NAFIPS 2005 - 2005 Annual Meeting of the North American Fuzzy Information Processing Society

An Algebraic Setting for Belief FunctionsElbert A. Walker

Department of Mathematical SciencesNew Mexico State University

Las Cruces, NM 88003elbert@nmsu.edu

Abstract-We detail a specific algebraic setting for the studyof functions from the set of subsets of a finite set into the realnumbers, in particular for the study of belief functions.

Keywords: allocations, belief functions, densities, incidencealgebras, monotonicity

I. INTRODUCTIONLet U be a finite set and R the real numbers. To give belief

functions a natural and convenient setting, we study them inthe context of the set of all functions 2u X-+ R, and developsome methods for computations with them. Functions 2u 1-+ Rarise in many contexts in reasoning under uncertainty. Somesets of such functions of interest are belief functions, variousmeasures, density functions on 2U, possibility functions, andmany others. Our purpose is to give a setting in which many ofthe properties of these functions can be derived efficiently. Thebasic tool is casting the vector space of all fimctions 2u -* Ras a module over a certain incidence algebra.

This is a work in progress. Here we will give the basicdefinitions, state some relevant theorems and indicate anoccasional proof. A detailed account is in progress and willappear elsewhere.

II. BELIEF FUNCTIONS AND INCIDENCE ALGEBRAS

Definition 1: Let U be a finite set, and F = { f: 2u t R}.For f,g E F and r E JR, let

(f +g) (X) = f(X) +g(X)(rf)(X) = r(f(X))

F is a vector space over JR. The proof is routine in allaspects. One basis for F as a vector space over JR is theset of fimctions {fy: Y C U} defined by fy (Y) = 1 andfy (X) = 0 if X :& Y. Thus F has dimension 21UI over JR.

Definition 2: Let A be the set of functions (2U) [2] __R,where (2U)[2] - {(X,Y): X C Y C U}. On A defineaddition pointwise and multiplication by the formula

(a * d) (X, Y) = E a(X, Z)/(Z, Y)xCZCY

A with these operations is the incidence algebra of U overthe field R.

This algebra was introduced by Rota in [6] as a setting forstudy of combinatorial problems, and has turned out to be anobject of interest in itself, and has been thoroughly studied.

Classically the definition is for a locally finite partially orderedset rather that for 2U, but we restrict ourselves here to thepartially ordered set 2u with the usual inclusion ordering. Thebook [9] is a good source, both for study of the algebra itselfand for fiurther references.

Theorem 3: A is a ring with identity. Its identity is thefunction given by 5(X, X) = 1 and 5(X, Y) = 0 if X 5# Y.

Pointwise addition is denoted by + and given by(a +3) (X, Y) = a (X, Y) + O3(X. Y). Let 0 denote themapping given by 0(X, Y) = 0 for all X C Y. To showthat A is a ring, the following properties must be verified forall a,/3.E A.

1) a+O=/+a2) (a +3) + =a +((/+±y)3) a+0=a4) For each a, there exists 3 such that a +/ = 05) (a */) * y =a * ( * -y)6) a?* (3+ y) =(a*)+( (*y)7) (at + 1)* -r' (at * ) + ( -y*)'8) a * d 6 * a = a

Properties 1-4 say that A with the operation + is an Abeliangroup. The properties 1-8 are routine to verify. The ring Ahas an identity 6, but it is not true that every non-zero elementa has an inverse. That is, there does not necessarily exist fora an element : such that a * /3 = = d * a. The followingtheorem characterizes those elements that do have inverses,and this characterization is a fundamental property of this ring.It is not totally obvious, and we include a proof to give someflavor of computations in this ring.

Theorem 4: In the ring A, an element a has an inverseif and only if for all X, a(X, X) :& 0. Its inverse is giveninductively by

a- (X,X) = (XX): and for X c Y,-1

a '(X,Y) = a(XX) E a(X,Z)a1(Z,Y)

Proof: If a has an inverse /3, then (a * /3) (X, X) =a (X, X) :(X, X) = 6(X, X) = 1, so that a(X, X) 0 0.Now suppose that for all X, a (X, X) :$ 0. We need anelement /3 such that /3 * a = a * 3 = 6. In particular, weneed (a * /) (X, Y) = 0 for X c Y and (a * 0) (Y, Y) = 1.We define (X, Y) inductively on the number of elementsbetween X and Y. If that number is 1, that is, if X = Y, let

0-7803-9187-X105/$20.00 ©2005 IEEE. 407

/(X, X) = 1/a(X, X), which is possible since a(X, X) :A 0.Assume that ,3(X, Z) has been defined for elements X andZ such that the number of elements between the two is < n,and suppose that the number of elements between X and Yis n > 1. We want

O = (a* )(X, Y)

ZE e(X,Z)3(Z,Y)xczc

XCZCY ~ CZCaa(X. X) (X,Y)+ a(X,Z) 0(Z,Y)xczcy

This equation can be solved for 3 (X, Y) since a (X, X) :A 0,yielding

1(X,Y) (X.,EZ),a)(Z,Y)XCZCY

Thus a * = 6. Similarly, there is an element -a such that-Y * a = 6. Then

The theorem follows. a

Elements in a ring that have an inverse are called units.There are two very special and important units in A.

,[(X, Y) = (-l)lY-xl is the Mobius function.* ((X, Y) = 1 is the Zeta function.

The element ( is easy to define: it is simply 1 everywhere.The element ,t is its inverse. That is, ,t * ( = ( * , = 6, whichis easy to check. These functions play an important role incombinatorics, and particularly in elementary number theory.They are also crucial in the study of belief functions and thelike, as we will see.

There is a natural operation on the elements of the vectorspace F by the elements of the incidence algebra A. Thisoperation is a common one in combinatorics, and we willformalize it here and apply it extensively in our development.

Definition 5: For a E A, f E F, and X E 2u, let

(f * a) (X) = E f(Z)a(Z. X)zcx

We have defined a "multiplication" of the elements ofF byelements ofA on the right. The analogy with vector spaces isthat F is the set of vectors, and A is the set of scalars, withmultiplication of scalars on the right of the vectors.

Proposition 6: F is a (right) module over the ring A. Thatis,

1) f*6=f2) (f*a)*i3=f*(a*3)3) (f+g)*a=f*a+g*a4) f *(a+i3)=f *a+f */The verifications are straightforward calculations. Notice

that for f E F, f * * = f * * = f.

With the operation f * a, elements ofA are linear transfor-mations on the real vector space F. So A is a ring of lineartransformations on F. Since U is finite, F is finite dimensionaland of dimension 12UJ, so A is isomorphic to a subring ofthe ring of 12U1 x 12 real matrices. With a basis orderedproperly, these matrices are upper triangular. Such a matrixhas an inverse if and only if its diagonal entries are non-zero.This corresponds to an element a E A having an inverse ifand only if a(X, X) $ 0. Following are some observations,elementary but significant.

. For each r E R, we identify r with the constant mapr E F defined by r(X) = r for all X E 2U.

. A is an algebra over R via the embedding JR -- Ar 4 r6, where r6(X,Y) = r (6 (X, Y)). That is, A isa vector space over JR and r6 * a = a * r6. Note that(r6 * a) (X, Y) = r (a (X, Y)).

. For r E R, f E F and a E A, r(f * a) = (rf) * a =f * (ra).. If a is a unit in A, then F -* F: f - f * a andF -F F: f -- f * a-1 are one to one maps ofF ontoF, and are inverses of one another.F- . ff-+f* pand F--F: f f-*f are one toone maps ofF onto F, and are inverses of one another.This case is of particular interest.f * At is called the Mobius inverse of f, or the Mobiusinversion of f.

The formulation of F as a module over A is implicitin many papers, but not formulated precisely. Sometimesthe definition is given for a * f rather than for f * a.In [3], the relevant operations are defined without some ofthe formal algebraic terminology. That paper is of particularinterest in that it investigates a pair of elements in A that areinverses of each other analogous to the Mobius transform andits inverse. The motivation for the study of this transform,called the interaction index, comes from cooperative gametheory and multicriteria decision making. That paper is highlyrecommended. Our motivation comes from the study of belieffunctions.

III. MONOTONICITY

We begin now with some facts that will be of particularinterest in the study of belief functions and certain measures.In [1], there are many fundamental results on monotonicity,and that excellent paper goes farther afield than we do here. Inparticular, the results on monotonicity here are in [1] in someform. Our purpose is to develop the theory through a moresystematic use of our algebraic setup.

Definition 7: Let k be > 2. An element f E F is monotoneof order k if for every non-empty subset S of 2U with ISI < k,

f u X)> E(-1)1'r1+1f n x)XES( ) T0CS XET

f is monotone of infinite order if monotone of order k forall k.

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Of course, monotone of order k implies monotone ofsmaller order. One goal is to identify those f that are M6biusinversions of maps that are monotone of order k. This is thesame as identifying those f such that f * ( is monotone oforder k. There is an alternate form for the right hand side ofthe inequality above which is convenient to have.Lemma 8: Let f: 2U -* R. Let S be a subset of 2U. Let

r = r (S) be the set of subsets that are contained in at leastone X in S. Then

T (-S ) TI+ (f * o (n x) = Zf(x)

Proof:

(-1),T,+l (f * X= Z (_~1)tT1+1 S (Y)

0o$TCS YCnXETX

This last expression is a linear combination of f(Y)'s, forY a subset of some elements of S. Fix Y. We will find thecoefficient of f(Y). Let Ty = {X E S: Y C X}. Then for

So#TCS

Y cnxCTX

IE(-1j)T1'+ f(Y)0o$TCTy

= f(Y)The result follows. UOf course, the result could have been stated as

5 (-_1) f+T ( n x) = Ej (f * ,t) (x)oATCS X XEF

The set F plays an important role in what follows. Let X CU with IXI > 2. Let S = {X -{}: x E X}. Then

* UyESY = X. Every subset Y not X itself is uniquely the intersection

of the sets in a subset of S. In fact

Y= nf(x-{x})xY

. The set F for this S is precisely the subsets Y ofX notX itself.

These facts are very useful. The following results are fairlyeasy consequences, making heavy use of our algebraic setup.

Theorem 9: f * ( is monotone of order k if and only if forall A, C with 2 <.CI < k, E f(X) > 0.

CCXCAAgain, the result could have been stated as f is monotone

of order k if and only if E (f *u) (X) .0 for all A,CCCXCA

with 2 < ICI < k. Taking A =- C, we getCorollary 10: If f * is monotone of order k, then f(X) >

0 for 2 < jXI < k.

Corollary 11: f*6 is monotone of infinite order if and onlyif f(X) > 0 for 2 < IXI.Some additional easy consequences of the theorem are

these.Corollary 12: The following hold.1) Constants are monotone of infinite order. In fact,

(r * y) (X) = 0 if X7&0, and (r * t) (0) = r.2) If f and g are monotone of order k, then so is f + g.3) If f is monotone of order k and r > 0 then rf is

monotone of order k.4) A function f is monotone of order k if and only if for

all r E R, f + r is monotone of order k.A connection with ordinary monotonicity is the following.Theorem 13: If f is monotone of order 2, then f is

monotone if and only if f(0) is the minimum value of f.Corollary 14: If f is monotone of order 2 and not

monotone, then f({x}) < f(0) for some x E U.

By choosing f appropriately, it is easy to get f * 6 that aremonotone of infinite order and not monotone.One cannot state the definition of ordinary monotonicity in

the form of Definition 7. Having the ISI = 1 imposes nocondition at all. However,

Theorem 15: f * ( is monotone if and only if for all A, CwithjCI=1, Z f(X)>0.

CCXCA

IV. BELIEFS, DENSITIES AND ALLOCATIONS

We now specialize to functions that are of particular interestto us, namely belief functions. There are several equivalentways to define them. We take as our definition that given byShafer [7].

Definition 16: A belief function on U is a function g2u '-+ [0,1] satisfying

1) g(0) = 02) g(U) = 13) g is monotone of infinite order.

Note the following.. A belief function is monotone of order k for any k.. A belief function is monotone by Theorem 13.There is an intimate connection between belief functions on

U and densities on 2U. We establish that connection now.Definition 17: A density on 2u is a function f : 2u

[0,1] such that Excu f(X) = 1.

Theorem 18: Let f be a density on 2u with f(0) = 0.Then

1) (f * ) (0) =02) (f 8 () (U) =13) f * 6 is monotone of infinite order.

Proof: The first two items are immediate. Since densitiesare non-negative, by Corollary 1 1, f * 6 is monotone of infiniteorder. U

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(- 1) Irl+1 f(y)

Now we get the precise correspondence between belieffunctions and densities.

Theorem 19: g is a belief function on U if and only if g *,uis a density on 2u with value 0 at 0.

Proof: If g * [t is a density with value 0 at 0, then theprevious theorem gets (g * bi) * ( = g to be a belief function.Assume that g is a belief function. Then

E (g*,u)(X) =((g* )*L (U)=g(U)=lxcU

We need g * tl > 0.

(g * g) (0) = g(0)(0, 0) = g(0) = 0

For {x},

(g * A) ({x}) = g(0)I(0, {x}) + g({x})1({x}, {x})= g({x}) > 0.

Since g is monotone of infinite order, (g * [b) (X) > 0 forlXI >2. U

Corollary 20: Let D be the set of densities on 2u withvalue 0 at 0, and B let be the set of belief functions on U.Then D-*B is a one-to-one correspondence with inverse ,u.

Corollary 21: A belief function g is a measure on U ifand only if g * At is a density on U, that is, if and only if(g * p) (X) =0 for all IXj > 2.

Iff is a density on 2u with f(0) = 0, then there is a naturalway to construct densities on the set U. These are allocations,assignments of non-negative values to elements of f(X) sothat their sum is f (X). Here is the precise definition.

Definition 22: Let f be a density on 2u with f(0) = 0.An allocation a of f is a function al: U x 2u - [0,1] suchthat ZuEx a(u, X) = f(X) for all X E 2u.The following proposition is obvious.Proposition 23: Let a be an allocation of a density on 2u.

Then U -- [0, 1] : u EuEX a(u, X), where the sum isover X, is a density on U.

Each belief function g on U gives a density f = g*( on 2u,there are many allocations of f, each such allocation givesa density on U, and each density on U gives a measure onU. There are some relations between these entities that are ofinterest. First here is some notation. Let g be a belief function.(We could start with a density f on 2u such that f(0) = 0.)

cf = g * A is the density on 2u associated with the belieffunction g on U.

* D is the set of densities on U arising from allocations aof the density f on 2u

* M is the set of probability measures P on U such thatP >g.

f g

1 MDAM

Theorem 24: Let g be a belief function on U. In thediagram above, let a be the map that takes a density on Uto the corresponding measure on U. Then

1) a maps D ontoM.2) g = inf{P: P E M}.

Proof: Let a be an allocation of the density f on 2u, andlet P be the probability measure it induces. First we show thatg(A) < P(A) for all A E 2U, that is, that a maps D intoM.

g(A) = (f*)(A)= E f(B)B:BCA

- Z Z ta(u, B)B:BCA u:uEB

P(A) = ZP({u})= cE a(u, B)uEA u:uEA B:uEB

Clearly g(A) < P(A).To show that g(A) = inf{P(A) : P E M}, let A E 2u.

Let a be an allocation of f such that for u E A, for all B notcontained in A, allocate 0 to u. Then g(A) = P(A), and itfollows that g(A) = inf{P(A) : P E M}.We have been unable to find an elementary proof that a is

onto M, or equivalently, that every P E M is the probabilitymeasure associated with an allocation of the density g * ,t. Werefer the reader to [1] and to [4]. U

In any case, M does consist of all probability measures Pon 2u coming from allocations of f. (This setM of measuresis often called the core of 9.)As we know, inf{P : P E C} = g. However, it is

not true that the inf of any set of probability measures on(U, 2U) is a belief function. There are easy examples to thecontrary. It seems to be a difficult problem to give a reasonableclassification of sets of probability measures on (U, 2U) whoseinf is a belief function.

There are some special cases of allocations that are ofinterest, especially in applications. We give two examples.

Example 25: For the Mobius inverse f = g * At of a belieffunction, an intuitively appealing allocation is a (u, A) =f(A)/ IAl . It is obviously an allocation.

Example 26: Let (Ul, U2, ..., un) be an ordering of the setU. For a density f on 2U, allocate all of f(X) to the largestelement of X. This is obviously an allocation a of f, andyields a density d on U. For the belief function g = f *define p: U -* [0, 1] by

p(Ui) = g{Ul, U2, ..., Ui} -g{u, U2, *--7 U-l}

for i = 1 2,...,rn. (If i = 1, then Y{U1lU2-.2.ui-l} =g(o) = 0.) Now p is obviously a density on U, and thecalculation below shows that p = d. So this allocation is easily

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described in terms of the belief function.

p(Ui) = g{Ul, U2,.. Ui}-9{U1, U2,*,Ui-1}= (f * U1) U,u2, ..., ui} -(f * ) Ul, U2, *--,i Ui'-l

= E f(X)- E f(X)XC{u1,U2,..,Ui} XC{uiIU2... iUi-1i

- E f(X)XC{u1,u2i ...,ui}

uiex

5E a(ui, X)XC{ul,u27,...,iUi}

uiEX- d(u-)

There are n! such orderings of U, so we get a set P of n!probability measures on 2u in this way. (There may be someduplications.) It is easy to check that g = min{P: P E P}.

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