Cumulative and Averaging Fission of Beliefs

Cumulativ e and A veraging Fission of Beliefs A udun J øsang UniK, Univ ersit y of Oslo Norway josang @ unik .no Abstract Belief fusion is the princi ple of com- bining separa te beliefs or bodies of e vidence origin ating from dif ferent source s. Dependin g on the situation to be modelled , differ ent belief fusion methods can be applied. Cumulati v e and a vera ging belief fusion is defined for fusing opini ons i n subjecti v e lo gic, and fo r fus ing belief fun ctions in gen- eral. The pri nciple of fi ssion is the oppos ite of fusion, namely to elimi- nate the con trib ution of a specific be - lief fro m an already fused be lief, w ith the purpose of derivi ng the remainin g belief. This paper describes fissi on of cumulati v e belie f as well as fission of av eraging b elief in subjec ti ve logi c. These operato rs can for exa mple be ap- plied t o belief re vision in Bayesian be- lief network s, where the belief contri- b ution of a gi v en evi dence source can be determine d as a f unction of a giv en fused beli ef and its other contr ib uting beliefs . Ke ywords: Fusio n, fission , sub jecti ve logic, belief , uncertai nty 1 Intr oduction Subjecti v e logic is a type of probabilis tic lo gic that expl icitly takes uncerta inty and belie f own- ership in to ac count. In genera l, subje cti ve logi c is suitabl e f or modeling and analysin g situat ions in v o lving uncertainty and inco mplete kno wledge [1, 2]. For example, it can be used for m odelin g trust netwo rks [6] an d f or analysi ng Bayesian net- works [5]. Arg uments in s ubjecti v e logic are su bjecti v e op in- ions ab out pro positio ns. The opinion spac e is a subset of the belief func tion space used in Dempster -Shafer belief theory . The term be- lief will be used interch angeab ly with opinio ns throug hout this paper . A binomial opinion ap- plies to a singl e propositio n, and can be rep- resent ed as a Beta distrib utio n. A m ultino mial opinio n applie s to a collection of propos itions, and ca n be represent ed as a Dirichlet di strib u- tion. Through the co rrespond ence between op in- ions and Beta/Dirichlet distr ib utions, sub jecti v e logic prov ides an algebra for these functions . The two types of fusion defined for subjecti ve logic a re cumul ative fusio n and avera ging fusio n [4]. Situations that can be m odelle d with th e cu- mulati v e operator are for exampl e when fusing beliefs of two observers w ho hav e assessed sepa- rate a nd ind epende nt eviden ce, such as when t hey ha ve obse rved the out comes of a gi ve n process ov er two separate non-ov erlapp ing time periods. Situation s that can be modelled with the av erag - ing operator are for e xample when fu sing bel iefs of two observ ers who hav e assess ed the same ev- idence and possib ly interpre ted it diff erently . Dempster’ s rule also represen ts a method com- monly applied for fusing beliefs. Ho wev er , it is not used in subjecti ve logic and w ill not be dis- cussed here. There are situatio ns where it is useful to split a fused b elief in its contrib uting belief compo nents, and this process is called belief fi ssion. This re- quires the already fused belief and one of its con- trib uting belie f componen ts a s input , and will pro- duce the remainin g co ntrib uting beli ef componen t as output. Fission is basically th e opposite of fu- sion, a nd the formal expres sions for fission can be deri ved by re arrangi ng the exp ression s for fu sion. This will be descri bed in the follo wing sections. 2 Fundamentals of Subjecti ve Logic Subjecti ve opinion s exp ress subjec tiv e beliefs about the truth of proposi tions with degrees of un- certain ty , and ca n indicate sub jecti ve be lief o wn- ership when ev er require d. An opinion is usu ally denote d as ω A x where A is the subje ct, also called the belief o wner , and x is the propos ition to which the opinion applies . An altern ati ve notation is ω ( A : x ) . The propos ition x is assumed to be- long to a frame of di scernment (also called state space) e.g. denoted as X , b ut the frame is usually not included in the opini on notat ion. The propo- sition s of a frame are normally assumed to be ex- hausti ve and mutually dis joint, and subjec ts are assumed to hav e a common semantic interpre ta- tion of proposition s. The subject, the propositio n and its frame are attrib utes of an opinio n. Indi- cation of sub jecti ve be lief o wnershi p is normally omitted whene ver irre le va nt. 2.1 Binomia l Opinions Let x be a proposi tion. Entity A ’ s binomial opin- ion abo ut the tr uth of a x is the ordere d quadruple ω A x = ( b, d , u, a ) with the compon ents: b : belief that the proposit ion is true d : d isbelief that the propositi on is true (i.e. th e belief that the proposition is false) u : uncertai nty about the probabi lity of x (i.e. th e amount of uncommitted belief) a : base rate of x (i.e. pr obabilit y of x in the absence of belief ) These compone nts satisfy: b, d, u, a ∈ [0 , 1] (1) and b + d + u = 1 (2) The characte ristics of variou s op inion classes are listed belo w . A n opini on w here: b = 1 : is equi valen t to binary logic TRUE, d = 1 : is equi valen t to binary logic F ALSE, b + d = 1 : is equi v alent to a probability , 0 < ( b + d ) < 1 : e xpresse s uncertain ty , and b + d = 0 : is va cuous (i.e. totally uncert ain). The probabilit y expe ctation v alue of a binomial opinio n is: p ( ω x ) = b x + a x u x . (3) The expressio n of Eq.(3) is equi v alent to the pig- nistic prob ability in traditiona l belie f function theory [10], and is based on the principle tha t the belie f mass assig ned to the whole frame is split equally among the singleto ns of the frame. In Eq.(3) the base rate a x must be interpre ted in the s ense th at the relativ e proportion of single tons contai ned in x is equal to a x . Binomial op inions can be represen ted on an equi - lateral tria ngle as sho wn in Fig.1 bel ow . A point inside the triangle represent s a ( b, d, u ) triple . The b,d,u-ax es run from one edge to the opposite ver - tex indicate d by the Belief, Disbelief or Uncer - tainty label. For examp le, a str ong positiv e opin- ion is repr esented by a point to ward s the bottom right Belief verte x. The base rate, also called rel- ati ve atomicity , is sho wn as a red pointer alon g the probabil ity base line, and the probabili ty ex- pectat ion, E, is formed by projecting the opin- ion ont o the base, paralle l to the base rate pro- jector line. As an exa mple, the opinio n ω x = (0 . 4 , 0 . 1 , 0 . 5 , 0 . 6) is sho w n on the figure. x ω x 0,5 0 0 1 0,5 0,5 Disbelief 1 Belief 1 0 0 1 Uncertainty 0,5 Probability base line Projector Director E( ) ω x a Figure 1: Opinio n triang le w ith exampl e opinion Uncertain ty about probab ility values c an be in ter - preted as ignora nce, or second order uncertain ty about the first order probab ilities. In this pa- per , the term “uncertai nty” will be used in the sense of “ uncert ainty abo ut the pr obability val- ues” . A p robabil istic logic based on belief theo ry therefo re repres ents a generalisatio n of tradit ional probab ilistic logic. 2.2 Multi nomial Opinions Let X be a frame, i.e. a set of e xhausti ve and mu- tually disjoi nt proposition s x i . Entity A ’ s multi- nomial opinion o ver X is the composite fun c- tion ω A X = ( ~ b, u, ~ a ) , where ~ b is a vec tor o f b elief masses ove r th e p ropositi ons of X , u is the u ncer - tainty m ass, and ~ a is a v ector of base rate va lues ov er the propo sitions of X . These components satisfy : ~ b ( x i ) , u, ~ a ( x i ) ∈ [0 , 1] , ∀ x i ∈ X (4) u + X x i ∈ X ~ b ( x i ) = 1 (5) X x i ∈ X ~ a ( x i ) = 1 (6) V isualisin g multin omial opinions is not tri vial. T rinomial opinion s can b e visu alised as p oints in- side a triangula r pyr amid as sho wn in Fig.2, bu t the 2D aspec t of printe d p aper and computer mon- itors make this impra ctical in general. Figure 2: Opinion pyramid with example trino- mial opini on Opinions with dimension s large r than trinomial do not lend themselv es to traditional visualisa- tion. 3 Fusion of Multinomial Opinions In man y situa tions there will b e mul tiple sou rces of evid ence, and fusion can be used to combine e vidence from dif ferent sources. In order to pro vide an interpret ation of fusion in subjec ti ve logic it is useful to consi der a proces s that is observ ed by two s ensors. A dis tinction can be made between two case s. 1. The two sen sors observ e the proce ss during disjoi nt time period s. In this case the ob- serv ations are indepen dent, and it is natu ral to simply add the observ ations from the two sensor s, and the resu lting fus ion is called cu- mulative fusion . 2. The two sen sors observ e the proce ss during the same time period . In this case the ob- serv ations are dependen t, and it is natural to tak e the av erage of the observ ation s by the two senso rs, an d the resulting fusion is called aver agin g fusion . 3.1 Cumulati ve Fusion Assume a frame X co ntaining k elements. As- sume two observ ers A an d B who hav e indepen- dent opinions o ver the frame X . This ca ne for ex- ample result from having ob serv ed the ou tcomes of a process ov er two sepa rate time period s. Let the two obse rvers ’ respecti ve opinions be exp ressed as ω A X = ( ~ b A X , u A X , ~ a A X ) an d ω B X = ( ~ b B X , u B X , ~ a B X ) . The cumulati ve fusion of the se two bod ies of e vi- dence is d enoted as ω A ⋄ B X = ω A X ⊕ ω B X . The symbol “ ⋄ ” d enotes the fusion of two observ ers A and B into a single imaginary observ er den oted as A ⋄ B . The mathema tical e xpressi ons for c umu- lati ve fusion is descri bed belo w . Theor em 1 The Cumulativ e Fusion Operator Let ω A X and ω B X be opinions res pectivel y held by a gents A and B over the same fr ame X = { x i | i = 1 , · · · , k } . Let ω A ⋄ B X be the opinion suc h that: Case I: F or u A X 6 = 0 ∨ u B X 6 = 0 :        b A ⋄ B x i = b A x i u B X + b B x i u A X u A X + u B X − u A X u B X u A ⋄ B X = u A X u B X u A X + u B X − u A X u B X (7) Case II: F or u A X = 0 ∧ u B X = 0 :    b A ⋄ B x i = γ b A x i + (1 − γ ) b B x i u A ⋄ B X = 0 (8) wher e γ = lim u A X → 0 u B X → 0 u B X u A X + u B X Then ω A ⋄ B X is called the cumulatively fused opin- ion of ω A X and ω B X , r epr esenting the combina tion of independ ent opinions of A and B . B y u sing t he symbol ‘ ⊕ ’ to designate this belief opera tor , we define ω A ⋄ B X ≡ ω A X ⊕ ω B X . The cumulat iv e fusion operato r is equi val ent to a poster iori updati ng of D irichle t dist rib utions . Its proof and deri v ation is based on the bijecti ve mapping between multinomial opinions and and an augmente d rep resentat ion of the Dirichlet dis- trib ution [4]. It can be verified that the cumula tiv e fusion operat or is commutati ve, associati ve and non- idempote nt. In Case II of Theorem 1, the asso- ciati vity depend s on the preserv ation of rel ati ve weights of inte rmediate resu lts, which requires the additi onal weigh t var iable γ . In this case, the cumulati ve operato r is equi valen t to the weighted a vera ge of probab ilities. The cumulati ve fus ion o perator represent s a gen- eralisa tion of the consensus operat or [3, 2] which emer ges direct ly from Theor em 1 by assuming a binary frame. 3.2 A veraging Fusion Assume a frame X co ntaining k elemen ts. A s- sume two observ ers A and B who ha ve dependen t opinio ns over the frame X . This can for example result from o bservin g the outcomes of th e pr ocess ov er the same time perio ds. Let the two obse rvers ’ respecti ve opinions be exp ressed as ω A X = ( ~ b A X , u A X , ~ a A X ) an d ω B X = ( ~ b B X , u B X , ~ a B X ) . The av eraging fusion of these two bodies of ev- idence is denoted as ω A ⋄ B X = ω A X ⊕ ω B X . T he symbol “ ⋄ ” denotes the av eraging fusion of two observ ers A and B into a single imaginar y ob- serv er denoted as A ⋄ B . The mathematic al ex- pressi ons for a veraging fusion is describ ed belo w . Theor em 2 The A v eraging Fusion Operator Let ω A X and ω B X be opinions res pectivel y held by a gents A and B o ver the sa me frame X = { x i | i = 1 , · · · , k } . Let ω A ⋄ B X be the opinion suc h that: Case I: F or u A X 6 = 0 ∨ u B X 6 = 0 :        b A ⋄ B x i = b A x i u B X + b B x i u A X u A X + u B X u A ⋄ B X = 2 u A X u B X u A X + u B X (9) Case II: F or u A X = 0 ∧ u B X = 0 :    b A ⋄ B x i = γ b A x i + (1 − γ ) b B x i u A ⋄ B X = 0 (10) wher e γ = lim u A X → 0 u B X → 0 u B X u A X + u B X Then ω A ⋄ B X is called the avera ged op inion of ω A X and ω B X , r epr esenting the combin ation of the de- pende nt opinio ns of A a nd B . B y using the s ym- bol ‘ ⊕ ’ to designat e th is belief operato r , we define ω A ⋄ B X ≡ ω A X ⊕ ω B X . The a vera ging operator is equi v alent to av eraging the eviden ce of Dirichlet distrib ution s. Its proof deri vation is based on the bijecti ve m apping be- tween multinomial opinio ns and an augment ed repres entation of Dirichlet distrib utions [4]. It can be verified that the a verag ing fusion opera- tor is c ommutati ve and idempotent, b ut not asso- ciati ve. The a veraging fusion ope rator repres ents a gener - alisati on of the consensu s ope rator for dependent opinio ns defined in [7]. 4 Fission of Multinomial Opinions The principle of belief fission is the opposite to belief fusion. This section describe s the fi ssion operat ors correspon ding to the cumulati ve and av- eragin g fusio n ope rators described i n the pre vious sectio n. 4.1 Cumulati ve Fiss ion Assume a frame X containing k elements. As- sume two observ ers A and B who ha ve observe d the outcomes of a process ov er two separa te time period s. Assume that the observ ers beliefs hav e been cumulati vely fused into ω A ⋄ B X = ω C X = ( ~ b C X , u C X , ~ a C X ) , and assume that entity B ’ s con- trib uting opinion ω B X = ( ~ b B X , u B X , ~ a B X ) is kno wn. The cumulati ve fi ssion of these two bodies of ev - idence is denote d as ω C ⋄ B X = ω A X = ω C X ⊖ ω B X , which repres ents ent ity A ’ s contrib uting opi nion. The mathematical exp ression s f or cumulat iv e fis- sion is describ ed belo w . Theor em 3 The Cumulativ e Fission Operator Let ω C X = ω A ⋄ B X be t he cumu latively fused opin- ion of ω B X and the un known opinion ω A X ove r the fra me X = { x i | i = 1 , · · · , k } . Let ω A X = ω C ⋄ B X be the opini on such tha t: Case I: F or u C X 6 = 0 ∨ u B X 6 = 0 :        b A x i = b C ⋄ B x i = b C x i u B X − b B x i u C X u B X − u C X + u B X u C X u A X = u C ⋄ B X = u B X u C X u B X − u C X + u B X u C X (11) Case II: F or u C X = 0 ∧ u B X = 0 :    b A x i = b C ⋄ B x i = γ B b C x i − γ C b B x i u A X = u C ⋄ B X = 0 (12) wher e              γ B = lim u C X → 0 u B X → 0 u B X u B X − u C X + u B X u C X γ C = lim u C X → 0 u B X → 0 u C X u B X − u C X + u B X u C X Then ω C ⋄ B X is called the cumulati vely fissioned opinio n of ω C X and ω B X , r epr esenting th e r esult of eliminati ng the opinion s of B fr om that of C . By using the symbo l ‘ ⊖ ’ to designate this belief op- era tor , we define ω C ⋄ B X ≡ ω C X ⊖ ω B X . Cumulati ve fission is the in ver se of cumulati ve fusion . Its proof and deri v ation is based on re - arrang ing the mathematical e xpres sions of T heo- rem 1 It can be verified tha t the cumulati ve rule is no n-commutati ve, n on-asso ciati ve and no n- idempote nt. In C ase II of Theorem 3, th e fissio n rule is equiv alent to the weighted subtrac tion of probab ilities. 4.2 A veraging Fission Assume a frame X co ntaining k elements. As- sume two observ ers A and B who ha ve observe d the same outco mes of a process over the same time perio d. Assume tha t the observ ers beliefs ha ve been a vera gely fused into ω C X = ω A ⋄ B X = ( ~ b C X , u C X , ~ a C X ) , and assume that entity B ’ s con- trib uting opinion ω B X = ( ~ b B X , u B X , ~ a B X ) is kno wn. The av eraging fission of these two bodies of ev- idence is denoted as ω A X = ω C ⋄ B X = ω C X ⊖ ω B X , which represen ts ent ity A ’ s contrib uting opi nion. The mathematical express ions for a vera ging fis- sion is describ ed belo w . Theor em 4 The A v eraging Fission Operator Let ω C X = ω A ⋄ B X be the fused aver ag e opin ion of ω B X and the un known opi nion ω A X ove r the frame X = { x i | i = 1 , · · · , k } . Let ω A X = ω C ⋄ B X be the opinio n suc h that: Case I: F or u C X 6 = 0 ∨ u B X 6 = 0 :        b A x i = b C ⋄ B x i = 2 b C x i u B X − b B x i u C X 2 u B X − u C X u A X = u C ⋄ B X = u B X u C X 2 u B X − u C X (13) Case II: F or u C X = 0 ∧ u B X = 0 :      b A x i = b C ⋄ B x i = γ B b C x i − γ C b B x i u A X = u C ⋄ B X = 0 (14) wher e              γ B = lim u C X → 0 u B X → 0 2 u B X 2 u B X − u C X γ C = lim u C X → 0 u B X → 0 u C X 2 u B X − u C X Then ω C ⋄ B X is called the aver ag e fission ed o pinion of ω C X and ω B X , r epr esenting the r esult of eliminat- ing t he o pinion s of B fr om t hat of C . By using the symbol ‘ ⊖ ’ to designate this belief opera tor , we define ω C ⋄ B X ≡ ω C X ⊖ ω B X . A veragin g fission is the in verse of av eragin g fu- sion. Its proof and deri vati on is base d on rear- rangin g the mathematica l exp ressions of T heo- rem 2 It can be verified that the a veraging fission op- erator is idempotent, non-co mmutati ve and non- associ ati ve. 5 Examples 5.1 Simple Belief Fission Assume that A has an unkno w n opinion about x . Let B ’ s opinion and the cumulati vel y fused opin- ion between A ’ s and B ’ s opinions be know as : ω A ⋄ B x = (0 . 90 , 0 . 05 , 0 . 05 , 1 2 ) ω B x = (0 . 70 , 0 . 10 , 0 . 20 , 1 2 ) and respec tiv ely . Using the cu mulati ve fission op era- tor it is possible to deri ve A ’ s opinion . This situ- ation is illustr ated in Fig.3. Figure 3: Princip le of belief fission By insertin g the opinio ns v alues into Eq.(7) the contri bu ting opinion from A can be deri ved as ω A x = (0 . 93 , 0 . 03 , 0 . 06 , 1 2 ) 5.2 I n verse Reasoning in Bayesian Networks Bayesian belief networks represent m odels of condit ional relationshi ps be tween propositio ns of interes t. Subjecti ve logic provid es operators for condit ional deductio n [8] an d con ditional abduc - tion [9] which allows reasonin g to take place in either direction along a con ditiona l edge. Fig.4 sho w s a simple Bayesian belief network where x and y are parent e vidence nodes and z is the ch ild node. Figure 4: Bayesi an network with belief fusion In order to deriv e the deduce d opin ions ω z k x and ω z k y using the deductio n operator , the opin ions ω x and ω y , as well as the condit ional opinio ns ω z | x , ω z | x , ω z | y and ω z | y are needed. Assuming that the co ntrib utions of ω z k x and ω z k y are ind e- pende nt, they can be fused with the cumula ti ve fusion operator to produc e the deriv ed opinion ω z k ( x,y ) . Belief rev ision based on the fission operator can be useful in case a very certain opinion about z has been determine d from other sources , and it is in conflict w ith the opinion der iv ed through the Bayesian netwo rk. In that ca se, the reasoni ng can be applied in the in verse directio n using the fis- sion opera tor to revis e the opin ions about x and y or about the condit ional relatio nships z | x and z | y . Opinion ownersh ip in the form of a supersc ript to the opinions is not exp ressed in this exa mple. It can be assumed that the analyst deriv es input opinio n v alues as a function of e vidence c ollected from differe nt source s. The orig in of the opin- ions are therefore implicitly repres ented as the e v- idence source s in this model. 6 Conclusion The principle of belief fusion is used in numero us applic ations. T he opposite p rinciple of belief fis- sion is less commonly used. Howe ver , there are situati ons w here fi ssion can be useful. In this pa- per we hav e describ ed the fission operators cor - respon ding to cumulati ve and av eraging fusion in subjec ti ve logic. 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Sm ets and R. Kenn es. The tra nsfer - able belie f model. Artificial Intellig ence , 66:19 1–234, 1994. 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 p Probability Probability density Beta( | 1,1 ) p 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 Probability density Beta( | 8,2 ) p p Probability