Why Dempster s Fusion Rule is not a Generalization of Bayes Fusion Rule

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1 Why Dempter Fuion Rule i not a Generalization of Baye Fuion Rule Jean Dezert Albena Tchamova Deqiang Han Jean-Marc Tacnet Originally publihed a Dezert J, Tchamova A, Han D, Tacnet J-M, Why Dempter fuion rule i not a generalization of Baye fuion rule, Proc Of Fuion 203 Int Conference on Information Fuion, Itanbul, Turkey, July 9-2, 203, and reprinted with permiion Abtract In thi paper, we analyze Baye fuion rule in detail from a fuion tandpoint, a well a the emblematic Dempter rule of combination introduced by Shafer in hi Mathematical Theory of evidence baed on belief function We propoe a new intereting formulation of Baye rule and point out ome of it propertie A deep analyi of the compatibility of Dempter fuion rule with Baye fuion rule i done We how that Dempter rule i compatible with Baye fuion rule only in the very particular cae where the baic belief aignment (bba ) to combine are Bayeian, and when the prior information i modeled either by a uniform probability meaure, or by a vacuou bba We how clearly that Dempter rule become incompatible with Baye rule in the more general cae where the prior i truly informative (not uniform, nor vacuou) Conequently, thi paper prove that Dempter rule i not a generalization of Baye fuion rule Keyword Information fuion, Probability theory, Baye fuion rule, Dempter fuion rule I INTRODUCTION In 979, Lotfi Zadeh quetioned in [] the validity of the Dempter rule of combination [2], [3] propoed by Shafer in Dempter-Shafer Theory (DST) of evidence [4] Since more than 30 year many trong debate [5], [6], [7], [8], [9], [0], [], [2], [3], [4], [5] on the validity of foundation of DST and Dempter rule have bloomed The purpoe of thi paper i not to dicu the validity of Dempter rule, nor the foundation of DST which have been already addreed in previou paper [6], [7], [8] In thi paper, we jut focu on the deep analyi of the real incompatibility of Dempter rule with Baye fuion rule Our analyi upport Mahler one briefly preented in [9] Thi paper i organized a follow In ection II, we recall baic of conditional probabilitie and Baye fuion rule with it main propertie In ection III, we recall the baic of belief function and Dempter rule In ection IV, we analyze in detail the incompatibility of Dempter rule with Baye rule in general and it partial compatibility for the very particular cae when prior information i modeled by a Bayeian uniform baic belief aignment (bba) Section V conclude thi paper II CONDITIONAL PROBABILITIES AND BAYES FUSION In thi ection, we recall the definition of conditional probability [20], [2] and preent the principle and the propertie of Baye fuion rule We preent the tructure of thi rule derived from the claical definition of the conditional probability in a new uncommon intereting form that will help u to analyze it partial imilarity with Dempter rule propoed by Shafer in hi mathematical theory of evidence [4] We will how clearly why Dempter rule fail to be compatible with Baye rule in general A Conditional probabilitie Let u conider two random event X and Z The conditional probability ma function (pmf) P(X Z) and P(Z X) are defined (auming P(X) > 0 and P(Z) > 0) by [20]: P(X Z) P(X Z) P(Z) and P(Z X) P(X Z) P(X) From Eq (), one get P(X Z) P(X Z)P(Z) P(Z X)P(X), which yield to Baye Theorem: P(X Z) P(Z X)P(X) P(Z) () and P(Z X) P(X Z)P(Z) P(X) (2) where P(X) i called the a priori probability of X, and P(Z X) i called the likelihood of X The denominator P(Z) play the role of a normalization contant warranting that N i P(X x i Z) In fact P(Z) can be rewritten a P(Z) N P(Z X x i )P(X x i ) (3) i The et of the N poible excluive and exhautive outcome of X i denoted Θ(X) x i,i,2,,n} B Baye parallel fuion rule In fuion application, we are often intereted in computing the probability of an event X given two event Z and Z 2 that have occurred More preciely, one want to compute P(X Z Z 2 ) knowing P(X Z ) and P(X Z 2 ), where X can take N ditinct exhautive and excluive tate x i, i, 2,, N Such type of problem i traditionally called a fuion problem The computation of P(X Z Z 2 ) from For convenience and implicity, we ue the notation P(X Z) intead of P(X x Z z), and P(Z X) intead of P(Z z X x) where x and z would repreent preciely particular outcome of the random variable X and Z 95

2 P(X Z ) and P(X Z 2 ) cannot be done in general without the knowledge of the probabilitiep(x) andp(x Z Z 2 ) which are rarely given However, P(X Z Z 2 ) become eaily computable by auming the following conditional tatitical independence condition expreed mathematically by: (A) : P(Z Z 2 X) P(Z X)P(Z 2 X) (4) With uch conditional independence condition (A), then from Eq () and Baye Theorem one get: P(Z Z2 X) P(Z Z2 X)P(X) P(X Z Z 2) P(Z Z 2) P(Z Z 2) P(Z X)P(Z 2 X)P(X) N ip(z X xi)p(z2 X xi)p(x xi) Uing again Eq (2), we have: P(Z X) P(X Z )P(Z ) P(X) andp(z 2 X) P(X Z 2)P(Z 2 ) P(X) and the previou formula of conditional probability P(X Z Z 2 ) can be rewritten a: P(X Z Z 2 ) P(X Z )P(X Z 2) P(X) N P(Xx i Z )P(Xx i Z 2) i P(Xx i) The rule of combination given by Eq (5) i known a Baye parallel (or product) rule and date back to Bernoulli [22] In the claification framework, thi formula i alo called the Naive Bayeian Claifier becaue it ue the aumption (A) which i often conidered a very unrealitic and too implitic, and that i why it i called a naive aumption The Eq (5) can be rewritten a: P(X Z Z 2 ) K(X,Z,Z 2 ) P(X Z ) P(X Z 2 ) (6) where the coefficient K(X,Z,Z 2 ) i defined by: K(X,Z,Z 2 ) P(X) N i C Symmetrization of Baye fuion rule (5) P(X x i Z )P(X x i Z 2 ) P(X x i ) (7) The expreion of Baye fuion rule given by Eq (5) can alo be ymmetrized in the following form that, quite urpriingly, rarely appear in the literature: P(X Z Z 2 ) N i or in an equivalent manner: P(X Z Z 2 ) P(X Z ) P(X Z2) P(X) P(X) P(Xx i Z ) P(Xxi Z2) P(Xxi) P(Xxi) (8) K (Z,Z 2 ) P(X Z ) P(X Z 2) (9) P(X) P(X) where the normalization contant K (Z,Z 2 ) i given by: K (Z,Z 2 ) N i P(X x i Z ) P(X xi ) P(X x i Z 2 ) P(X xi ) (0) We call the quantity A 2 (X x i ) P(Xxi Z) P(Xxi) entering in Eq (0) the Agreement Factor on P(Xx i Z 2) P(Xxi) X x i of order 2, becaue only two poterior pmf are ued in the derivation A 2 (X x i ) correpond to the poterior conjunctive conenu on the eventx x i taking into account the prior pmf of X The denominator of Eq (8) meaure the level of the Global Agreement (GA) of the conjunctive conenu taking into account the prior pmf of X It i denoted 2 GA 2 N P(X x i Z ) GA 2 i P(X xi ) P(X x i 2 Z 2 ) P(X xi2 ),i 2 i i 2 N P(X x i Z ) P(X xi ) P(X x i Z 2 ) P(X xi ) K (Z,Z 2 ) i () In fact, with aumption (A), the probability P(X Z Z 2 ) given in Eq (9) i nothing but the imple ratio of the agreement factor A 2 (X) (conjunctive conenu) on X over the global agreement GA 2 N i A 2(X x i ), that i: P(X Z Z 2 ) A 2(X) GA 2 (2) The quantity GC 2 given in Eq (3) meaure the global conflict (ie the total conjunctive diagreement) taking into account the prior pmf of X N P(X x i Z ) GC 2 i P(X xi ) P(X x i 2 Z 2 ) P(X xi2 ),i 2 i i 2 Generalization to P(X Z Z 2 Z ) (3) It can be proved that, when auming conditional independence condition, Baye parallel combination rule can be generalized for combining > 2 poterior pmf a: P(X Z Z ) K(X,Z,,Z ) k where the coefficient K(X,Z,,Z ) i defined by: K(X,Z,,Z ) P(X) N i The ymmetrized form of Eq (4) i: P(X Z Z ) P(X Z k ) (4) ( k P(X x i Z k )) P(X x i ) (5) K (Z,,Z ) k P(X Z k ) P(X) (6) with the normalization contant K (Z,,Z ) given by: K (Z,,Z ) N i k P(X x i Z k ) P(X xi ) (7) 2 The index 2 i introduced explicitly in the notation becaue we conider only the fuion of two poterior pmf 96

3 The generalization of A 2 (X), GA 2, and GC 2 provide the agreement A (X) of order, the global agreement GA and the global conflict GC for ource a follow: A (X x i ) P(X x i Z k ) P(X xi ) k N P(X x i Z ) GA i P(X xi ) P(X x i Z ) P(X xi ),,i i i GC N i,,i P(X x i Z ) P(X xi ) P(X x i Z ) P(X xi ) GA Symbolic repreentation of Baye fuion rule The (ymmetrized form of) Baye fuion rule of two poterior probability meaurep(x Z ) andp(x Z 2 ), given in Eq (9), require an extra knowledge of the prior probability of X For convenience, we denote ymbolically thi fuion rule a: P(X Z Z 2 ) Baye(P(X Z ),P(X Z 2 );P(X)) (8) Similarly, the (ymmetrized) Baye fuion rule of 2 probability meaure P(X Z k ), k,2,, given by Eq (6), which require alo the knowledge of P(X), will be denoted a: P(X Z Z ) Baye(P(X Z ),,P(X Z );P(X)) Particular cae: Uniform a priori pmf If the random variable X i aumed a a priori uniformly ditributed over the pace of it N poible outcome, then the probability of X i equal to P(X x i ) /N for i,2,,n In uch particular cae, all the prior probabilitie value P(X x i ) /N and P(X x i ) /N can be implified in Baye fuion formula Eq (9) and Eq (0) Therefore, Baye fuion formula (9) reduce to: P(X Z Z 2 ) P(X Z )P(X Z 2 ) N i P(X x i Z )P(X x i Z 2 ) (9) By convention, Eq (9) i denoted ymbolically a: P(X Z Z 2 ) Baye(P(X Z ),P(X Z 2 )) (20) Similarly, Baye(P(X Z ),,P(X Z )) rule defined with an uniform a priori pmf of X will be given by: k P(X Z Z ) P(X Z k) N i k P(X x (2) i Z k ) When P(X) i uniform and from Eq (9), one can redefine the global agreement and the global conflict a: GA unif 2 GC unif 2 N i,j ij N i,j i j P(X x i Z )P(X x j Z 2 ) (22) P(X x i Z )P(X x j Z 2 ) (23) Becaue N i P(X x i Z ) and N j P(X x j Z 2 ), then N N ( P(X x i Z ))( P(X x j Z 2 )) i j N P(X x i Z )P(X x j Z 2 ) i,j N i,j ij + P(X x i Z )P(X x j Z 2 ) N i,j i j P(X x i Z )P(X x j Z 2 ) Therefore, one ha alway GA unif 2 +GC unif 2 whenp(x) i uniform, and Eq (9) can be expreed a: P(X Z Z 2 ) P(X Z )P(X Z 2 ) GA unif P(X Z )P(X Z 2 ) 2 GC unif 2 (24) By a direct extenion, one will have: k P(X Z Z ) P(X Z k) k GA unif P(X Z k) GC unif (25) GA unif N i,,i i i P(X x i Z )P(X x i Z ) GC unif GA unif Remark : The normalization coefficient correponding to the global conjunctive agreement GA unif can alo be expreed uing belief function notation [4] a: GA unif P(X x i Z )P(X x i Z ) x i,,x i Θ(X) x i x i and the global diagreement, or total conflict level, i given by: GC unif P(X x i Z )P(X x i Z ) x i,,x i Θ(X) x i x i D Propertie of Baye fuion rule In thi ubection, we analyze Baye fuion rule (auming condition (A) hold) from a pure algebraic tandpoint In fuion jargon, the quantitie to combine come from ource of information which provide input that feed the fuion rule In the probabilitic framework, a ource to combine correpond to the poterior pmf P(X Z ) In thi ubection, we etablih five intereting propertie of Baye rule Contrary to Dempter rule, we prove that Baye rule i not aociative in general (P) : The pmf P(X) i a neutral element of Baye fuion rule when combining only two ource Proof: A ource i called a neutral element of a fuion rule if and only if it ha no influence on the fuion reult P(X) i a neutral element of Baye rule if and only if 97

4 Baye(P(X Z ),P(X);P(X)) P(X Z ) It can be eaily verified that thi equality hold by replacing P(X Z 2 ) by P(X) and P(X x i Z 2 ) by P(X x i ) (a if the conditioning term Z 2 vanihe) in Eq (5) One can alo verify that Baye(P(X),P(X Z 2 );P(X)) P(X Z 2 ), which complete the proof Remark 2: When conidering Baye fuion of more than two ource, P(X) doen t play the role of a neutral element in general, except if P(X) i uniform For example, let u conider 3 pmfp(x Z ),P(X Z 2 ) andp(x Z 3 ) to combine with formula (4) with P(X) not uniform When Z 3 vanihe o that P(X Z 3 ) P(X), we can eaily check that: Baye(P(X Z ),P(X Z 2 ),P(X);P(X)) Baye(P(X Z ),P(X Z 2 );P(X)) (26) (P2) : Baye fuion rule i in general not idempotent Proof: A fuion rule i idempotent if the combination of all ame input i equal to the input To prove that Baye rule i not idempotent it uffice to prove that in general: Baye(P(X Z ),P(X Z );P(X)) P(X Z ) From Baye rule (5), when P(X Z 2 ) P(X Z ) we clearly get in general P(X) N i P(X Z )P(X Z ) P(Xx i Z )P(Xx i Z ) P(Xx i) P(X Z ) (27) but when Z and Z 2 vanih, becaue in uch cae Eq (27) reduce to P(X) on it left and right ide Remark 3: In the particular (two ource) degenerate cae where Z and Z 2 vanih, one ha alway: Baye(P(X),P(X);P(X)) P(X) However, in the more general degenerate cae (when conidering more than 2 ource), one will have in general: Baye(P(X),P(X),,P(X);P(X)) P(X), but when P(X) i uniform, or when P(X) i a determinitic probability meaure uch that P(X x i ) for a given x i Θ(X) and P(X x j ) 0 for all x j x i (P3) : Baye fuion rule i in general not aociative Proof: A fuion rule f i called aociative if and only if it atifie the aociative law: f(f(x,y),z) f(x,f(y,z)) f(y,f(x,z)) f(x,y,z) for all poible input x, y and z Let u prove that Baye rule i not aociative from a very imple example Example : Let u conider the implet et of outcome x,x 2 } for X, with prior pmf: P(X x ) and P(X x 2 ) and let u conider the three given et of poterior pmf: P(X x Z ) 0 and P(X x 2 Z ) 09 P(X x Z 2 ) 05 and P(X x 2 Z 2 ) 05 P(X x Z 3 ) 06 and P(X x 2 Z 3 ) 04 Baye fuion Baye(P(X Z ),)P(X Z 2 ),P(X Z 3 );P(X)) of the three ource altogether according to Eq (6) provide: P(X x Z Z 2 Z 3 ) K P(X x 2 Z Z 2 Z 3 ) K where the normalization contant K 23 i given by: K Let u compute the fuion of P(X Z ) with P(X Z 2 ) uing Baye(P(X Z ),P(X Z 2 );P(X)) One ha: P(X x Z Z 2 ) K P(X x 2 Z Z 2 ) K where the normalization contant K 2 i given by: K Let u compute the fuion of P(X Z 2 ) with P(X Z 3 ) uing Baye(P(X Z 2 ),P(X Z 3 );P(X)) One ha P(X x Z 2 Z 3 ) K P(X x 2 Z 2 Z 3 ) K where the normalization contant K 23 i given by: K Let u compute the fuion of P(X Z ) with P(X Z 3 ) uing Baye(P(X Z ),P(X Z 3 );P(X)) One ha: P(X x Z Z 3 ) K P(X x 2 Z Z 3 ) K where the normalization contant K 3 i given by: K Let u compute the fuion of P(X Z Z 2 ) with P(X Z 3 ) uing Baye(P(X Z Z 2 ),P(X Z 3 );P(X)) One ha P(X x (Z Z 2 ) Z 3 ) K (2) P(X x 2 (Z Z 2 ) Z 3 ) K (2)3 727 where the normalization contant K (2)3 i given by K (2) Let u compute the fuion of P(X Z ) with P(X Z 2 Z 3 ) uing Baye(P(X Z ),P(X Z 2 Z 3 );P(X)) One ha P(X x Z (Z 2 Z 3 )) 0 57 K (23) P(X x 2 Z (Z 2 Z 3 )) K (23) 727 where the normalization contant K (23) i given by K (23) Let u compute the fuion of P(X Z Z 3 ) with P(X Z 2 ) uing Baye(P(X Z Z 3 ),P(X Z 2 );P(X)) One ha P(X x (Z Z 3 ) Z 2 ) K (3) P(X x 2 (Z Z 3 ) Z 2 ) K (3)

5 where the normalization contant K (3)2 i given by K (3) Therefore, one ee that even if in our example one ha f(x,f(y,z)) f(f(x,y),z) f(y,f(x,z)) becaue P(X (Z Z 2 ) Z 3 ) P(X Z (Z 2 Z 3 )) P(X Z 2 (Z Z 3 )), Baye fuion rule i not aociative ince: P(X (Z Z 2 ) Z 3 ) P(X Z Z 2 Z 3 ) P(X Z (Z 2 Z 3 )) P(X Z Z 2 Z 3 ) P(X Z 2 (Z Z 3 )) P(X Z Z 2 Z 3 ) (P4) : Baye fuion rule i aociative if and only if P(X) i uniform Proof: If P(X) i uniform, Baye fuion rule i given by Eq (2) which can be rewritten a: P(X Z ) k P(X Z Z ) P(X Z k) N i P(X xi Z) k P(X xi Z k) By introducing the term / N i k P(X x i Z k ) in numerator and denominator of the previou formula, it come: k P(X Z k) Ni P(X Z) P(X Z Z ) k P(Xx i Z k ) N k P(Xx i Z k ) i Ni P(X xi Z) k P(Xx i Z k ) which can be imply rewritten a: P(X Z Z )P(X Z ) P(X Z Z ) N i P(X xi Z Z )P(X xi Z) Therefore when P(X) i uniform, one ha: Baye(P(X Z ),,P(X Z )) Baye(Baye(P(X Z ),,P(X Z )),P(X Z )) The previou relation wa baed on the decompoition of k P(X Z k) a P(X Z ) k P(X Z k) Thi choice of decompoition wa arbitrary and choen only for convenience In fact k P(X Z k) can be decompoed in different manner, a P(X Z j ) k k j P(X Z k), j,2, and the imilar analyi can be done In particular, when 3, we will have: Baye(P(X Z ),P(X Z 2 ),P(X Z 3 )) Baye(Baye(P(X Z ),P(X Z 2 )),P(X Z 3 )) Baye(P(X Z ),Baye(P(X Z 2 ),P(X Z 3 ))) which complete the proof (P5) : The level of global agreement and global conflict between the ource do not matter in Baye fuion rule Proof: Thi property eem urpriing at firt glance, but, ince the reult of Baye fuion i nothing but the ratio of the agreement on x i (i,2,,n) over the global agreement factor, many ditinct ource with different global agreement (and thu with different global conflict) can yield ame Baye fuion reult Indeed, the ratio i kept unchanged when multiplying it numerator and denominator by ame non null calar value Conequently, the abolute level of global agreement between the ource (and therefore of global conflict alo) do not matter in Baye fuion reult What really matter i only the proportion of relative agreement factor Example 2: To illutrate thi property, let u conider Baye fuion rule applied to two ditinct et 3 of ource repreented by Baye(P(X Z ),P(X Z 2 );P(X)) and by Baye(P (X Z ),P (X Z 2 );P(X)) with the following prior and poterior pmf: P(X x ) and P(X x 2 ) P(X x Z ) and P(X x 2 Z ) P(X x Z 2 ) and P(X x 2 Z 2 ) P (X x Z ) 360 and P (X x 2 Z ) 0640 P (X x Z 2 ) and P (X x 2 Z 2 ) Applying Baye fuion rule given by Eq (5), one get for Baye(P(X Z ),P(X Z 2 );P(X)): P(X x Z Z 2 ) +04 /3 P(X x 2 Z Z 2 ) /3 (28) Similarly, one get for Baye(P (X Z ),P (X Z 2 );P(X)) P (X x Z Z 2 ) 0 0+ /3 P (X x 2 Z Z 2 ) 0+ 2/3 (29) Therefore, one ee thatbaye(p(x Z ),P(X Z 2 );P(X)) Baye(P (X Z ),P (X Z 2 );P(X)) even if the level of global agreement (and global conflict) are different In thi particular example, one ha: (GA2 060) (GA 2 030) (GC 2 60) (GC 2 (30) 205) In ummary, different et of ource to combine (with different level of global agreement and global conflict) can provide exactly the ame reult once combined with Baye fuion rule Hence the different level of global agreement and global conflict do not really matter in Baye fuion rule What really matter in Baye fuion rule i only the ditribution of all the relative agreement factor defined a A (X x i )/GA III BELIEF FUNCTIONS AND DEMPSTER S RULE The Belief Function (BF) have been introduced in 976 by Glenn Shafer in hi mathematical theory of evidence [4], alo known a Dempter-Shafer Theory (DST) in order to reaon under uncertainty and to model epitemic uncertaintie We will not preent in detail the foundation of DST, but only the baic mathematical definition that are neceary for the cope of thi paper The emblematic fuion rule propoed by Shafer to combine ource of evidence characterized by their baic belief aignment (bba) i Dempter rule that will be analyzed in detail in the equel In the literature over the year, DST ha been widely defended by it proponent in arguing that: ) Probability meaure are particular cae of Belief 3 The value choen for P(X Z ), P(X Z 2 ), P (X Z ), P (X Z 2 ) here have been approximated at the fourth digit They can be preciely determined uch that the expreion for P(X Z Z 2 ) and P (X Z Z 2 ) a given in Eq (28) and (29) hold For example, the exact value of P(x Z 2 ) i obtained by olving a polynomial equation of degree 2 having a a poible olution P(x Z 2 ) 2 ( ) , etc 99

6 function; and 2) Dempter fuion rule i a generalization of Baye fuion rule Although the tatement ) i correct becaue Probability meaure are indeed particular (additive) Belief function (called a Bayeian belief function), we will explain why the econd tatement about Dempter rule i incorrect in general A Belief function Let Θ be a frame of dicernment of a problem under conideration More preciely, the et Θ θ,θ 2,,θ N } conit of a lit of N exhautive and excluive element θ i, i,2,,n Each θ i repreent a poible tate related to the problem we want to olve The exhautivity and excluivity of element of Θ i referred a Shafer model of the frame Θ A baic belief aignment (bba), alo called a belief ma function, m() : 2 Θ [0,] i a mapping from the power et of Θ (ie the et of ubet of Θ), denoted 2 Θ, to [0,], that verifie the following condition [4]: m( ) 0 and X 2 Θ m(x) (3) The quantity m(x) repreent the ma of belief exactly committed to X An element X 2 Θ i called a focal element if and only if m(x) > 0 The et F(m) X 2 Θ m(x) > 0} of all focal element of a bba m() i called the core of the bba A bba m() i aid Bayeian if it focal element are ingleton of 2 Θ The vacuou bba characterizing the total ignorance denoted 4 I t θ θ 2 θ N i defined by m v () : 2 Θ [0;] uch that m v (X) 0 if X Θ, and m v (I t ) From any bba m(), the belief function Bel() and the plauibility function Pl() are defined for X 2 Θ a: Bel(X) Y 2 Θ Y X m(y) Pl(X) Y 2 Θ X Y m(y) (32) Bel(X) repreent the whole ma of belief that come from all ubet of Θ included in X It i interpreted a the lower bound of the probability of X, ie P min (X) Bel() i a ubadditive meaure ince θ Bel(θ i Θ i) Pl(X) repreent the whole ma of belief that come from all ubet of Θ compatible with X (ie, thoe interecting X) Pl(X) i interpreted a the upper bound of the probability of X, ie P max (X) Pl() i a uperadditive meaure ince θ Pl(θ i Θ i) Bel(X) and Pl(X) are claically een [4] a lower and upper bound of an unknown probability P(), and one ha the following inequality atified X 2 Θ : Bel(X) P(X) Pl(X) The belief function Bel() (and the plauibility function P l()) built from any Bayeian bba m() can be interpreted a a (ubjective) conditional probability meaure provided by a given ource of evidence, becaue if the bba m() i Bayeian the following equality alway hold [4]: Bel(X) Pl(X) P(X) 4 The et θ,θ 2,,θ N } and the complete ignorance θ θ 2 θ N are both denoted Θ in DST B Dempter rule of combination Dempter rule of combination, denoted DS rule 5 i a mathematical operation, repreented ymbolically by, which correpond to the normalized conjunctive fuion rule Baed on Shafer model of Θ, the combination of > independent and ditinct ource of evidence characterized by their bba m (),, m () related to the ame frame of dicernment Θ i denoted m DS () [m m ]() The quantity m DS () i defined mathematically a follow: m DS ( ) 0 and X 2 Θ m DS (X) m 2(X) K 2 (33) where the conjunctive agreement on X i given by: m 2 (X) m (X )m 2 (X 2 )m (X ) X,X 2,,X 2 Θ X X 2 X X (34) and where the global conflict i given by: K 2 m (X )m 2 (X 2 )m (X ) (35) X,X 2,,X 2 Θ X X 2 X When K 2, the ource are in total conflict and their combination cannot be computed with DS rule becaue Eq (33) i mathematically not defined due to 0/0 indeterminacy [4] DS rule i commutative and aociative which make it very attractive from engineering implementation tandpoint It ha been proved in [4] that the vacuou bba m v () i a neutral element for DS rule becaue [m m v ]() [m v m]() m() for any bba m() defined on 2 Θ Thi property look reaonable ince a total ignorant ource hould not impact the fuion reult becaue it bring no information that can be helpful for the dicrimination between the element of the power et 2 Θ IV ANALYSIS OF COMPATIBILITY OF DEMPSTER S RULE WITH BAYES RULE To analyze the compatibility of Dempter rule with Baye rule, we need to work in the probabilitic framework becaue Baye fuion rule ha been developed only in thi theoretical framework So in the equel, we will manipulate only probability ma function (pmf), related with Bayeian bba in the Belief Function framework Thi perfectly jutifie the retriction of ingleton bba a a prior bba ince we want to manipulate prior probabilitie to make a fair comparion of reult provided by both rule If Dempter rule i a true (conitent) generalization of Baye fuion rule, it mut provide ame reult a Baye rule when combining Bayeian bba, otherwie Dempter rule cannot be fairly claimed to be a generalization of Baye fuion rule In thi ection, we analyze the real (partial or total) compatibility of Dempter rule with Baye fuion rule Two important cae mut be analyzed depending on the nature of the prior information P(X) one ha in hand for performing the fuion of the ource Thee 5 We denote it DS rule becaue it ha been propoed hitorically by Dempter [2], [3], and widely promoted by Shafer in the development of DST [4] 200

7 ource to combine will be characterized by the following Bayeian bba : m () m (θ i ) P(X x i Z ),i,2,,n} m () m (θ i ) P(X x i Z ),i,2,,n} (36) The prior information i characterized by a given bba denoted a m 0 () that can be defined either on 2 Θ, or only on Θ if we want to deal for the need of our analyi with a Bayeian prior In the latter cae, ifm 0 () m 0 (θ i ) P(X x i ),i,2,,n} then m 0 () play the ame role a the prior pmf P(X) in the probabilitic framework When conidering a non vacuou prior m 0 () m v (), we denote Dempter combination of ource ymbolically a: m DS () DS(m (),,m ();m 0 ()) When the prior bba i vacuou m 0 () m v () then m 0 () ha no impact on Dempter fuion reult, and o we denote ymbolically Dempter rule a: m DS () DS(m (),,m ();m v ()) DS(m (),,m ()) A Cae : Uniform Bayeian prior It i important to note that Dempter fuion formula propoed by Shafer in [4] and recalled in Eq (33) make no real ditinction between the nature of ource to combine (if they are poterior or prior information) In fact, the formula (33) reduce exactly to Baye rule given in Eq (25) if the bba to combine are Bayeian and if the prior information i either uniform or vacuou Stated otherwie the following functional equality hold DS(m (),,m ();m 0 ()) Baye(P(X Z ),,P(X Z );P(X)) (37) a oon a all bba m i (), i,2,, are Bayeian and coincide with P(X Z i ), P(X) i uniform, and either the prior bbam 0 () i vacuou (m 0 () m v ()), orm 0 () i the uniform Bayeian bba Example 3: Let u conider Θ(X) x,x 2,x 3 } with two ditinct ource providing the following Bayeian bba m (x ) P(X x Z ) m 2 (x ) 05 m (x 2 ) P(X x 2 Z ) 03 and m 2 (x 2 ) 0 m (x 3 ) P(X x 3 Z ) 05 m 2 (x 3 ) 04 If we chooe a prior m 0 () the vacuou bba, that i m 0 (x x 2 x 3 ), then one will get m DS (x ) K m vacuou (x )m 2 (x )m 0 (x x 2 x 3 ) m DS (x 2 ) K m 2 vacuou (x 2 )m 2 (x 2 )m 0 (x x 2 x 3 ) m DS (x 3 ) K m 2 vacuou (x 3 )m 2 (x 3 )m 0 (x x 2 x 3 ) with K2 vacuou m (x )m 2 (x )m 0 (x x 2 x 3 ) m (x 2 )m 2 (x 2 )m 0 (x x 2 x 3 ) m (x 3 )m 2 (x 3 )m 0 (x x 2 x 3 ) 067 If we chooe a prior m 0 () the uniform Bayeian bba given by m 0 (x ) m 0 (x 2 ) m 0 (x 3 ) /3, then we get m DS (x ) m K uniform (x )m 2 (x )m 0 (x ) 2 00/ / m DS (x 2 ) m K uniform (x 2 )m 2 (x 2 )m 0 (x 2 ) 2 003/ / m DS (x 3 ) m K uniform (x 3 )m 2 (x 3 )m 0 (x 3 ) 2 0/ / where the degree of conflict when m 0 () i Bayeian and uniform i now given by K uniform 2 9 Clearly K uniform 2 K2 vacuou, but the fuion reult obtained with two ditinct prior m 0 () (vacuou or uniform) are the ame becaue of the algebraic implification by /3 in Dempter fuion formula when uing uniform Bayeian bba When combining Bayeian bba m () andm 2 (), the vacuou prior and uniform prior m 0 () have therefore no impact on the reult Indeed, they contain no information that may help to prefer one particular tate x i with repect to the other one, even if the level of conflict i different in both cae So, the level of conflict doen t matter at all in uch Bayeian cae A already tated, what really matter i only the ditribution of relative agreement factor It can be eaily verified that we obtain ame reult when applying Baye Eq (4), or (6) Only in uch very particular cae (ie Bayeian bba, and vacuou or Bayeian uniform prior), Dempter rule i fully conitent with Baye fuion rule So the claim that Dempter i a generalization of Baye rule i true in thi very particular cae only, and that i why uch claim ha been widely ued to defend Dempter rule and DST thank to it compatibility with Baye fuion rule in that very particular cae Unfortunately, uch compatibility i only partial and not general becaue it i not longer valid when conidering the more general cae involving non uniform Bayeian prior bba a hown in the next ubection B Cae 2: Non uniform Bayeian prior Let u conider Dempter fuion of Bayeian bba with a Bayeian non uniform prior m 0 () In uch cae it i eay to check from the general tructure of Baye fuion rule (6) and Dempter fuion rule (33) that thee two rule are incompatible Indeed, in Baye rule one divide each poterior ource m i (x j ) by m 0 (x j ), i,2,, wherea the prior ource m 0 () i combined in a pure conjunctive manner by Dempter rule with the bba m i (),i,2,, a ifm 0 () wa a imple additional ource Thi difference of proceing prior information between the two approache explain clearly the incompatibility of Dempter rule with Baye rule when Bayeian prior bba i not uniform Thi incompatibility i illutrated in the next imple example Mahler and Fixen have already propoed in [23], [24], [25] a modification of 20

8 Dempter rule to force it to be compatible with Baye rule when combining Bayeian bba The analyi of uch modified Dempter rule i out of the cope of thi paper Example 4: Let u conider the ame frame Θ(X), and ame bba m () and m 2 () a in the Example 3 Suppoe that the prior information i Bayeian and non uniform a follow: m 0 (x ) P(X x ) 06, m 0 (x 2 ) P(X x 2 ) 03 and m 0 (x 3 ) P(X x 3 ) 0 Applying Baye rule (2) yield: P(x Z Z 2 ) A2(x) GA 2 05/ P(x 2 Z Z 2 ) A2(x2) 03 0/ GA 2 P(x 3 Z Z 2 ) A2(x3) GA / Applying Dempter rule yield m DS (x i ) P(x i Z Z 2 ) becaue: 0060 m DS (x ) m DS (x 2 ) m DS (x 3 ) Therefore, one ha in general 6 : DS(m (),,m ();m 0 ()) Baye(P(X Z ),,P(X Z );P(X)) (38) V CONCLUSIONS In thi paper, we have analyzed in detail the expreion and the propertie of Baye rule of combination baed on tatitical conditional independence aumption, a well a the emblematic Dempter rule of combination of belief function introduced by Shafer in hi Mathematical Theory of evidence We have clearly explained from a theoretical tandpoint, and alo on imple example, why Dempter rule i not a generalization of Baye rule in general The incompatibility of Dempter rule with Baye rule i due to it impoibility to deal with non uniform Bayeian prior in the ame manner a Baye rule doe Dempter rule turn to be compatible with Baye rule only in two very particular cae: ) if all the Bayeian bba to combine (including the prior) focu on ame tate (ie there i a perfect conjunctive conenu between the ource), or 2) if all the bba to combine (excluding the prior) are Bayeian, and if the prior bba cannot help to dicriminate a particular tate of the frame of dicernment (ie the prior bba i either vacuou, or Bayeian and uniform) Except in thee two very particular cae, Dempter rule i totally incompatible with Baye rule Therefore, Dempter rule cannot be claimed to be a generalization of Baye fuion rule, even when the bba to combine are Bayeian ACKNOWLEDGMENT Thi tudy wa co-upported by Grant for State Key Program for Baic Reearch of China (973) (No 203CB329405), National NSF of China (No60424, No ), and alo partly upported by the project AComIn, grant 36087, funded by the FP7 Capacity Programme REFERENCES [] LA Zadeh, On the validity of Dempter rule of combination, Memo M79/24, Univ of California, Berkeley, CA, USA, 979 [2] A Dempter, Upper and lower probabilitie induced by a multivalued mapping, Ann Math Statit, Vol 38, pp , 967 [3] A Dempter, A generalization of bayeian inference, J R Stat Soc B 30, pp , 968 [4] G Shafer, A Mathematical theory of evidence, Princeton Univerity Pre, Princeton, NJ, USA, 976 [5] LA Zadeh, Book review: A mathematical theory of evidence, The Al Magazine, Vol 5, No 3, pp 8-83, 984 [6] LA Zadeh, A imple view of the Dempter-Shafer theory of evidence and it implication for the rule of combination, The Al Magazine, Vol 7, No 2, pp 85 90, 986 [7] J Lemmer, Confidence factor, empiricim and the Dempter-Shafer theory of evidence, Proc of UAI 85, pp 60 76, Lo Angele, CA, USA, July 0 2, 985 [8] J Pearl, Do we need higher-order probabilitie, and, if o, what do they mean?, Proc of UAI 87, pp 47 60, Seattle, WA, USA, July 0 2, 987 [9] F Voorbraak, On the jutification of Dempter rule of combination, Dept of Phil, Utrecht Univ, Logic Group, Preprint Ser, No 42, Dec 988 [0] G Shafer, Perpective on the theory and practice of belief function, IJAR, Vol 4, No 5 6, pp , 990 [] J Pearl, Reaoning with belief function: an analyi of compatibility, IJAR, Vol 4, No 5 6, pp , 990 [2] J Pearl, Rejoinder of comment on Reaoning with belief function: An analyi of compatibility, IJAR, Vol 6, No 3, pp , May 992 [3] GM Provan, The validity of Dempter-Shafer belief function, IJAR, Vol 6, No 3, pp , May 992 [4] P Wang, A defect in Dempter-Shafer theory, Proc of UAI 94, pp , Seattle, WA, USA, July 29 3, 994 [5] A Gelman, The boxer, the wretler, and the coin flip: a paradox of robut bayeian inference and belief function, American Statitician, Vol 60, No 2, pp 46 50, 2006 [6] F Smarandache, J Dezert (Editor), Application and advance of DSmT for information fuion, Vol 3, ARP, USA, [7] J Dezert, P Wang, A Tchamova, On the validity of Dempter-Shafer theory, Proc of Fuion 202 Int Conf, Singapore, July 9 2, 202 [8] A Tchamova, J Dezert, On the Behavior of Dempter rule of combination and the foundation of Dempter-Shafer theory, (bet paper award), Proc of 6th IEEE Int Conf on Intelligent Sytem IS 2, Sofia, Bulgaria, Sept 6 8, 202 [9] RP Mahler, Statitical multiource-multitarget information fuion, Chapter 4, Artech Houe, 2007 [20] XR Li, Probability, random ignal and tatitic, CRC Pre, 999 [2] PE Pfeiffer, Applied probability, Connexion Web ite, Aug 3, [22] G Shafer, Non-additive probabilitie in the work of Bernoulli and Lambert, in Archive for Hitory of Exact Science, C Truedell (Ed), Springer-Verlag, Berlin, Vol 9, No 4, pp , 978 [23] RP Mahler, Uing a priori evidence to cutomize Dempter-Shafer theory, Proc of 6th Nat Symp on Senor Fuion, Vol, pp , Orlando, FL, USA, April 3 5, 993 [24] RP Mahler, Claification when a priori evidence i ambiguou, Proc SPIE Conf on Opt Eng in Aeropace Sening (Automatic Object Recognition IV), pp , Orlando, FL, USA, April 4 8, 994 [25] D Fixen, RP Mahler, The modified Dempter-Shafer approach to claification, IEEE Tran on SMC, Part A, Vol 27, No, pp 96 04, but in the very degenerate cae when manipulating determinitic Bayeian bba, which i of little practical interet from the fuion tandpoint 202

9 On the Conitency of PCR6 with the Averaging Rule and it Application to Probability Etimation Florentin Smarandache Jean Dezert Originally publihed a Smarandache F, Dezert J, On the conitency of PCR6 with the averaging rule and it application to probability etimation, Proc of Fuion 203 Int Conference on Information Fuion, Itanbul, Turkey, July 9-2, 203, and reprinted with permiion (with typo correction) Abtract Since the development of belief function theory introduced by Shafer in eventie, many combination rule have been propoed in the literature to combine belief function pecially (but not only) in high conflicting ituation becaue the emblematic Dempter rule generate counter-intuitive and unacceptable reult in practical application Many attempt have been done during lat thirty year to propoe better rule of combination baed on different framework and jutification Recently in the DSmT (Dezert-Smarandache Theory) framework, two intereting and ophiticate rule (PCR5 and PCR6 rule) have been propoed baed on the Proportional Conflict Reditribution (PCR) principle Thee two rule coincide for the combination of two baic belief aignment, but they differ in general a oon a three or more ource have to be combined altogether becaue the PCR ued in PCR5 and in PCR6 are different In thi paper we how why PCR6 i better than PCR5 to combine three or more ource of evidence and we prove the coherence of PCR6 with the imple Averaging Rule ued claically to etimate the probability baed on the frequentit interpretation of the probability meaure We how that uch probability etimate cannot be obtained uing Dempter-Shafer (DS) rule, nor PCR5 rule Keyword: Information fuion, belief function, PCR6, PCR5, DSmT, frequentit probability I INTRODUCTION In thi paper, we work with belief function [] defined from the finite and dicrete frame of dicernment Θ θ,θ 2,,θ n } In Dempter-Shafer Theory (DST) framework, baic belief aignment (bba ) provided by the ditinct ource of evidence are defined on the fuion pace 2 Θ (Θ, ) coniting in the power-et of Θ, that i the et of element of Θ and thoe generated from Θ with the union et operator Such fuion pace aume that the element of Θ are non-empty, exhautive and excluive, which i called Shafer model of Θ More generally, in Dezert-Smarandache Theory (DSmT) [2], the fuion pace denoted G Θ can alo be either the hyper-power et D Θ (Θ,, ) (Dedekind lattice), or uper-power et S Θ (Θ,,,c()) depending on the underlying model of the frame of dicernment we chooe to fit with the nature of the problem Detail on DSm model are given in [2], Vol We aume that 2 baic belief aignment (bba ) m i (), i,2,, provided by ditinct ource of evidence defined on the fuion pace G Θ are available and we need to combine them for a final deciion-making purpoe and c() are repectively the et interection and complement operator For doing thi, many rule of combination have been propoed in the literature, the mot emblematic one being the imple Averaging Rule, Dempter-Shafer (DS) rule, and more recently the PCR5 and PCR6 fuion rule The contribution of thi paper i to analyze in deep the behavior of PCR5 and PCR6 fuion rule and to explain why we conider more preferable to ue PCR6 rule rather than PCR5 rule for combining everal ditinct ource of evidence altogether We will how in detail the trong relationhip between PCR6 and the averaging fuion rule which i commonly ued to etimate the probabilitie in the claical frequentit interpretation of probabilitie Thi paper i organized a follow In ection II, we briefly recall the background on belief function and the main fuion rule ued in thi paper Section III demontrate the conitency of PCR6 fuion rule with the Averaging Rule for binary mae in total conflict a well a the ability of PCR6 to dicriminate aymmetric fuion cae for the fuion of Bayeian bba Section IV how that PCR6 can alo be ued to etimate empirical probability in a imple (coin toing) random experiment Section V will conclude and open challenging problem about the recurivity of fuion rule formula that are ought for efficient implementation II A Baic belief aignment BACKGROUND ON BELIEF FUNCTIONS Let conider a finite dicrete frame of dicernment Θ θ,θ 2,,θ n },n > of the fuion problem under conideration and it fuion paceg Θ which can be choen either a 2 Θ, D Θ or S Θ depending on the model that fit with the problem A baic belief aignment (bba) aociated with a given ource of evidence i defined a the mapping m() : G Θ [0,] atifying m( ) 0 and A GΘ m(a) The quantity m(a) i called ma of belief of A committed by the ource of evidence If m(a) > 0 then A i called a focal element of the bba m() When all focal element are ingleton and G Θ 2 Θ then m() i called a Bayeian bba [] and it i homogeneou to a (poibly ubjective) probability meaure The vacuou bba repreenting a totally ignorant ource i defined a m v (Θ) Belief and plauibility function are defined by Bel(A) B A B G Θ m(b) and Pl(A) B A B G Θ m(b) () 203

10 B Fuion rule The main information fuion problem in the belief function framework (DST or DSmT) i how to combine efficiently everal ditinct ource of evidence repreented by m (), m 2 (),, m () ( 2) bba defined on G Θ Many rule have been propoed for uch tak ee [2], Vol 2, for a detailed lit of fuion rule and we focu here on the following one: ) the Averaging Rule becaue it i the implet one and it i ued to empirically etimate probabilitie in random experiment, 2) DS rule becaue it wa hitorically propoed in DST, and 3) PCR5 and PCR6 rule becaue they were propoed in DSmT and have hown to provide better reult than the DS rule in all application where they have been teted o far So we jut briefly recall how thee rule are mathematically defined Averaging fuion rule,2,, () For any X in G Θ,,2,, (X) i defined by,2,, (X) Average(m,m 2,,m ) m i (X) i (2) Note that the vacuou bba m v (Θ) i not a neutral element for thi rule Thi Averaging Rule i commutative but it i not aociative becaue in general,2,3 (X) 3 [m (X)+m 2 (X)+m 3 (X)] i different from (,2),3 (X) 2 [m (X)+m 2 (X) +m 3 (X)] 2 which i alo different from,(2,3) (X) 2 [m (A)+ m 2(X)+m 3 (X) ] 2 and alo from (,3),2 (X) 2 [m (X)+m 3 (X) +m 2 (X)] 2 In fact, it i eay to prove that the following recurive formula hold,2,, (X),2,, (X)+ m (X) (3) Thi imple averaging fuion rule ha been ued ince more than two centurie for etimating empirically the probability meaure in random experiment [3], [4] Dempter-Shafer fuion rule m DS,2,,() In DST framework, the fuion pace G Θ equal the poweret 2 Θ becaue Shafer model of the frame Θ i aumed The combination of 2 ditinct ource of evidence characterized by the bba m i (), i,2,,, i done with DS rule a follow []: m DS,2,,( ) 0 and for all X in 2 Θ m DS,2,, (X) K,2,, X,X 2,,X 2 Θ i X X 2 X X m i (X i ) (4) where the numerator of (4) i the ma of belief on the conjunctive conenu on X, and where K,2,, i a normalization contant defined by K,2,, m i (X i ) m,2,, ( ) X,X 2,,X 2 Θ i X X 2 X The total degree of conflict between the ource of evidence i defined by m,2,, ( ) m i (X i ) X,X 2,,X 2 Θ i X X 2 X The ource are aid in total conflict when m,2,, ( ) The vacuou bba m v (Θ) i a neutral element for DS rule and DS rule i commutative and aociative It remain the miletone fuion rule of DST The doubt on the validity of uch fuion rule ha been dicued by Zadeh in 979 [5] [7] baed on a very imple example with two highly conflicting ource of evidence Since 980, many criticim have been done about the behavior and jutification of uch DS rule More recently, Dezert et al in [8], [9] have put in light other counter-intuitive behavior of DS rule even in low conflicting cae and howed eriou flaw in logical foundation of DST PCR5 and PCR6 fuion rule To work in general fuion pace G Θ and to provide better fuion reult in all (low or high conflicting) ituation, everal fuion rule have been developed in DSmT framework [2] Among them, two fuion rule called PCR5 and PCR6 baed on the proportional conflict reditribution (PCR) principle have been proved to work efficiently in all different application where they have been ued o far The PCR principle tranfer the conflicting ma only to the element involved in the conflict and proportionally to their individual mae, o that the pecificity of the information i entirely preerved The general principle of PCR conit: ) to apply the conjunctive rule; 2) calculate the total or partial conflicting mae; 3) then reditribute the (total or partial) conflicting ma proportionally on non-empty et according to the integrity contraint one ha for the frame Θ Becaue the proportional tranfer can be done in two different way, thi ha yielded to two different fuion rule The PCR5 fuion rule ha been propoed by Smarandache and Dezert in [2], Vol 2, Chap, and PCR6 fuion rule ha been propoed by Martin and Owald in [2], Vol 2, Chap 2 We will not preent in deep thee two fuion rule ince they have already been dicued in detail with many example in the aforementioned reference but we only give their expreion for convenience here 204

11 The general formula of PCR5 for the combination of 2 ource i given by m PCR5,2,,( ) 0 and for X in G Θ m PCR5,2,, (X) m,2,,(x)+ 2 t X j2,,x jt G Θ \X} r,,r t r j <r 2<<r t <(r t) 2,,j t} P t (,,n}) X X j2 X j i,,i } P (,,}) ( r k m i k (X) 2 ) [ t l2 ( r l k l r l + m i kl (X jl )] ( r k m i k (X))+[ t l2 ( r l k l r l + m i kl (X jl )] where i, j, k, r, and t in (5) are integer m,2,, (X) correpond to the conjunctive conenu on X between ource and where all denominator are different from zero If a denominator i zero, that fraction i dicarded; P k (,2,,n}) i the et of all ubet of k element from,2,,n} (permutation of n element taken by k), the order of element doen t count The general formula of PCR6 propoed by Martin and Owald for the combination of 2 ource i given by ( ) 0 and for X in GΘ,2,,,2,,(X) m,2,, (X)+ m i (X) 2 i k Y σ i (k) X (Y σi (),,Y σi ( )) (G Θ ) j m i (X)+ m σi(j)(y σi(j)) j where σ i count from to avoiding i: σi (j) j if j < i, σ i (j) j + if j i, m σi(j)(y σi(j)) Since Y i i a focal element of expert/ource i, m i(x)+ m σi(j)(y σi(j)) 0 j The general PCR5 and PCR6 formula (5) (6) are rather complicate and not very eay to undertand From the implementation point of view, PCR6 i much imple to implement than PCR5 For convenience, very baic (not optimized) Matlab code of PCR5 and PCR6 fuion rule can be found in [2], [0] and from the toolboxe repoitory on the web [] The PCR5 and PCR6 fuion rule are commutative but not aociative, like the averaging fuion rule, but the vacuou belief aignment i a neutral element for thee PCR fuion rule (5) (6) (7) The PCR5 and PCR6 fuion rule implify greatly and coincide for the combination of two ource ( 2) In uch implet cae, one alway get the reulting bbam PCR5/6 (),2 () m PCR5,2 () expreed a m PCR5/6 ( ) 0 and for all X in G Θ m PCR5/6 (X) Y G Θ \X} X Y X,X 2 G Θ X X 2 X m (X )m 2 (X 2 )+ [ m (X) 2 m 2 (Y) m (X)+m 2 (Y) + m 2(X) 2 m (Y) m 2 (X)+m (Y) ] (8) where all denominator in (8) are different from zero If a denominator i zero, that fraction i dicarded All propoition/et are in a canonical form Example : See [2], Vol2, Chap for more example Let conider the frame of dicernment Θ A,B} of excluive element Here Shafer model hold o that G Θ 2 Θ,A,B,A B} We conider two ource of evidence providing the following bba m (A) 06 m (B) 03 m (A B) 0 m 2 (A) m 2 (B) 03 m 2 (A B) 05 Then the conjunctive conenu yield : m 2 (A) 044 m 2 (B) 7 m 2 (A B) 005 with the conflicting ma m 2 (A B ) m (A)m 2 (B)+m (B)m 2 (A) One ee that only A and B are involved in the derivation of the conflicting ma, but not A B With PCR5/6, one reditribute the partial conflicting ma 08 to A and B proportionally with the mae m (A) and m 2 (B) aigned to A and B repectively, and alo the partial conflicting ma 006 to A and B proportionally with the mae m 2 (A) and m (B) aigned to A and B repectively, thu one get two weighting factor of the reditribution for each correponding et A and B repectively Let x be the conflicting ma to be reditributed to A, and y the conflicting ma reditributed to B from the firt partial conflicting ma 08 Thi firt partial proportional reditribution i then done according x 06 y 03 x +y whence x 06 02, y Now let x 2 be the conflicting ma to be reditributed to A, and y 2 the conflicting ma reditributed to B from the econd the partial conflicting ma 006 Thi econd partial proportional reditribution i then done according x 2 y 2 03 x 2 +y

12 whence x , y Thu one finally get: m PCR5/6 (A) m PCR5/6 (B) m PCR5/6 (A B) The difference between PCR5 and PCR6 fuion rule For the two ource cae, PCR5 and PCR6 fuion rule coincide A oon a three (or more) ource are involved in the fuion proce, PCR5 and PCR6 differ in the way the proportional conflict reditribution i done For example, let conider three ource with bba m (), m 2 () and m 3 (), A B for the model of the frame Θ, and m (A) 06, m 2 (B) 03, m 3 (B) 0 With PCR5, the ma m (A)m 2 (B)m 3 (B) correponding to a conflict i reditributed back to A and B only with repect to the following proportion repectively: x PCR5 A 0074 and x PCR5 B becaue the proportionalization require that i Thu x PCR5 A m (A) x PCR5 B m 2 (B)m 3 (B) m (A)m 2 (B)m 3 (B) m (A)+m 2 (B)m 3 (B) x PCR5 A 06 x PCR5 A x PCR5 B xpcr5 B With the PCR6 fuion rule, the partial conflicting ma m (A)m 2 (B)m 3 (B) i reditributed back toaandb only with repect to the following proportion repectively: x PCR6 A 0008 and x PCR6 B becaue the PCR6 proportionalization i done a follow: x PCR6 A m (A) that i x PCR6 B m 2 (B)+m 3 (B) x PCR6 A 06 m (A)m 2 (B)m 3 (B) m (A)+(m 2 (B)+m 3 (B)) xpcr6 B (03+0) 008 and therefore with PCR6, one get finally the following reditribution to A and B: x PCR6 A x PCR6 B (03+0) In [2], Vol 2, Chap 2, Martin and Owald have propoed PCR6 baed on intuitive conideration and the author have hown through imulation that PCR6 i more table than PCR5 in term of deciion for combining > 2 ource of evidence Baed on thee reult and the relative implicity of implementation of PCR6 over PCR5, PCR6 ha been conidered more intereting/efficient than PCR5 for combining 3 (or more) ource of evidence III CONSISTENCY OF PCR6 WITH THE AVERAGING RULE In thi ection we how why we alo conider PCR6 a better than PCR5 for combining bba But here, our argumentation i not baed on particular imulation reult and deciion-making a done by Martin and Owald, but on a theoretical analyi of the tructure of PCR6 fuion rule itelf In particular, we how the full conitency of PCR6 rule with the averaging fuion rule ued to empirically etimate probabilitie in random experiment For doing thi, it i neceary to implify the original PCR6 fuion formula (6) Such implification ha already been propoed in [2] and the PCR6 fuion rule can be in fact rewritten a,2,,(x) m,2,,(x)+ k X i,x i2,,x ik G Θ \X (i,i 2,,i k ) P (,,}) ( k j X ij ) X [m i (X)+m i2 (X)++m ik (X)] m i (X)m ik (X)m ik+ (X ik+ )m i (X i ) m i (X)++m ik (X)+m ik+ (X ik+ )++m i (X i ) (9) where P (,,}) i the et of all permutation of the element,2,,} It hould be oberved that X i, X i2,,x i may be different from each other, or ome of them equal and other different, etc We wrote thi PCR6 general formula (9) in the tyle of PCR5, different from Arnaud Martin & Chritophe Owald notation, but actually doing the ame thing In order not to complicate the formula of PCR6, we did not ue more ummation or product after the third Sigma We now are able to etablih the conitency of general PCR6 formula with the Averaging fuion rule for the cae of binary bba through the following theorem Theorem : When 2 ource of evidence provide binary bba on G Θ whoe total conflicting ma i, then the PCR6 fuion rule coincide with the averaging fuion rule Otherwie, PCR6 and the averaging fuion rule provide in general different reult Proof : All 2 bba are aumed binary, ie m(x) 0 or (two numerical value 0 and only are allowed) for any bba m() and for any et X in the focal element A focal element in thi cae i an element X uch that at leat one of the binary ource aign a ma equal to to X Let uppoe the focal element are F, F 2,, F n Then the et of bba to combine can be expreed a in the Table I where Table I LIST OF BBA S TO COMBINE bba \ Focal elem F F 2 F n m () m 2() m () all are 0 or ; on each row there i only a (ince the um of all mae of a bba i equal to ) and all the other element are 0 ; 206