements there are, smaller the contradictio n of th e b ba will be , and mor e the focal elements a re pr ecise, higher the contrad iction of the bb a will be . I V . D E G R E E S O F U N C E RTA I N T Y W e have seen in the section II that the measure non- specificity given b y the equation (6) take its v alues in a space depend ent o n the size of the discernm ent sp ace Θ . I ndeed, th e measure of non-sp ecificity takes its values in [0 , log 2 ( | Θ | )] . In order to compare some non-specificity me asures in an absolute space, we define a degree of non-specificity from the equation (6) by: δ NS ( m ) = X X ∈ G Θ , X 6≡∅ m ( X ) log 2 ( | X | ) log 2 ( | Θ | ) = X X ∈ G Θ , X 6≡∅ m ( X ) log | Θ | ( | X | ) . (25) Therefo re, this degree takes its values into [0 , 1] for all b ba’ s m , ind ependen tly of the size of discern ment. W e still have δ NS ( m Θ ) = 1 , whe re m Θ is the catego rical bba giving th e total igno rance. Moreover, we obtain δ NS ( m ) = 0 f or all Bayesian bba’ s. From the d efinition of the degree of no n-specificity , we can propo se a degree o f specificity su ch as: δ B ( m ) = 1 − X X ∈ G Θ , X 6≡∅ m ( X ) log 2 ( | X | ) log 2 ( | Θ | ) = 1 − X X ∈ G Θ , X 6≡∅ m ( X ) log | Θ | ( | X | ) . (26) As we observe already the degree of n on-specificity given by th e eq uation (2 6) doe s no t really measure the specificity but the Bayesianity of the co nsidered bba. T his degree is equal to 1 for Bayesian bba’ s and no t o ne for oth er bba’ s. Definition The B ayesianity in the theo ry of b elief fun ctions quantify how fa r a bba m is fr om a pr ob ability . W e illustrate these degrees in the next subsectio n. A. Illustration In o rder to illustrate an d discu ss the p revious introduced degrees we take some examples given in the table I. The bba’ s a re defined o n 2 Θ where Θ = { θ 1 , θ 2 , θ 3 } . W e only consider here non-Bay esian bba’ s, else the values are still giv en h ereinbef ore. W e can o bserve fo r a given sum o f basic b elief on the singletons o f Θ the Bayesianity degree can chan ge acco rding to the basic belief on th e other focal elements. For examp le, the de grees are the sam e f or m 2 and m 3 , but dif f erent for m 4 . For the bba m 4 , n ote that the sum of the basic beliefs o n the singletons is equal to th e b asic b elief on the ignorance. In this case the Bayesianity degree is exactly 0.5. That is conf orm to the intuiti ve signification of the Bayesianity . If we look m 5 and m 6 , we fir st note that ther e is no basic be lief on th e singleton s. As a consequ ence, the Bayesian ity is weaker . Moreover, the bba m 5 is natu rally mor e Bayesian than m 6 because of the basic b elief on Θ . W e must ad d that these d egrees are dep endent on th e cardinality of the frame of discernment for non Bayesian bba’ s. Indeed , if we consider the given bb a m 1 with Θ = { θ 1 , θ 2 , θ 3 } , the degree δ B = 0 . 75 . Now if we con sider the same bba with Θ = { θ 1 , θ 2 , θ 3 , θ 4 } (n o belief s are given on θ 4 ), the Bayesianity degre e is δ B = 0 . 8 0 . The larger is the frame, the larger beco mes the Bayesianity degree . V . D E G R E E O F S P E C I FI C I T Y In the previous section, we showed the importan ce to con- sider a degree instead o f a measure. Moreover , the measures T able I E V A L UAT I O N O F B AY E S I A N I T Y O N E X A M P L E S m 1 m 2 m 3 m 4 m 5 m 6 m Θ θ 1 0.4 0.3 0.1 0.3 0 0 0 θ 2 0.1 0.1 0.3 0.1 0 0 0 θ 3 0.1 0.1 0.1 0.1 0 0 0 θ 1 ∪ θ 2 0.3 0.3 0.5 0 0.6 0.6 0 θ 1 ∪ θ 3 0.1 0.2 0 0 0.4 0 0 θ 2 ∪ θ 3 0 0 0 0 0 0 0 Θ 0 0 0 0.5 0 0.4 1 δ B 0.75 0.68 0.68 0.5 0.37 0.23 0 δ NS 0.25 0.32 0.32 0.5 0.63 0.77 1 of specificity and non-sp ecificity given by the eq uations (7) and (6) gi ve the same values fo r e very Bayesian bb a’ s as seen on the examples of the sectio n II. W e introduc e here a degree of specificity based on com par- ison with the bba th e most specific. This d egree of specificity is giv en 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 d efined o nto [0 , 1] . Here we also c hoose th e Jousselme distance, the most appropriated on the bba’ s space, with values onto [0 , 1 ] . This distance is dep enden t on the size of the space G Θ , we have to compare the de grees of specificity for bba’ s defined from the same space. Accord ingly , the main problem is to define the bba th e m ost specific associated to m . A. The most sp ecific bba In the theor y of belief func tions, several pa rtial order s have been propo sed in or der to comp are the b ba’ s [3]. These partial o rdering are generally b ased on th e comparison s of their plausibilities or their communalities, specially in or der to find th e least-committed b ba. However , comp aring bba’ s with plausibilities or communality can be complex and without unique solution. The problem to find the most specific bba associated to a bba m does n ot need to use a pa rtial order ing. W e limit the specific bba’ s to the categorical bb a’ s: m X ( X ) = 1 where X ∈ G Θ and we will use the following axiom and pr oposition: Axiom F or two cate gorical bba’s m X and m Y , m X is more specific than m Y if and only if | X | < | Y | . In case of equality , th e both bba’ s are isospecific . Proposition If we c onsider two isospe cific bb a’s m X and m Y , the Jousselme distan ce between every bba m and m X is equal to the Jousselme distance between m an d m Y : ∀ m, d ( m, m X ) = d ( m, m Y ) (28) if and only if m ( X ) = m ( Y ) . Proof The pr oof is o bvious considering the eq uations (21) and (22 ) . As the bo th bba ’ s m X and m Y ar e categoric ther e is only one non null term in the differ ence of vectors m − m X and m − m Y . These b oth terms a X and a Y ar e equ al, because m X and m Y ar e isospecific and so acco r d ing to the equation (22) D ( Z, X ) = D ( Z , Y ) ∀ Z ∈ G Θ . Therefor e m ( X ) = m ( Y ) , that pr oves the pr op osition  W e define the most specific b ba m s associated to a bba m as a c ategorical bb a as f ollows: m s ( X max ) = 1 where X max ∈ G Θ . Therefo re, the m atter is no w h ow to find X max . W e p ropo se two a pproac hes: First approach, Bayesian ca se If m is a Bayesian bba we define X max such as: X max = ar g max ( m ( X ) , X ∈ Θ) . (29 ) If th ere exist many X max ( i.e. having th e same maximal bb a and giving many isospecific bba’ s), we can take any of them. I ndeed, accord ing to the previous propo sition, the degree of specificity of m will be the same with m s giv en b y either X max satisfying (29). First approach, non-Bayesian c ase If m is a n on-Bayesian bb a, we can define X max in a similar way such as: X max = ar g max  m ( X ) | X | , X ∈ G Θ , X 6≡ ∅  . (30 ) In fact, this equ ation gener alizes the equ ation (29). Howe ver, in this c ase we can also have sev eral X max not giving isospecific bba’ s. Th erefore , we cho ose one of the m ore specific bba, i.e. b elieving in the element with th e sm allest cardinality . Note also that we keep th e ter ms of Y ager from th e equation (7). Second approach Another way in th e case of non -Bayesian bb a m is to transfo rm m into a Bay esian bba , tha nks to one of the probability tr ansforma tion such as th e pignistic probab ility . Afterwards, we can apply th e previous Bayesian case. W ith this approa ch, the mo st specific bba associated to a bba m is always a categorical bba such as: m s ( X max ) = 1 wh ere X max ∈ Θ an d not in G Θ . B. Illustration In order to illustrate this degree of specificity we gi ve som e examples. The table II gives the degree of specificity for some Bayesian b ba’ s. The smallest degree o f specificity of a Bayesian bb a is obtained for the u niform d istribution ( m 1 ), and the largest degree of specificity is of course obtain for categorical bb a ( m 8 ). The degree of specificity in creases when the dif ferences between th e mass of the largest singleton and the masses of other singleton s ar e getting big ger: δ S ( m 3 ) < δ S ( m 4 ) < δ S ( m 5 ) < δ S ( m 6 ) . I n the case when one has three disjoint singletons an d the largest mass of on e of them is 0.45 (on θ 1 ), the m inimum d egree of specificity is reached when th e masses of θ 2 and θ 3 are getting further from the mass of θ 1 ( m 6 ). If T able II I L L U S T R A T I O N O F T H E D E G R E E O F S P E C I F I C I T Y O N B AY E S I A N B B A . θ 1 θ 2 θ 3 δ S m 1 1/3 1/3 1/3 0.423 m 2 0.4 0. 4 0.2 0.471 m 3 0.45 0.45 0.10 0.493 m 4 0.45 0.40 0.15 0.508 m 5 0.45 0.3 0.25 0.523 m 6 0.45 0.275 0.275 0.524 m 7 0.6 0. 3 0.1 0.639 m 8 1 0 0 1 two s ingleton s ha ve the same max imal mass, b igger this m ass is and bigger is the degree of sp ecificity: δ S ( m 2 ) < δ S ( m 3 ) . In the case of no n-Bayesian bba, we first take a simple example: m 1 ( θ 1 ) = 0 . 6 , m 1 ( θ 1 ∪ θ 2 ) = 0 . 4 (31) m 2 ( θ 1 ) = 0 . 5 , m 2 ( θ 1 ∪ θ 2 ) = 0 . 5 . (32) For these two b ba’ s m 1 and m 2 , b oth proposed ap proach es giv e the same mo st specific bba: m θ 1 . W e ob tain δ S ( m 1 ) = 0 . 7172 and δ S ( m 2 ) = 0 . 6 465 . Therefore, m 1 is more specific than m 2 . Remark th at these degrees are the same if we consider the bba’ s defined on 2 Θ 2 and 2 Θ 3 , with Θ 2 = { θ 1 , θ 2 } and Θ 3 = { θ 1 , θ 2 , θ 3 } . If we now consider Bayesian bba m 3 ( θ 1 ) = m 3 ( θ 2 ) = 0 . 5 , the associated degree of specificity is δ S ( m 3 ) = 0 . 5 . As expected by intuition, m 2 is more specific than m 3 . W e con sider now th e f ollowing bba: m 4 ( θ 1 ) = 0 . 6 , m 1 ( θ 1 ∪ θ 2 ∪ θ 3 ) = 0 . 4 . (33) The mo st specific bb a is also m θ 1 , and we have δ S ( m 4 ) = 0 . 6734 . Th is degree of specificity is n aturally smaller than δ S ( m 1 ) because o f the m ass 0. 4 on a mor e imprecise f ocal element. Let’ s now consid er the fo llowing example: m 5 ( θ 1 ∪ θ 2 ) = 0 . 7 , m 5 ( θ 1 ∪ θ 3 ) = 0 . 3 . (34) W e do not obtain the same most specific bba with both propo sed approach es: the first one will give the categorical bba m θ 1 ∪ θ 2 and the second o ne m θ 1 . Hence , the fir st d egree of spec ificity is δ S ( m 5 ) = 0 . 755 an d the second on e is δ S ( m 5 ) = 0 . 1 11 . W e note that th e seco nd ap proach pro duces naturally some smaller degrees of sp ecificity . C. Applicatio n to measur e the specificity of a combinatio n rule W e propose in this section to u se th e pro posed degree of specificity in order to measu re the qu ality of the result of a c ombinatio n ru le in the th eory of belief f unctions. Ind eed, many combin ation rules have been developed to merge th e bba’ s [10], [1 9]. The choice of on e of them is n ot always obvious. For a special app lication, we can co mpare the p ro- duced results of several rules, or try t o choo se according to the special proprieties wanted for an application. W e also proposed to stud y the comportm ent of the rules on g enerated bb a’ s [12]. However , no real measures have been used to ev a luate the com bination rules. Her eafter, w e only show h ow we can use the degree o f specificity to e valuate an d compare the combinatio n rules in the theory of belief functions. A complete study could then be done f or examp le on gener ated bba’ s. W e recall here the used comb ination ru les, see [10] f or th eir description. The Demp ster rule is the no rmalized co njunctive co mbi- nation rule of the equation (16) given for two basic belief assignments m 1 and m 2 and for all X ∈ G Θ , X 6≡ ∅ by : m DS ( X ) = 1 1 − k X A ∩ B = X m 1 ( A ) m 2 ( B ) . (35) where k is eith er m c ( ∅ ) or the sum of the m asses of the elements of ∅ equiv alence class fo r D Θ . The Y ager rule transfers the g lobal conflict on the total ignoran ce Θ : m Y ( X ) =    m c ( X ) if X ∈ 2 Θ \ {∅ , Θ } m c (Θ) + m c ( ∅ ) if X = Θ 0 if X = ∅ (36) The disjun ctiv e comb ination rule is giv en fo r two b asic belief assignments m 1 and m 2 and for all X ∈ G Θ by: m Dis ( X ) = X A ∪ B = X m 1 ( A ) m 2 ( B ) . (37) The Du bois and Prade r ule is given for two basic belief assignments m 1 and m 2 and for all X ∈ G Θ , X 6≡ ∅ by : m DP ( X ) = X A ∩ B = X m 1 ( A ) m 2 ( B ) + X A ∪ B = X A ∩ B ≡∅ m 1 ( A ) m 2 ( B ) . (38) The PCR rule is given for two basic belief assignme nts m 1 and m 2 and for all X ∈ G Θ , X 6≡ ∅ by : m PCR ( X ) = m c ( X ) + X Y ∈ G Θ , X ∩ Y ≡∅  m 1 ( X ) 2 m 2 ( Y ) m 1 ( X ) + m 2 ( Y ) + m 2 ( X ) 2 m 1 ( Y ) m 2 ( X ) + m 1 ( Y )  , (39) The principle is very simple: compu te the degree of speci- ficity of the bb a’ s yo u want co mbine, then c alculate the degree of specificity obtain ed on the bba afte r the chosen com bination rule. T he degree of spec ificity ca n be compared to the d egrees of specificity o f the combin ed bb a’ s. In the following example giv en in the tab le III we com- bine two Bayesian bba’ s with the d iscernment frame Θ = { θ 1 , θ 2 , θ 3 } . Both bba’ s a re very contradictor y . The v alues are roun ded up . The first ap proach gives the same d egree of specificity than the seco nd on e except for the ru les m Dis , m DP and m Y . W e observe that the smallest degree of specificity is obtained for the co njunctive rule because o f the accumulated mass o n the em pty set not con sidered in the calculus of the degree. The highest degree of specificity is r eached for the T able III D E G R E E S O F S P E C I FI C I T Y F O R C O M B I NAT I O N R U L E S O N B AYE S I A N B B A ’ S . m 1 m 2 m c m DS m Y m Dis m DP m PCR ∅ 0 0 0.76 0 0 0 0 0 θ 1 0.6 0.2 0.12 0.50 0.12 0.12 0.12 0.43 θ 2 0.1 0.6 0.06 0.25 0.06 0.06 0.06 0.37 θ 3 0.3 0.2 0.06 0.25 0.06 0.06 0.06 0.20 θ 1 ∪ θ 2 0 0 0 0 0 0.38 0.38 0 θ 1 ∪ θ 3 0 0 0 0 0 0.18 0.18 0 θ 2 ∪ θ 3 0 0 0 0 0 0.20 0.20 0 Θ 0 0 0 0 0.76 0 0 0 m s 1- m θ 1 m θ 2 m θ 1 m θ 1 m Θ m θ 1 ∪ θ 2 m θ 1 ∪ θ 2 m θ 1 m s 2- m θ 1 m θ 2 m θ 1 m θ 1 m θ 1 m θ 1 m θ 1 m θ 1 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 Y ager ru le, for the same reason. That is the o nly r ule given a degree of specificity superior to δ S ( m 1 ) an d to δ S ( m 2 ) . The second approach shows well the loss of specificity with the rules m Dis , m Y and m DP having a cautiou s co mportm ent. W ith the example, the degree o f specificity o btained by the combinatio n rules a re almo st all inf erior to δ S ( m 1 ) and to δ S ( m 2 ) , because the bb a’ s are very conflicting. If the degrees of specificity of the rule such as m DS and m PCR are superior to δ S ( m 1 ) and to δ S ( m 2 ) , th at mean s th at the bba’ s are n ot in conflict. Let’ s consider now a simp le non -Bayesian example in table IV. Figure 4. T wo non-Bayesian bba ’ s θ 1 0 . 4 θ 2 0 . 1 θ 3 0 . 3 0 . 2 m 1 : θ 1 0 . 2 θ 2 0 . 3 θ 3 0 . 1 0 . 1 0 . 2 0 . 1 m 2 : V I . C O N C L U S I O N First, we pro pose in this ar ticle a r eflection on th e mea- sures on uncertainty in the theory o f belief function s. A lot of me asures have bee n pr oposed to q uantify different kin d of uncertainty such as the specificity - very linked to the imprecision - and th e d iscord. Th e discor d, we do no t have to con fuse with the co nflict, is f or us a contrad iction o f a source (g iving info rmation with a bb a in the th eory of belief T able IV D E G R E E S O F S P E C I FI C I T Y F O R C O M B I N ATI O N RU L E S O N N O N - B AY E S I A N B B A ’ S . m 1 m 2 m c m DS m Y m Dis m DP m PCR ∅ 0 0 0.47 0 0 0 0 0 θ 1 0.4 0.2 0.2 0.377 0.2 0.08 0.2 0.39 θ 2 0.1 0.3 0.17 0.321 0.17 0.03 0.17 0.28 θ 3 0.3 0.1 0.12 0.226 0.12 0.03 0.12 0.24 θ 1 ∪ θ 2 0.2 0.1 0.04 0.076 0.04 0.31 0.18 0.06 θ 1 ∪ θ 3 0 0 0 0 0 0.1 0.1 0 θ 2 ∪ θ 3 0 0.2 0 0 0 0.18 0.1 0.03 Θ 0 0.1 0 0 0.47 0.27 0.13 0 m s 1- m θ 1 m θ 2 m θ 1 m θ 1 m θ 1 m θ 1 ∪ θ 2 m θ 1 m θ 1 m s 2- m θ 1 m θ 2 m θ 1 m θ 1 m θ 1 m θ 1 m θ 1 m θ 1 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 function s) with one self. W e disting uish the contrad iction and the conflict that is the co ntradiction between 2 or more bb a’ s. W e introdu ce a measure of contrad iction f or a b ba b ased on the w eighted average of the conflict betwe en the b ba and the categorical bb a’ s o f th e focal elements. The previous p roposed specificity or non-specificity mea- sures are no t define d on th e same space. Theref ore th at is difficult to co mpare them. That is the reason why we prop ose the use of degree of uncertainty . Moreover these measures give some cou nter-intuitiv e results on Bayesian bba’ s. W e pro pose a degree of spe cificity based on the distance b etween a mass and its m ost specific associated mass that we in troduce . This most specific associated mass can be obtaine d by two ways and giv e the nearest categorical bba f or a given bba. W e pro pose also to use the degree of specificity in or der to measur e the specificity of a fusion ru le. That is a tool to com pare and ev alu ate the several c ombinatio n rules given in the the ory of belief function s. 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