toward a general theory of conditional beliefs

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Toward a General Theory of Conditional Beliefs Giulianella Coletti, 1,† Romano Scozzafava 2, * 1 Dipartimento Matematica e Informatica, University of Perugia, Italy 2 Dipartimento Metodi e Modelli Matematici, University “La Sapienza,” Roma, Italy We consider a class of general decomposable measures of uncertainty, which encompasses ~as its most specific elements, with respect to the properties of the rules of composition! probabil- ities, and ~as its most general elements! belief functions. The aim, using this general context, is to introduce ~in a direct way! the concept of conditional belief function as a conditional gener- alized decomposable measure w~{6{!, defined on a set of conditional events. Our main tool will be the following result, that we prove in the first part of the article and which is a sort of con- verse of a well-known result ~i.e., a belief function is a lower probability!: a coherent condi- tional lower probability t P~{6 K ! extending a coherent probability P~ H i ! —where the events H i s are a partition of the certain event V and K is the union of some ~ possibly all! of them—is a belief function. © 2006 Wiley Periodicals, Inc. 1. INTRODUCTION The starting point to introduce conditional belief functions is the consider- ation of a class of general decomposable measures of uncertainty, which encom- passes ~as its most specific elements, with respect to the properties of the rules of composition! probabilities, and ~as its most general elements! belief functions. Our main aim, using this general context, is to introduce ~in a direct way! the concept of conditional belief function as a generalized decomposable conditional measure, a suitable real function w~{6{!, defined on a set of conditional events. This requires, first of all, a deepening of this concept: We adopt a definition of conditional event E 6 H that differs from many seemingly “similar” ones appear- ing in the relevant literature since 1935, starting with de Finetti. 1 In fact, if we do not assign the same “third” value u ~“undetermined,” when the conditioning event is false! to all conditional events, but make it depend on the given E 6 H, it turns out ~as shown in Refs. 2 and 3; for a complete account of *Author to whom all correspondence should be addressed: e-mail: romscozz@dmmm. uniroma1.it. e-mail: [email protected]. INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 21, 229–259 ~2006! © 2006 Wiley Periodicals, Inc. Published online in Wiley InterScience ~www.interscience.wiley.com!. DOI 10.1002/ int.20133

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Page 1: Toward a general theory of conditional beliefs

Toward a General Theory ofConditional BeliefsGiulianella Coletti,1,† Romano Scozzafava2,*1Dipartimento Matematica e Informatica, University of Perugia, Italy2Dipartimento Metodi e Modelli Matematici, University “La Sapienza,” Roma,Italy

We consider a class of general decomposable measures of uncertainty, which encompasses ~asits most specific elements, with respect to the properties of the rules of composition! probabil-ities, and ~as its most general elements! belief functions. The aim, using this general context, isto introduce ~in a direct way! the concept of conditional belief function as a conditional gener-alized decomposable measure w~{6{!, defined on a set of conditional events. Our main tool willbe the following result, that we prove in the first part of the article and which is a sort of con-verse of a well-known result ~i.e., a belief function is a lower probability!: a coherent condi-tional lower probability tP~{6K ! extending a coherent probability P~Hi !—where the events Hi sare a partition of the certain event V and K is the union of some ~possibly all! of them—is abelief function. © 2006 Wiley Periodicals, Inc.

1. INTRODUCTION

The starting point to introduce conditional belief functions is the consider-ation of a class of general decomposable measures of uncertainty, which encom-passes ~as its most specific elements, with respect to the properties of the rules ofcomposition! probabilities, and ~as its most general elements! belief functions.

Our main aim, using this general context, is to introduce ~in a direct way! theconcept of conditional belief function as a generalized decomposable conditionalmeasure, a suitable real function w~{6{!, defined on a set of conditional events.

This requires, first of all, a deepening of this concept: We adopt a definitionof conditional event E6H that differs from many seemingly “similar” ones appear-ing in the relevant literature since 1935, starting with de Finetti.1

In fact, if we do not assign the same “third” value u ~“undetermined,” whenthe conditioning event is false! to all conditional events, but make it depend onthe given E6H, it turns out ~as shown in Refs. 2 and 3; for a complete account of

*Author to whom all correspondence should be addressed: e-mail: [email protected].

†e-mail: [email protected].

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 21, 229–259 ~2006!© 2006 Wiley Periodicals, Inc. Published online in Wiley InterScience~www.interscience.wiley.com!. • DOI 10.1002/int.20133

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the relevant theory, see Ref. 4! that this function t~E6H ! is a general conditionaluncertainty measure, and we get ~through an appropriate—in a sense “compul-sory” ~so to say!—choice of the relevant operations � and � between condi-tional events! the “natural” axioms for many different ~besides probability!conditional measures.

By resorting to this kind of procedure, we propose a general definition ofconditional belief functions, driven by the choice of a pair ~�,�! of locally asso-ciative and commutative operations, with � not necessarily increasing and �locally distributive with respect to �. So we need to single out the “right” choiceof � and �, able to distinguish belief functions inside the class of uncertaintymeasures.

This goal will be reached also by means of the following result that we givein Section 3 ~Theorem 2!. While it is well known ~since the pioneering work byDempster5 !, that the class of belief functions is a subclass of that of lower proba-bilities, we prove here a sort of converse. Precisely, by resorting to a general con-cept of conditional probability, whose peculiar aspects refer to coherence ~whichallows the assessment of conditional probability on an arbitrary family of condi-tional events! and to its interpretation as degree of belief, we are able to showthat—under weak conditions strictly related to Dempster’s framework—a coher-ent conditional lower probability tP~{6K ! extending a coherent probability P~Hi !—where the events Hi s are a partition of the certain event V and K is the union ofsome ~possibly all! of them—is a belief function.

Our ~general! framework does not require distinguishing available informa-tion—for example, evidence coming from statistical data—from any other poten-tial or assumed ~i.e., even if not yet observed! information: in this way, by suitablyexploiting the aforementioned “status” of all available information as being of thesame quality and nature, we can put on the same frame all the relevant events—looked on as propositions—of both the “evidential frame” and the “frame of dis-cernment.” It turns out that belief functions actually depend also on a suitableunderlying probabilistic conditioning.

For the main concepts concerning belief functions, we may refer to the alreadymentioned pioneering work by Dempster5 and to Shafer’s book.6

2. PRELIMINARIES

2.1. Events and Conditional Events

An event can be singled out by a ~nonambiguous! proposition E, that is astatement that can be either true or false ~corresponding to the two values 1 or 0 ofits indicator!. Because in general it is not known whether E is true or not, we areuncertain about E.

The probabilistic approach adopted here differs radically from the usual theorybased on a measure-theoretic framework, which assumes that a unique probabilitymeasure is defined on an algebra ~or s-algebra! of sets constituting the so-calledsample space V. Directing attention to events as subsets of the sample space ~and

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to algebras of sets! may be unsuitable for many real situations, which makes itinstead very significant both to give events a more general meaning and to assumethat there is no given specific structure for the family E where probability isassessed. Each event can be obviously represented as a set of points, neverthelessthis may lead to a too rigid framework.

In fact, even if any Boolean algebra is in a one-to-one correspondence ~byStone’s theorem! with an algebra of subsets of a given set V, the ensuing “anal-ogy” between events and sets is nothing more than an analogy: A set is actuallycomposed of elements ~or points!, and so its subdivision into subsets necessarilystops when the subdivision reaches its “constituent” points; conversely, events canbe allowed to go on in the subdivision by introducing suitable “new” events, sin-gled out by relevant propositions, not to mention that we need to consider alsoarbitrary families of events, not only Boolean algebras.

We will refer to the state of information ~at a given moment! of a real ~orfictitious! person that will be denoted by “You”: For example, the truth value of anevent is well determined in itself, but You are uncertain about it.

Notice that a proposition—which can be either true or false—need not to belooked on as an assertion: So, even if beliefs may come from various sources, theycan be treated in the same way, because the relevant events ~including possiblystatistical data!may always be considered as being assumed and not asserted prop-ositions ~cf. also our comments in the final part of the Introduction!.

To deal with conditional probability we need also to introduce conditionalevents, which are ~first of all! ordered pairs ~E, H !, with H � À, where À is theimpossible event ~we will adopt the usual notation E6H, and E6V—where V is thecertain event—reduces to the event E !. An interpretation of E6H in terms of abetting scheme may help in defining its truth values.

If an amount t—which should suitably depend on E6H—is paid to bet onE6H, we get, when H is true, either an amount 1 if also E is true ~the bet is won! oran amount 0 if E is false ~the bet is lost!, and we get back the amount t if H turnsout to be false ~the bet is called off !.

In short, introducing the truth value T ~E6H ! of a conditional event—recallthat, for an ~unconditional! event E, this is just its indicator IE � T ~E6V!, equaleither to 1 or to 0 according to whether E is, respectively, true ~the bet is won! orfalse ~the bet is lost!—we may write

T ~E6H ! � 1{IE∧H � 0{IE c∧H � t{IH c

Then, by elementary properties of indicators and introducing, for the “third”value of T ~E6H !, the symbol t~E6H ! in place of t, to point out its dependence onthe pair E, H ~in fact it depends on the ordered triple ~E ∧ H, E c ∧ H, H c !!, wehave

T ~E6H ! � 1{IE IH � 0{IE c IH � t~E6H !{IH c ~1!

In conclusion, a conditional event E6H ~or, better, its truth value! can be seen as adiscrete random quantity ~with range $ yk � R, k � 1,2, . . . ,n%!

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Y � (k�1

n

yk IEk

taking n� 3, E1 � E ∧ H, E2 � E c ∧ H, E3 � H c , and y1 �1, y2 � 0, y3 � t~E6H !.We have shown in Ref. 3 that, by introducing suitable ~partial! operations

among conditional events ~looked on as particular discrete random quantities!, thechoice of these operations determines the various conditional measures t~E6H !representing uncertainty.

2.2. From Conditional Events to Conditional Measures

The main aim of this section is to give an outline on how we will introduce~in a direct way! the concept of conditional belief function, in our general context,as a generalized decomposable conditional measure ~a suitable real function w~{6{!defined on a set of conditional events!.

Notation. From now on, to reduce cumbersome expressions, any conjunctionA ∧ B will be often denoted simply by AB.

We start from the definition of conditional event E6H recalled from Sec-tion 2.1. The class of conditional events is the set T of particular random quanti-ties ~1!; so, given two commutative, associative, and increasing operations � and �from R

� � R� to R

�, with R� � $x � R : x � 0% , we suitably define correspond-

ing operations among the random quantities of T: The “result” is a random quan-tity, but it does not, in general, belong to T. In fact, if � is any operation fromR � R to R, and

X � (h�1

n1

xh IEhY � (

k�1

n2

yk IFk

are two discrete random quantities in canonical form, define X � Y as the randomquantity

Z � (j�1

n

zj IGj

where each event Gj is of the form Gj � Eh Fk and zj � xh � yk . Therefore anoperation between random quantities can be defined by means of a relevant oper-ation between real numbers.

Example 1. Consider � and �, two commutative, associative, and increasingoperations from R

� � R� to R

�, having 0 and 1, respectively, as neutral element,and use the same symbols for the relevant operations on T � T. If we operatebetween two elements of T, in general we do not obtain an element of T. We havein fact, by Equation 1, for any pair of conditional events E6H, A6K ~to improvereadability of the following two formulas, for any event E we put IE � E !:

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T ~E6H ! � T ~A6K ! � @1 � 1#EHAK � @1 � 0#EHAcK � @1 � 0#E cHAK

� @0 � 0#E cHAcK � @1 � t~A6K !#EHK c

� @0 � t~A6K !#E cHK c� @1 � t~E6H !#AKH c

� @0 � t~E6H !#AcKH c � @t~E6H ! � t~A6K !#H cK c

and

T ~E6H ! � T ~A6K ! � @1 � 1#EHAK � @1 � 0#EHAcK � @1 � 0#E cHAK

� @0 � 0#E cHAcK � @1 � t~A6K !#EHK c

� @0 � t~A6K !#E cHK c� @1 � t~E6H !#AKH c

� @1 � t~E6H !#AcKH c � @t~E6H ! � t~A6K !#H cK c

As is easily seen, both right-hand sides of the latter two expressions are not of thekind ~1!, that is, there does not exist a conditional event A6B such that they can bewritten ~using the simplified notation! as

1{AB � 0{AcB � t~A6B!{Bc

On the other hand, if we operate only with those elements of T � T such thatthe range of each operation is T, we get “automatically” ~so to say!—as shown inRef. 3—conditions on t~E6H ! that can be regarded as the “natural” axioms for aconditional measure w defined on C � E � H0 , with H0 � H �$À% , that is,

~C1! w~E6H !� w~E ∧ H 6H !, for every E � E and H � H0.~C2! For any given H � H0 and for any E, A � E, with A ∧ E ∧ H � À, we have

w~~E ∨ A!6H ! � w~E6H ! � w~A6H ! w~V6H !� 1 w~À6H !� 0

~C3! For every A � E and E, H, E ∧ H � H0,

w~~E ∧ A!6H ! � w~E6H ! � w~A6~E ∧ H !!

These properties hold also for arbitrary sets E and H, but a sensible defini-tion of conditional measure requires putting “natural” algebraic structures on thesesets.

Definition 1. Let w be a real function defined on C � E � H0, with E a Booleanalgebra, H � E an additive set (i.e., closed with respect to finite logical sums),and H0 � H �$À%, and denote by w~C ! the range of w. Then the function w is a~�,�!-decomposable conditional measure if there exist two commutative, asso-ciative, and increasing operations �,� from w~C !� w~C ! to R

�, having, respec-tively, 0 and 1 as neutral elements, and with � distributive over �, such that (C1),(C2), (C3) hold.

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Then different ~decomposable! conditional measures can be obtained by par-ticular choices of the two operations � and �. For example, choosing ordinarysum and product ~see Ref. 2!, or max and min ~see Ref. 7!, we get, respectively,conditional probability or conditional possibility.

Let us recall the case of conditional probability: We proved in Ref. 2 that thefunction t~{6{! satisfies “familiar” rules, that coincide with the usual axioms ~asgiven by de Finetti8 !, and can be expressed as follows, putting t~{6{!� P~{6{!:

~i! P~~E ∧ H !6H !� P~E6H !, for every E � E and H � H0.~ii! P~{6H ! is a ~finitely additive! probability on E for any given H � H0.~iii! P~~E ∧ A!6H !� P~E6H !P~A6~E ∧ H !!, for A � E and E, H � H0, E ∧ H � À.

The function P~{6{! is called a conditional probability on E � H0 .Axioms ~i! and ~iii! can be replaced by

~i' ! P~H 6H !� 1, for every H � H0.~iii' ! P~A6C!� P~A6B!P~B 6C! if A � B � C.

2.3. Toward Conditional Beliefs as Generalized Conditional Measures

In the second part of the article, to introduce conditional beliefs we need amore general class of conditional measures. So we now give the definition of thegeneralized ~�,�!-decomposable conditional measure, which extends ~not onlyDefinition 1 above, but also! that of the weakly ~�,�!-decomposable conditionalmeasure given in Ref. 3, in the sense that here we do not require the operation � tobe increasing.

Definition 2. Given a family C � E � H0 of conditional events, where E is aBoolean algebra, H an additive set, with H � E and H0 � H �$À%, a real functionw defined on C is a generalized ~�,�!-decomposable conditional measure if

~g1) w~E6H !� w~E ∧ H 6H ! , for every E � E and H � H0.~g2) For any given H � H0, it is w~V6H ! � 1, w~À6H ! � 0, and w~{6H ! is a capacity;

there exists an operation � : $w~C !%2 r w~C ! whose restriction to the set

D � $~w~Ei 6H !,w~Ej 6H !! : Ei , Ej � E, H � H0, Ei ∧ Ej ∧ H � À%

is such that

w~Ei ∨ Ej 6H ! � w~Ei 6H ! � w~Ej 6H !

for every Ei , Ej � E, with Ei ∧ Ej � À.~g3) There exists an operation � : $w~C !%2 r w~C ! whose restriction to the set

G � $~w~E6H !,w~A6~E ∧ H !!! : A � E, E, H, E ∧ H � H0 %

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is increasing, admits 1 as a neutral element, and is such that, for every A � E and E,H � H0, E ∧ H � À,

w~~E ∧ A!6H ! � w~E6H ! � w~A6~E ∧ H !!

~g4) The operation � is distributive over � only for relations of the kind

w~H 6K ! � ~w~E6~H ∧ K !! � w~F 6~H ∧ K !!!

with K, H ∧ K � H0, E ∧ F ∧ H ∧ K � À.

Remark 1. It is easily seen that, with respect to the elements of D and G, theoperations � and � are commutative and associative.

2.4. Coherent Conditional Probability

Going back to probability, we need to cope with the problem of partial assess-ments, because it is not always appropriate to give every event of all possibleenvisaged situations a probability. So the question is: Is it possible to assess P onan arbitrary set C of conditional events?

We say that the assessment P~{6{! on C is coherent if there exists C ' � C, withC ' � E � H0 ~E a Boolean algebra, H an additive set!, such that P~{6{! can beextended from C to C ' as a conditional probability. Coherent conditional probabil-ities play a fundamental role for the first main result of this article, that is Theo-rem 2 in Section 3.

Remark 2. All concepts concerning conditional probabilities can be obviouslyreferred also to unconditional ones, just taking equal to the certain event V allconditioning events of the given family of conditional events.

Remark 3. A characterization of coherence for conditional probability has beendiscussed in many previous papers, starting with Ref. 9. For the sake of brevity,we do not report here the relevant theorem: See the formulation given in Ref. 10 orin Ref. 4 ~as Theorem 4, p. 81!. It essentially shows the complexity and subtletiesof the concept of coherent conditional probability. In fact, whereas in the classicapproach it is expected ~by definition! to represent a conditional probability byjust one unconditional probability, in our context it is instead necessary—to rep-resent P~E6H !—a suitable Pa depending on H, so getting a class $Pa% of proba-bilities ~agreeing with the given conditional probability!.

A fundamental result concerning coherent conditional probabilities is the fol-lowing, essentially due ~for unconditional events, i.e., the particular case in whichall conditioning events coincide with V! to de Finetti.8

Theorem 1. Let C be a family of conditional events and P a corresponding assess-ment; then there exists a (possibly not unique) coherent extension of P to an arbi-trary family K � C, if and only if P is coherent on C.

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This theorem can be seen as a ~potent! generalization of the well-known theo-rem ~see Horn and Tarski11 ! that every finitely additive probability on a subalge-bra of a Boolean algebra can be extended to the whole algebra. Notice that itcorresponds to a twofold generalization, that is, not only from events to condi-tional events, but also replacing Boolean algebras by arbitrary families.

We need to consider also coherent lower probabilities ~conditional or not!:Given an arbitrary set C � E � H of conditional events ~with À � H!, a coherentlower conditional probability on C is a nonnegative function tP such that thereexists a nonempty dominating family P � $P~{6{!% of coherent conditional proba-bilities on C whose lower envelope is tP, that is, for every E6H � C,

tP~E6H ! � infP

P~E6H !

In particular, by taking H � $V% , we get a coherent lower probability on E.

2.5. Belief Functions

It is well known that the class of belief functions is a subclass of that of lowerprobabilities. In general, a lower probability is not a belief function.

Example 2. Given a partition $A, B, C, D% of V, consider two probabilityassessments

P1~A! � 0 P1~B!� P1~C!�1

2P1~D!� 0

P2~A! �1

4P2~B!� 0 P2~C!�

1

4P2~D!�

1

2

on the algebra generated by these four events. The lower probability obtained aslower bound of the class $P1, P2 % has, in particular, the following values:

tP~A! � 0 tP~A ∨ B!�1

4tP~A ∨ C!�

1

2tP~A ∨ B ∨ C!�

1

2

so that

1

2� tP~A ∨ B ∨ C! � tP~A ∨ C!� tP~A ∨ B!� tP~A!�

1

2�

1

4� 0

So a lower probability may not be 2-monotone, that is, it may not satisfy

tP~A ∨ B! � tP~A!� tP~B!� tP~A ∧ B!

Obviously, all the more reason a lower probability is not necessarily an n-monotonefunction for any n � IN.

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A way of introducing belief functions is, in fact, through the last conceptmentioned in the previous example.

Definition 3. A belief function on an algebra E is, for any n � IN, an n-monotonelower probability, that is,

tP~A1 ∨ {{{ ∨ An ! � (i�1

n

tP~Ai !�(i�j

tP~Ai ∧ Aj !

� {{{� ~�1!k�1( tP~Ai1 ∧ {{{ ∧ Ain !

� {{{� ~�1!n�1 tP~A1 ∧ {{{ ∧ An !

We recall now briefly how belief functions were introduced by Dempster.Consider a four-tuple D � $S, T,G,m% , where S and T are two different sets ofatoms ~i.e., two different finite partitions of V! and to each element of s � S therecorresponds an element G~s! belonging to the algebra A generated by the ele-ments of T ; moreover, m is a probability distribution on S such that m~S0 ! � 0,where S0 is the set of regular points s � S, that is, those such that G~s!�À ~whereasan element s � S is called singular if G~s!� À!.

The regularization of m is defined, for s � S0, as

m0~s! �m~s!

(s�S0

m~s!

Starting from D, a basic probability assignment is a function m : Ar @0,1# :

m~A! � �m$s : G~s!� A%, if A � À

0, if A � À

This function does not share the usual properties of a measure: It satisfies only, forany A � A,

m~A! � 0 and (A�A

m~A!� 1

A belief function can be defined, for E � A, as

Bel~E ! � (B�E

m~B!

and this function turns out to be n-monotone for every n � IN and satisfying Bel~À!�0 and Bel~V! � 1. In particular, the function Bel reduces to a probability if andonly if the function m is different from 0 only on the atoms of A ~i.e., on the atomsof T !.

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3. LOWER COHERENT CONDITIONAL PROBABILITIESAND BELIEF FUNCTIONS

Now we are going to show that a coherent conditional lower probability tP~{6K !extending a coherent probability P~Hi !—where the events Hi s are a partition ofthe certain event V and K is the union of some ~possibly all! of them—is a belieffunction.

This result is, in fact, not surprising, because this has been shown for partic-ular classic examples: see, for example, Section 18.9 of Ref. 4 ~containing a dis-cussion of Shafer’s well-known example of Fred and slippery streets; this exampleis reported below, at the beginning of Section 3.1, as Example 3!, and Ref. 12,which was challenging the inevitability claimed by Smets13 to resort to belief func-tions for dealing with the so-called “Mr. Jones’ murder case.”

Let us recall the latter example: Big Boss has decided that Mr. Jones has to bemurdered by one of the three persons present in his waiting room and whose namesare Peter, Paul, and Mary. Big Boss has decided that the killer on duty will beselected according to the result of a dice-tossing experiment: If the result is even,the killer will be a female; if the result is odd, the killer will be a male. We knowwho was in the waiting room and the story of the dice-tossing experiment, but wedo not know what its result was and who was selected. We also do not know howBig Boss would have decided between Peter and Paul if the result given by the dicehad been odd. Given the available information, the probability that the killer was amale or a female is equal, that is, 1/2. Then we learn that if Peter was not the killer,he would go to the police station at the time of the killing in order to get a perfectalibi. Peter went indeed to the police station, so he is not the killer. The problem isto assess now the belief that the killer is a female or a male.

This example is just a version of a classic one, expressed in various similarforms in the relevant literature ~the three prisoners, the two boys in a family withtwo children one of which is a boy, the car and the goats, the puzzle of the twoaces, etc.!. We will put forward the following ~“isomorphic”! abstract version ~oneblack and two white balls!. Three balls are given: Two of them are white and dis-tinguishable ~marked 1 and 2!, the third one is black. One out of the three corre-sponding events B,W1,W2 is the possible outcome of the following experiment: Areferee tosses a dice and puts in a box the black ball or the two white ones, accord-ing to whether the result is “even” ~event E ! or “odd” ~event O!; in the formercase the final outcome of the experiment is B, whereas in the latter the refereechooses (as the final outcome of the experiment) one of the two white balls ~andwe do not know how the choice is done!. Then we learn that, if W1 was not thefinal outcome, “the referee shows 1 as one of the two remaining balls” ~denote byK the event expressed by this statement!. Actually, the referee shows indeed thatone of the two remaining balls is 1: What is the probability ~or the belief ! that Bwas the final outcome of the experiment?

Clearly, there are the following correspondences with Smets’ formulation:The three final outcomes correspond to the three possible killers, that is,

W1 � Peter W2 � Paul B � Mary

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Moreover “ball 1 is one of the two remaining balls” corresponds to “Peter was notthe killer,” and the event K � “the referee shows ball 1 as one of the two remainingballs” corresponds to “Peter goes to the police station at the time of the killing inorder to get a perfect alibi.”

For a complete discussion, we refer to the aforementioned paper; here wesimply claim that Bel~B!� 1

2_ can be expressed ~taking into account that we do not

know how the choice between the balls 1 and 2 is made when the toss of the dicegives O! as the lower conditional probability tP~B 6K !, and we show, in turn, thatthe latter lower probability can be expressed as

tP~B 6K ! � (Hi�B∧K

P~Hi 6K !

where

H1 � K ∧ E H2 � K ∧ O

In fact, because H1 � B ∧ K, whereas H2 � B ∧ K, we get easily

tP~B 6K ! � P~H16K !� P~E6K !�1

2

What it is important to stress is that ~besides holding in this specific example! thelatter result is true in general, as expressed in Theorem 2 below ~in particular, cf.formula ~*! in the proof !.

First of all, we recall an easy consequence of the characterization theoremmentioned in Remark 3: It shows that a coherent assignment of P~{6{! to a familyof conditional events whose conditioning ones are a partition of V is essentiallyunbound ~cf. Ref. 4, Theorem 5!.

Lemma 1. Let C be a family of conditional events $Ei 6Hi %i�I , with card~I ! arbi-trary and events Hi s a partition of V. Then any function f : Cr @0,1# such that

f ~Ei 6Hi ! � 0 if Ei ∧ Hi � À and f ~Ei 6Hi !� 1 if Hi � Ei

is a coherent conditional probability.

We give now the main result of this section.

Theorem 2. Let

H � $H1, . . . , Hn %

be a finite set of pairwise incompatible events. Denoting by K the additive setspanned by them, and given an algebra A of events, put C � A � K.

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If P~{! is a coherent probability on H, let P be the class of coherent condi-tional probabilities P~{6{! extending P~{! on C. Consider, for E6K � C, the lowerprobability

tP~E6K ! � infP

P~E6K ! ~2!

Then for any K � K the function tP~{6K ! is a belief function on A.

Proof. Since P is coherent, Theorem 1 warrants the existence of a coherent exten-sion to A � K, and so, by the same theorem, also to F � K, where F is the algebraspanned by A � K.

Given E6K � A � K, the axioms of conditional probability ~that are neces-sary conditions for coherence! give easily

P~E6K ! � (Hi�K

P~E6Hi !P~Hi 6K !

By the previous Lemma, any function f : C r @0,1#—with the only conditionf ~Ei 6Hi !� 0 if Ei Hi � À, and f ~Ei 6Hi !� 1 if Hi � Ei —is a coherent conditionalprobability; then the P~E6Hi !s such that E and Hi do not satisfy the above logicalrelations can take any value in @0,1# .

Then, because all P~E6Hi !s such that Hi � E may assume the value zero, itfollows, for the lower probability obtained from the class P of conditional proba-bilities extending P~{6{!, that

tP~E6K ! � (Hi�EK

P~Hi 6K ! ~*!

We prove now that tP~{6K ! is, for any given K � K, that is

K � ∨i

Hi

a belief function Bel on A, that is, that there exists a function mK : Ar @0,1# , withmK ~À!� 0 and (E�A mK ~E !� 1, such that

tP~E6K ! � (F�E

mK ~F!� BelK ~E !

Take, in fact, for any F � A,

mK ~F! � (Hi�FK

Hi�G�F

P~Hi 6K !

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where G is any event of A implying F: We show that mK is a basic probabilityassignment on A restricted to K. We have

mK ~À! � (Hi�À

P~Hi 6K !� tP~À6K !� 0

(E�A

mK ~E ! � (E�A(

Hi�EK

P~Hi 6K !� (Hi�K

P~Hi 6K !� tP~V6K !� 1

and

(F�E

mK ~F! � (F�E(

Hi�FKHi�G�F

P~Hi 6K !� (Hi�EK

P~Hi 6K ! �

Corollary 1. Under the same conditions of the previous theorem, let H be, inparticular, a partition of V, and take K �V. Then the lower probability

tP~E ! � infP

P~E !

is a belief function on A.

3.1. Interpretation and Remarks

In the framework of Theorem 2, let us examine the connections with Demp-ster’s representation of a belief function, which is ~as we show below! a particularcase with the conditioning event K equal to the union of all Hi s.

In our context the two spaces S and T essentially merge into only one familyof events: Moreover in this scenario there are other events, like those of the alge-bra A ~which corresponds to the algebra generated by the elements of T !, and thepartition $Hi % can be related to the function G ~which introduces, under a suitableconditioning, logical relations among the relevant events!.

To clarify these claims, let us first discuss some ~well-known! examples.

Example 3. In the aforementioned classic example of Fred, take

s1 � truthful s2 � careless

Fred announces that E is true, and then we have

G~s1! � E � slippery streets G~s2 !�V

Because Fred’s announcements are truthful reports on what he knows 80% of thetime and are careless statements the other 20% of the time, Shafer actually assumes

P~s1! � 0.8 P~s2 !� 0.2 ~3!

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The set T is $E, E c % , whereas m is the P given by Equation 3; it follows thatm~À!� 0, m~E !� m~s1!� 0.8, m~E c !� m~À!� 0, m~V!� m~s2 !� 0.2, so thatthe usual belief function argument recalled above gives

Bel~E ! � 0.8 Bel~E c !� 0 ~4!

On the other hand, in our context we start from an explicit formulation of theevent

K � Fred announces that E is true

turning our attention to the conditional events E6K and E c 6K. The family H ofpairwise incompatible events is

Hj � K ∧ $sj % ~ j � 1,2! ~5!

so that the union H1 ∨ H2 of all of them is K. All these conjunctions are possible,and this corresponds ~in Dempster’s framework! to G~sj !� À for j � 1,2.

Now we need to find the logical relations among Hi � K ∧ $si % and the eventsF ∧ K, for any F � $E, E c,V,À% . Because E may be true also in the case that Fred’sannouncement is a careless statement, we have H1 � EK and H2 � EK, whereasHi � E cK ~for i � 1,2!.

So the function G is nothing else that an “external tool” to single out explic-itly ~assuming K true! these relations, pointing out which is the “minimum” ~withrespect to �! event of the algebra A implied by $si % .

The belief values in Equation 4 can be seen, referring to the conditional eventsE6K and E c 6K, as lower and upper conditional probability assessments; in fact, bya simple application of formula ~*!—see the proof of Theorem 2—we have

tP~E6K ! � (Hi�EK

P~Hi 6K !� P~H16K !� P~$s1%6K !� 0.8

whereas

tP~À6K ! � tP~E c 6K !� 0, tP~V6K !� 1

Example 4. A digitalized map shows areas of land and water, with 0.8 of the areaof the map being visible; the visible area is subdivided ~with proportions 0.4 and0.6! into land area and water area. Taking

s1 � visible land s2 � visible water s3 � invisible part

we have

G~s1! � E � land G~s2 !� E c � water G~s3 !�V

with m~E !� m~s1!� 0.32, m~E c !� m~s2 !� 0.48, m~V!� m~s3 !� 0.2, so that

Bel~E ! � 0.32 Bel~E c !� 0.48 ~6!

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On the other hand, in our context we turn our attention, assuming

K � visible part is 0.8 of the area

to the conditional events E6K and E c 6K. The family H of pairwise incompatibleevents is

Hj � K ∧ $sj % ~ j � 1,2,3!

so that the union H1 ∨ H2 ∨ H3 of all of them is K.Moreover, because E and E c may be true also in the invisible part of the map,

we have, again by ~*! and taking into account that H1 � EK and Hi � EK ~fori � 2,3!, whereas Hi � E cK ~for i � 1,3! and H2 � E cK,

tP~E6K ! � P~H16K !� 0.32 tP~E c 6K !� P~H2 6K !� 0.48 ~7!

Example 5. There are symptoms s1, s2, s3, s4 ~set S ! related to diseases E1, E2,E3, E4 ~set T ! in the following way: Symptom s1 may come only from disease E1,symptom s2 may come from either E1 or E2 ~but we are not able to distinguishbetween them!, s3 disconfirms E1, and s4 tells nothing. These logical relations canbe expressed, in terms of G, as

G~s1! � E1 G~s2 !� $E1 ∨ E2 % G~s3 !� $E2 ∨ E3 ∨ E4 % G~s4 !� T

Suppose now that we have on S the probability distribution

m~s1! � 0.42 m~s2 !� 0.18 m~s3 !� 0.28 m~s4 !� 0.12

so that the corresponding basic probability assignment on the algebra A allows usto compute, for example,

Bel~E1! � 0.42 Bel~E2 !� Bel~E4 !� 0

Bel~E1 ∨ E2 !� 0.42 � 0.18 � 0.60 Bel~E1 ∨ E3 !� 0.42

Bel~E2 ∨ E3 ∨ E4 !� 0.28, . . .

and so on.Arguing as in the previous examples, we consider now the event

K � $s1, s2, s3, s4 are the only symptoms taken into account%

and the family H

Hj � K ∧ $sj % ~ j � 1,2,3,4!

So we have, for example

tP~E16K ! � P~H16K !� P~$s1%6K !� 0.42

because H1 � E1 K, whereas Hi � E1 K ~for i � 2,3,4!.

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Moreover, because Hi � Ej K ~for i � 1,2,3,4 and j � 1,2!, we have

tP~E2 6K ! � tP~E4 6K !� 0

Concerning E1 ∨ E2, we have

Hi � ~E1 ∨ E2 !K ~for i � 1,2! Hi � ~E1 ∨ E2 !K ~for i � 3,4!

so that

tP~~E1 ∨ E2 !6K ! � P~H16K !� P~H2 6K !� 0.42 � 0.18 � 0.60

and so on.

Remark 4. Summing up: In general, we single out explicitly an event K such thatfor any s � S the conjunction K ∧ $s% implies G~s!� B and not any other event Fimplied by B. Precisely,

• K ∧ $s% � G~s!� B � A• K ∧ $s% � F for any F � A, with F � B

The events Hs � K ∧ $s%� À are those mentioned in Theorem 2 ~the union of all ofthem is K !. The events Hi s are, in the previous examples, not only mutually exclu-sive, but also exhaustive, but in Theorem 2 this assumption is not needed, so a“dynamical” approach is possible in our framework ~a sort of “open world”!, asshown in the following example.

Example 6 (going on with Example 5). Suppose we take now into account a newsymptom s5 that is incompatible with each of the given diseases ~i.e., G~s5 !� À!.The circumstance that the events Hi s are not ~necessarily! exhaustive makes itpossible to consider successively “new” events—incompatible with the former—giving them either a positive probability, if the sum of their probabilities is lessthan 1, or equal to zero otherwise. In other words, there is no need to take

(i�1

n

P~Hi ! � 1

but rather we can take

(i�1

n

P~Hi 6K ! � 1

Then the probabilities of all symptoms, including the new one, must satisfy

(i�1

n�1

P~Hi 6K ' ! � 1

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with K '� $s1, s2, s3, s4, s5 are the symptoms taken into account%, and they can takeany value ~possibly zero!.

We recall that giving zero probability to an event H does not mean that H isimpossible or that we should give zero probability—violating coherence—to con-ditional events such as Ei 6H.

Remark 5. In many approaches to belief functions, a situation like that justdescribed in the latter example ~concerning a symptom H that is possibly incom-patible with all the considered diseases! takes the “logical” form G~H !� À, so thatsometimes it is suggested ~in the current literature! to give positive belief to theimpossible event. This—unnatural ~and incoherent!—choice is not necessary ~andnot allowed! in our framework ~in the previous example we have $s5 % � K c !,where a conditioning event does not necessarily reduce to a given fact ~of course,it can also be taken—provisionally, in a given circumstance—as “asserted”: seealso our comments in the second part of the Introduction!. Anyway, conditioningwith respect to the event K corresponds precisely to the concept of regularization.

3.2. Combining Evidence

When new information is taken into account ~by means of another event K '

different from K !, we can obviously interpret in the same way the lower probabil-ity tP~E6K ' !, that is, we get BelK ' ~E !.

Then, to combine the two pieces of information we need to extend the lowerprobability assessment on $E6K, E6K ' % to the new conditional event E6~KK ' !, show-ing that tP~E6~KK ' !! is a belief function—to be denoted as BelKK ' ~E !.

Let the new information be represented by a further partition $Hj' % of V, and

let K ' be an event of the relevant algebra ~possibly, K ' � V!. We assign a proba-bility to the events Hj

' , and, of course, there are also logical relations among themand the events of the algebra A. This gives rise ~proceeding along the same linesdiscussed at length above! to a new BelK ' ~{! in A.

So the problem now is how to merge the two beliefs BelK ~{! and BelK ' ~{! toobtain a belief function to be called BelKK ' ~{!.

First of all, notice that when KK '� À it is useless to consider the joint parti-tion $Hi Hj

' % . Conversely, when KK '� À, the aforementioned logical relations rel-ative to the two partitions $Hi % and $Hj

' % induce logical relations for the partition$Hi Hj

' % . Then, going on as in the proof of Theorem 2, we get, for E � A,

BelKK ' ~E ! � tP~E6KK ' !� infP

P~E6KK ' !� (Hi Hj

'�EKK 'P~Hi Hj

' 6KK ' ! ~8!

In particular, we recall that we put ~in the proof of Theorem 2!, for any F � A,

mK ~F! � (Hi�FK

Hi�G�F

P~Hi 6K !

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where G is any event of A implying F, and that we have shown that mK is a basicprobability assignment on A restricted to K. So the right-hand side of Equation 8can be written in the form

(FG�E

mKK ' ~FG!

In conclusion, the problem reduces to the assessments of the conditional prob-abilities P~Hi Hj

' 6KK ' ! or—simply ~if K � K '�V!—of P~Hi Hj' !. Then the gen-

eral formula expressing a belief function that combines two pieces of evidence is

BelKK ' ~E ! � (Hi Hj

'�EKK 'P~Hi 6KK ' !P~Hj

' 6Hi KK ' !

And what about Dempster’s rule? This can be obtained as a particular caseby suitable assumptions:

• The right-hand side of P~Hi Hj' ! � P~Hi !P~Hj

' 6Hi ! reduces, for every i and j, to theproduct of the two unconditional probabilities, that is, the two partitions are stochasti-cally independent ~which implies—in our framework, see Ref. 14—that they are alsologically independent, so that all events Hi Hj

' are possible!.• P~KK ' ! � 0, so that, from the previous equality

BelKK ' ~E ! � (Hi Hj

'�EKK 'P~Hi Hj

' 6KK ' !

taking into account that

P~KK ' ! � (Hi Hj

'�KK 'P~Hi Hj

' !� (Hi Hj

'�KK 'P~Hi !P~Hj

' !

we get the following expression:

BelKK ' ~E ! �

(Hi Hj

'�EKK 'P~Hi !P~Hj

' !

(Hi Hj

'�KK 'P~Hi !P~Hj

' !�

(FG�E

mK ~F!mK ' ~G!

(FG�
mK ~F!mK ' ~G!

which is Dempster’s rule.

4. CONDITIONAL BELIEFS

First of all, we need to recall a definition of inclusion between conditionalevents ~see Ref. 4, p. 68!:

A6H �� B 6K? T ~A6H !� T ~B 6K ! ~9!

where T is the truth value defined by Equation 1 and the inequality refers to thenumerical values corresponding to every atom generated by $A, B, H, K % , that is,to every element of the partition obtained as the intersection of the two partitions

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$AH, AcH, H c % and $BK, BcK, K c % . As shown in Ref. 4, this definition is equiva-lent, when the “truth value” t of T is a conditional probability, to that given inRef. 15, that is,

A6H �� B 6K ? AH � BK BcK � AcH ~10!

Consider now a set C � G � B of conditional events A6B, with G a Booleanalgebra and B � G closed with respect to ~finite! logical sums ~with À � B!. Weare going to introduce on C a conditional belief through a procedure similar to thatfollowed in Section 3, where we proved that a belief function Bel can be obtainedas a lower probability by starting from a class P of probabilities extending anassessment P0 given on a partition of an event K ~possibly, K �V!: Here we needto consider a class P* “locally” extending P0, in the sense of the next definition.

To simplify notation ~and without lack of generality! we refer to the caseK �V. Recall that, given a partition H of V, a probability P0 : Hr @0,1# , a finitealgebra A, and the class P of probabilities P on A extending P0, we have

Bel~A! � infP

P~A!� P0� ∨Hi�A

Hi�and so, similarly, for a plausibility,

Pl~A! � supP

P~A!� P0� ∨Hi A�À

Hi�� 1 � Bel~Ac !

Definition 4. Given the set C � G � B of conditional events and a partition Hof V, denote by G ' the algebra spanned by G and H and by B ' the additive setspanned by B and H. If P0 : H r @0,1# is a probability on H, for every B � Bdenote by PB

' the class of probabilities P ' on H such that

• Bel~B!� P '~∨Hi�B Hi !� P0~∨Hi�B Hi !� infP P~B!• Pl~B!� P '~∨Hi B�À Hi !� P0~∨Hi B�À Hi !� supP P~B!• P '~Hi !� P0~Hi ! for Hi B � À

for any P ' � PB' . We call belief generating class the set P* of all conditional

probabilities on G' � B ' extending every P � P '� �B�B PB' .

Theorem 3. Let H � $H1, . . . , Hn % be a partition of V, C � G � B the set ofconditional events, P0 a probability on H, and P* the relevant belief generatingclass. Putting, for every A6B � C,

Bel~A6B! � infP�P*

P~A6B!

one has

Bel~A6B! � �1, if AB � B

P� ∨Hi�AB

Hi� ∨Hi B�À

Hi� , otherwise ~11!

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Proof. Notice that, to simplify notation, we are using the same symbol P forall probabilities, including the relevant extensions. First of all, if AB � B, for anyP � P* we have, because P~B 6B!� 1,

Bel~B 6B! � 1

Consider now the case AB � B: One has

P~A6B! �(i

P~A6BHi !P~Hi 6B!� (BHi�A

P~Hi 6B!� (BHi�A

P~BHi 6B!

On the other hand, by Equation 10,

∨BHi�A

BHi�B �� ∨Hi�AB

BHi�B �� ∨Hi�AB

Hi� ∨Hi B�À

Hi

and, because A6B �� C 6D implies P~A6B!� P~C 6D!, it follows that

P~A6B! � P� ∨Hi�AB

Hi� ∨Hi B�À

Hi�The right member of the previous inequality is precisely infP�P* P~A6B!, because

P~A6B! � P� ∨BHi�A

BHi �B�� P�� ∨Hi�AB

Hi� ∨ � ∨BHi�ABHi�Hi

BHi �B��� P�� ∨

Hi�AB

Hi� ∨ � ∨BHi�ABHi�Hi

BHi��� ∨Hi�B

Hi� ∨ � ∨BHi�Hi

BHi��Then the minimum of P is obtained by giving value zero to

P� ∨BHi�ABHi�Hi

BHi �B�and the whole probability of Hi , for Hi � A, to Hi BAc .

Therefore

Bel~A6B! � P� ∨Hi�AB

Hi� ∨Hi B�À

Hi� �

We derive now some properties of such function Bel~{6{!, which will turn outto be useful for looking at a belief function as a generalized decomposable uncer-tainty measure ~Section 2.3, Definition 2!.

Proposition 1. For a function Bel given by Equation 11, the following proper-ties hold:

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• Bel~E6H !� Bel~EH 6H !.• Bel~{6H ! is a belief on G for any H � B.• Given the events E � H � K, with E � G, H, K � B, then from Bel~E6H !� 0 it follows

Bel~E6K !� 0.• Suppose that Bel~E6H !{Bel~H 6K ! � 0. Then

Bel~E6H !Bel~H 6K ! � Bel~E6K !Bel~H 6H !

with

Bel~H 6H ! � P� ∨Hi�H

Hi� ∨Hi H�À

Hi�Proof. The first property is obvious, because it refers to the same conditionalevent, since P~EH 6H ! � P~E6H !. For the second property, notice that Equa-tion 11 gives in particular Bel~V6H !� Bel~H 6H !�1. Moreover, the proof of then-monotonicity for any n can be done along the same lines followed to prove themain theorem relative to unconditional beliefs.

To prove the third property, start from

Bel~E6H ! � P� ∨Hi�E

Hi� ∨Hi H�À

Hi�Bel~E6K ! � P� ∨

Hi�E

Hi� ∨Hi K�À

Hi�and since

∨Hi�E

Hi� ∨Hi H�À

Hi �� ∨Hi�E

Hi� ∨Hi K�À

Hi

the monotonicity of P with respect to �� gives Bel~E6K !� 0. Analogously, it canbe proved that Bel~H 6K !� 0 implies Bel~E6K !� 0.

Finally, by axiom ~iii' ! of conditional probability we get

Bel~E6H ! � P� ∨Hi�E

Hi� ∨Hi�H

Hi�P� ∨Hi�H

Hi� ∨Hi H�À

Hi�Bel~H 6K ! � P� ∨

Hi�H

Hi� ∨Hi�K

Hi�P� ∨Hi�K

Hi� ∨Hi K�À

Hi�and, by multiplying these equalities,

Bel~E6H !Bel~H 6K ! � P� ∨Hi�E

Hi� ∨Hi�H

Hi�{P� ∨Hi�H

Hi� ∨Hi H�À

Hi�{P� ∨

Hi�H

Hi� ∨Hi�K

Hi�{P� ∨Hi�K

Hi� ∨Hi K�À

Hi�� Bel~E6K !Bel~H 6H !

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with

Bel~H 6H ! � P� ∨Hi�K

Hi� ∨Hi K�À

Hi� �

Corollary 2. Given a conditional belief as that of Equation 11, consider thelast property of Proposition 1 for K �V: then

Bel~E6H !Bel~H ! � Bel~EH !Bel~H 6H !� Bel~E !Bel~H 6H !

Corollary 3. If Bel~H c !� 1 (so that 1� Bel~H c !� Pl~H !� 0) and Bel~H !� 0,we get

Bel~E6K ! � Bel~E6H !Bel~H 6K !Pl~H !

Bel~H !

and, for K �V,

Bel~E ! � Bel~EH !� Bel~E6H !Pl~H !

so that it follows, for E � H, the well-known relation

Bel~E6H ! �Bel~EH !

Pl~H !

If Bel~H 6H !� 0, then Bel~E6H !� 0, because

Bel~E6H ! � P� ∨Hi�EH

Hi� ∨Hi H�À

Hi� � P� ∨Hi�H

Hi� ∨Hi H�À

Hi�� 0

and so also Bel~E6K !� 0.

5. BELIEF FUNCTIONS AS GENERALIZED���-DECOMPOSABLE MEASURES

The previous results will enable us to introduce ~in this section! suitable oper-ations � and � in the family C.

Given a Boolean algebra E, a function w : E r @0,1# is a weakly �-decomposable measure if

w~V! � 1 w~À!� 0

and there exists an operation � from w~E! � w~E! to @0,1# , whose restriction tothe following subset of w~E! � w~E!

K � $~w~A!,w~B!! : A, B � E, AB � À%

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is increasing and is such that the following condition holds: For every A, B � E,with AB � À,

w~A ∨ B! � w~A! � w~B! ~12!

Even if the operation � is, with respect to the elements of K, commutative,associative, and admits 0 as neutral element, nevertheless it need not be extensibleto a function defined on the whole w~E! � w~E! ~and so neither on @0,1# 2 !, andsatisfying the same properties.

We recall that if the operation � can be extended to all pairs of @0,1# 2 ~main-taining the aforementioned properties and including that of being strictly increas-ing or increasing and continuous!, then w is a quasilinear mean of a probability;that is, there exists a continuous and strictly increasing function c : @0,1#r @0,1#such that, for AB � À,

c~w~A ∨ B!! � c~w~A!!� c~w~B!!

We want to suitably relax the above definition, by giving up the requirementfor � of being increasing and to have 0 as the neutral element. The followingdefinition is, essentially, for H �V, condition ~g2 ! of Definition 2 ~Section 2.3!.

Definition 5. Let E be a Boolean algebra of events and w a function from E to@0,1# . We say that w is a generalized �-decomposable measure if it is a capacityand there exists a binary operation � from w~E!� w~E! to w~E! such that condi-tion 12 holds for every A, B � E, with AB � À.

Obviously the operation � is, with respect to the elements of K, commutativeand associative. Moreover it admits w~À! as a neutral element in K ~notice that inthis case it is not assured that w~E ! � w~F!� w~F!, if w~E !� w~À!� 0!. Finally,the condition that w is a capacity implies that � is increasing restricted to the pairs$~w~E !,w~À!!% : In fact, for every F � E, FE � À we have

w~E ! � w~À! and w~E ! � w~F!� w~E ∨ F!� w~F!� w~F! � w~À!

Theorem 4. Given an algebra E, let Bel be a belief function on E and denote byBel~E! its range. Then there exist an operation � from Bel~E!� Bel~E! to Bel~E! ,with � increasing only with respect to pairs of events ordered by implication, thatis, in the set

$~w~A!,w~C!!, ~w~B!,w~C!! : A, B,C � E, A � B, BC � À, Bel~B! � Bel~A!%

such that Bel is a generalized �-decomposable measure on E.

Proof. First of all, the function Bel is a capacity. We prove now that there existsan operation � with the required features. Let m be the basic probability assign-ment associated to the given Bel, and take A and B, with AB � À. Then

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Bel~A ∨ B! � Bel~A!� Bel~B!� (C�A∨B

C�A,C�B

m~C!

� Bel~A!� Bel~B!� k~A, B!

� Bel~A! � Bel~B!

The function k~A, B! depends only on the pair ~A, B! and is such that ~see theproof of associativity below!

k~A, B,C! � k@~A ∨ B!,C#� k~A, B!� k@A, ~B ∨ C!#� k~B,C!

Commutativity of � is obvious; we show now its associativity. Let A, B, C bethree incompatible events; then

Bel~A ∨ B ∨ C! � Bel~A ∨ B!� Bel~C!� k~A ∨ B,C!

� Bel~A ∨ B!� Bel~C!� (D�A∨B∨C

D�A∨B, D�C

m~D!

� Bel~A!� Bel~B!� Bel~C!� (D�A∨B

D�A, D�B

m~D!

� (D�A∨B∨C

D�A∨B, D�C

m~D!

� Bel~A!� Bel~B!� Bel~C!� (D�A∨B∨C

D�A, D�B, D�C

m~D!

� Bel~A!� Bel~B!� Bel~C!� k~A, B,C!

Clearly, � has the neutral element Bel~À!� 0.Actually, � is “locally” strictly increasing: It is, in fact, strictly increasing

with respect to the chain of events ordered by implication ~corresponding to setinclusion!, because, if A � B and Bel~A! � Bel~B!, then

Bel~A! � Bel~C! � Bel~B! � Bel~C!

for any C incompatible with B; so the right-hand side is equal ~by associativity! to

Bel~A! � Bel~AcB! � Bel~C! � Bel~A!� Bel~AcB!� Bel~C!� k~A, B,C!

with

k~A, B,C! � k~~A ∨ C!, AcB!� k~A,C! �

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Remark 6. Notice that, in our framework, the function k~A, B! can be written as

k~A, B! � P� ∨Hi�AHi�B

Hi�A∨B

Hi�Remark 7. The operation � it is not necessarily increasing: Given, in fact, A, B,C, with C~A ∨ B!� À, let Bel~A! � Bel~B!; then

Bel~A ∨ C! � Bel~A! � Bel~C!� Bel~A!� Bel~C!� (D�A∨C

D�A, D�C

m~D!

Bel~B ∨ C! � Bel~B! � Bel~C!� Bel~B!� Bel~C!� (D�B∨C

D�B, D�C

m~D!

The two sums on the right-hand sides can take quite arbitrary values, because thefunction m obeys very weak conditions: For example, the second sum can be greaterthan the first one in such a way as to change the order of the inequality.

6. CONDITIONAL BELIEF FUNCTIONS AS GENERALIZED~���,���!-DECOMPOSABLE CONDITIONAL MEASURES

It is time now to introduce, in the set

K* � $~Bel~E6H !, Bel~F 6H !!, H � B, E, F � G, EFH � À%

the operation � for conditional beliefs defined on C � G � B, with G a Booleanalgebra and B � G closed with respect to ~finite! logical sums, in the followingway:

Bel~E ∨ F 6H ! � Bel~E6H ! � Bel~F 6H !� Bel~E6H !� Bel~F 6H !� kH ~E ∨ F!

with

kH ~E ∨ F! � (C�EHC�FH

C�~E∨F!H

m~C!� P� ∨Hi�EHHi�FH

Hi�~E∨F!H

Hi� ∨Hi H�À

Hi�Moreover, the results of Section 4 allow us to introduce an operation

� : @0,1# 2 r @0,1#

such that for E6H, H 6K, E6K � C, with E � H � K one has

Bel~E6H ! � Bel~H 6K ! � Bel~E6K !

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The above relation holds for the operation � defined as follows:

Bel~E6H ! � Bel~H 6K ! � �0, if Bel~H 6H !� 0

Bel~E6H !Bel~H 6K !

Bel~H 6H !, elsewhere

~13!

as can be easily seen taking into account the second part of Corollary 3 and the lastitem of Proposition 1 ~Section 4!.

Theorem 5. The operation � defined by Equation 13 turns out to be associativeand commutative in the set

G � $~Bel~E6H !, Bel~H 6K !! : E6H, H 6K � C, E � H � K %

and has 1 as a neutral element. Moreover � is monotone with respect to the setG ' � G of the pairs

G ' � $~Bel~E6H !, Bel~H 6K !!, ~Bel~F 6H !, Bel~H 6W !!%

and it is distributive with respect to � in K* � G.

Proof. The proof of commutativity is trivial. To prove the associativity, let E �H � K � W and consider first the case Bel~H 6H !Bel~K 6K ! � 0. The quantity insquare brackets of the following expression,

@Bel~E6H ! � Bel~H 6K !# � Bel~K 6W !

is equal to

Bel~E6K ! �Bel~E6H !Bel~H 6K !

Bel~H 6H !

Then

@Bel~E6H ! � Bel~H 6K !# � Bel~K 6W ! � Bel~E6K ! � Bel~K 6W !

�Bel~E6K !Bel~K 6W !

Bel~K 6K !

�Bel~E6H !Bel~H 6K !Bel~K 6W !

Bel~H 6H !Bel~K 6K !

Clearly, we get the same result starting from

Bel~E6H ! � @Bel~H 6K ! � Bel~K 6W !#

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Consider now the case Bel~H 6H !� 0 and Bel~K 6K !� 0: From the definition itfollows that Bel~E6K !� 0, so that

@Bel~E6H ! � Bel~H 6K !# � Bel~K 6W ! �Bel~E6K !Bel~K 6W !

Bel~K 6K !� 0

On the other hand,

Bel~E6H ! � @Bel~H 6K ! � Bel~K 6W !# � Bel~E6H ! � Bel~H 6W !� 0

The proof is similar for Bel~H 6H !� 0 and Bel~K 6K !� 0, and even simpler whenboth the latter quantities are equal to zero.

To prove that 1 is the neutral element, consider a conditional event E6H suchthat Bel~E6H !� 1. If Bel~H 6H ! � 0, then

Bel~E6H ! � Bel~H 6K ! �Bel~E6H !Bel~H 6K !

Bel~H 6H !�

Bel~H 6K !

Bel~H 6H !� Bel~H 6K !

the latter equality coming from Bel~H 6H !� 1; in fact, E � H implies

∨Hi�H

Hi� ∨Hi H�À

Hi �� ∨Hi�E

Hi� ∨Hi H�À

Hi

and because

Bel~E6H ! � P� ∨Hi�E

Hi� ∨Hi H�À

Hi� Bel~E6H !� P� ∨Hi�H

Hi� ∨Hi H�À

Hi�from the monotonicity of P with respect to ��, the conclusion follows.

From the definition of � it follows immediately that � is monotone withrespect to the set G ' , because

Bel~E6H ! � Bel~F 6H ! and Bel~H 6K !� Bel~H 6W !

imply

Bel~E6H ! � Bel~H 6K ! � Bel~F 6H ! � Bel~H 6W !

We prove now distributivity of � with respect to � in K* � G, that is, thatfrom E6H, F 6H � K* , H 6K � C, with H � K, it follows that

@Bel~E6H ! � Bel~F 6H !# � Bel~H 6K !

� @Bel~E6H ! � Bel~H 6K !# � @Bel~F 6H ! � Bel~H 6K !# ~14!

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Consider first the case Bel~H 6H ! � 0. The quantity in square brackets in the left-hand side of Equation 14 can be written, by the definition of �,

Bel~E ∨ F 6H ! � Bel~E6H !� Bel~F 6H !� kH ~E ∨ F!

so that this equality and the definition of � imply

Bel~~E ∨ F!H 6K ! � Bel~E ∨ F 6H ! � Bel~H 6K !�Bel~E ∨ F 6H !Bel~H 6K !

Bel~H 6H !

�Bel~E6H !� Bel~F 6H !� kH ~E ∨ F!

Bel~H 6H !Bel~H 6K !

�Bel~E6H !Bel~H 6K !

Bel~H 6H !�

Bel~F 6H !Bel~H 6K !

Bel~H 6H !

� kH ~~EH ! ∨ ~FH !!Bel~H 6K !

Bel~H 6H !

Consider now the right-hand side of Equation 14: The first term in squarebrackets is

Bel~EH 6K ! �Bel~E6H !Bel~H 6K !

Bel~H 6H !

and the second gives the same equality, with F in place of E. By operating with �

between the first members of these equalities, one has

Bel~EH 6K ! � Bel~FH 6K ! � Bel~EH 6K !� Bel~FH 6K !� kK ~~EH ! ∨ ~FH !!

� Bel~~EH ! ∨ ~FH !6K !

with

kK ~~EH ! ∨ ~FH !! � P� ∨Hi�EHHi�FH

Hi�~E∨F!H

Hi� ∨Hi K�À

Hi�Considering now also the second members, we have

Bel~~E ∨ F!H 6K ! �Bel~E6H !Bel~H 6K !

Bel~H 6H !�

Bel~F 6H !Bel~H 6K !

Bel~H 6H !

� kK ~~EH ! ∨ ~FH !!

Then we need only, to end the proof, to show that

kK ~~EH ! ∨ ~FH !! � kH ~~EH ! ∨ ~FH !!Bel~H 6K !

Bel~H 6H !

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In terms of P, the second member is

P� ∨Hi�~EH !∨~FH !

Hi� ∨Hi H�À

Hi�P� ∨Hi�H

Hi� ∨Hi K�À

Hi�P� ∨

Hi�H

Hi� ∨Hi H�À

Hi�and, by axiom ~iii' ! of probability, taking into account that H � K, it can be written

P� ∨Hi�~EH !∨~FH !

Hi� ∨Hi H�À

Hi�P� ∨Hi�H

Hi� ∨Hi H�À

Hi�P� ∨Hi H�À

Hi� ∨Hi K�À

Hi�P� ∨

Hi�H

Hi� ∨Hi H�À

Hi�� P� ∨

Hi�~EH !∨~FH !

Hi� ∨Hi K�À

Hi�� kK ~~EH ! ∨ ~FH !!

The proof in the case Bel~H 6H !� 0 is trivial ~we get 0 � 0!. �

In conclusion, we have shown ~according to what has been announced in Sec-tion 2.3! that a conditional belief function can be obtained as a generalized ~�,�!-decomposable conditional measures, where the two operations � and � are thosedefined at the beginning of Section 6.

7. CONCLUSIONS

In a sense, belief functions have been regarded as a different way of interpret-ing lower ~conditional! probabilities, through our merging of the two spaces S andT ~of the classic Dempster’s framework! into only one family of events.

In our context, events are looked on as propositions and, consequently, weavoid any distinction between available information—for example, evidence com-ing from statistical data or, more generally, from any observation—and any otherpotential ~or assumed, i.e., not yet observed! information. So, by exploiting theaforementioned “status” of all available information as being of the same qualityand nature, we can put on the same frame all the relevant events of both the “evi-dential frame” and the “frame of discernment.”

For these families of events ~conditional or not!, the presence of an algebraicstructure is not required, thanks to the fundamental concept of coherence, and inthis framework a main role is played by Theorem 1, which ~essentially! allows usto extend suitably ~and coherently! any probability assessment.

This way of reading belief functions leads to the introduction of a conditionalbelief as the lower envelope of a class of ~coherent! conditional probabilities and,then, to its characterization in terms of two operations �,� ruling

Bel~~A ∨ B!6H ! � Bel~A6H ! � Bel~B 6H !

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and, when E � H � K,

Bel~E6H ! � Bel~H 6K ! � Bel~E6K !

At this point a “natural” conclusion is to introduce axiomatically a conditionalbelief function as a particular element of a general class of generalized ~�,�!-decomposable conditional measures.

Moreover, it turns out that the results of this article permit us to get a charac-terization of a conditional belief function in terms of a class of unconditional ones.In fact, a conditional measure w~E6H ! may not be deducible from ~or expressedby! a unique ~unconditional! measure w~{!� w~{6V! evaluated on the two eventsE ∧ H and H, but it is possible to find a class of unconditional measures that singleit out as the unique solution of a suitable equation. This procedure is wholly sim-ilar to the one followed for a conditional probability P~E6H !, which, in fact, turnsout to be ~for each conditional event E6H of a given family! the unique solution xof the equation

Pa~E ∧ H ! � x � Pa~H !

where the probabilities Pa constitute ~in the characterization theorem for a coher-ent conditional probability, mentioned in Remark 3, Section 2.4! the so-called agree-ing class.

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