Fuzzy Systems. Possibility Theory.

Fuzzy Systems Possibility Theory Rudolf Kruse Christian Moewes {kruse,cmoewes}@iws.cs.uni-magdeburg.de Otto-von-Guericke University of Magdeburg Faculty of Computer Science Department of Knowledge Processing and Language Engineering R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 1 / 61

Outline 1. Introduction Problems with Probability Theory Upper and Lower Probabilities 2. Dempster-Shafer Theory of Evidence 3. Possibility Theory R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 2 / 61

Problems with Probability Theory Representation of Ignorance standard method to handle uncertainty is probability theory however, it has some problems regarding ignorance we are given one die with faces 1,..., 6 what is the certainty of showing up face i? 1. conduct statistical survey (roll the die 10,000 times) and estimate relative frequency: P({i}) = 1 6 2. use subjective probabilities (i.e., often normal case): we don t know anything (especially and explicitly we don t have any reason to assign unequal probabilities), so most plausible distribution is uniform problem: uniform distribution because of ignorance or extensive statistical tests R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 3 / 61

Problems with Probability Theory Representation of Ignorance experts analyze aircraft shapes 3 aircraft types A, B, C It is type A or B with 90% certainty. About C, I don t have any clue and I don t want to commit myself. No preferences for A or B. problem: propositions hard to handle with Bayesian theory R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 4 / 61

Upper and Lower Probabilities a way out to solve this problem are upper and lower probabilities It is type A or B with 90% certainty. About C, I don t have any clue and I don t want to commit myself. No preferences for A or B. A {A} {B} {C} {A, B} {A, C} {A, B, C} P (A) 0 0 0 0.9 0 1 P (A) 0 1 1.1 1 1 1 consider P (A) and P (A) as lower and upper probability bounds total ignorance: P (A) = 0 and P (A) = 1 for A, A Ω R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 5 / 61

Properties of Upper and Lower Probabilities I 1. P : P(Ω) [0, 1] 2. 0 P P 1, P ( ) = P ( ) = 0, P (Ω) = P (Ω) = 1 3. A B P (A) P (B) and P (A) P (B) 4. A B = P (A)+P (B) = P (A B) 5. P (A B) P (A)+P (B) P (A B) 6. P (A B) P (A)+P (B) P (A B) 7. P (A) = 1 P (Ω\A) R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 6 / 61

Properties of Upper and Lower Probabilities II one can prove the following generalized equation n P ( A i ) i=1 =I:I {1,...,n}( 1) I +1 P ( A i ) i I these set functions play important role in theoretical physics (i.e., capacities [Choquet, 1954]) thoughts have been generalized and developed into theory of belief functions [Shafer, 1976] R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 7 / 61

Outline 1. Introduction 2. Dempster-Shafer Theory of Evidence Belief Measure Plausibility Measure Basic Belief Assignment Focal Element Total Ignorance Dempster s Rule of Combination Example Conversion Formulas 3. Possibility Theory R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 8 / 61

Dempster-Shafer Theory of Evidence like Bayesian approaches, the Dempster-Shafer theory of evidence [Shafer, 1976] wants to model and quantify uncertainty by degrees of belief Bayesian approaches assign degrees of belief to single hypothesis theory of evidence assigns degrees of belief to sets of hypotheses let X be finite frame of discernment each proposition x is x 0 can be represented by A X sets including one element correspond to elementary propositions proposition is true if it contains the true elementary proposition belief in x is x 0 related to A X is measured by Bel(A) [0, 1] R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 9 / 61

Belief Measure Definition Given a finite universe X, a belief measure is a function s.t. Bel( ) = 0, Bel(X) = 1 and Bel : P(X) [0, 1] n Bel(A 1 A 2... A n ) Bel(A j ) Bel(A i A j ) i=1 i,j:1 i<j n for all possible families of subsets of X. +...+( 1) n+1 Bel(A 1 A 2... A n ) e.g., Bel(A 1 A 2 ) Bel(A 1 )+Bel(A 2 ) Bel(A 1 A 2 ) difficult to test basic belief assignment is introduced R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 10 / 61

Belief Measure for each A P(X), Bel(A) is interpreted as degree of belief (based on evidence) that given x X belongs to A if sets A 1,..., A n are pairwise disjoint, then n Bel(A 1,..., A n ) Bel(A i ) i=1 weaker version of additivity property of probability measures probability measures are special case of belief measures where equality in definition of Bel is always satisfied fundamental property: let n = 2, A 1 = A and A 2 = A C, then Bel(A)+Bel(A C ) 1 R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 11 / 61

Plausibility Measure associated with each Bel is plausibility measure Pl defined by Pl(A) = 1 Bel(A C ), A P(X) Definition Given a finite universe X, a plausibility measure is a function Pl : P [0, 1] s.t. Pl( ) = 0, Pl(X) = 1 and Pl(A 1 A 2... A n ) j Pl(A j ) j<k Pl(A j A k ) for all possible families of subsets of X. +...+( 1) n+1 Pl(A 1 A 2... A n ) R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 12 / 61

Basic Belief Assignment fundamental property: let n = 2, A 1 = A and A 2 = A C, then Pl(A)+Pl(A C ) 1 Bel and Pl can be characterized by basic belief assignment s.t. m( ) = 0 and m : P(X) [0, 1] A P(X) m(a) = 1 for each A P(X), value m(a) expresses proportion to which all evidence supports particular element of X belongs to A m(a) does not imply any claim regarding subsets of A definition of m resembles probability distribution function, but probability distribution functions are defined on X basic belief assignments are defined on P(X) R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 13 / 61

Basic Belief Assignment observe the following facts 1. m(x) need not to be 1 2. if A B, then m(a) need not to be less or equal m(b) 3. there does not need to be any relation between m(a) and m(a C ) given m, Bel and Pl are uniquely determined for all A P(X) by Bel(A) = m(b) and Pl(A) = m(b) B B A B A B note that inverse is also possible e.g., m(a) = B B A ( 1) A\B Bel(B) each of three functions is sufficient to determine other two R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 14 / 61

Basic Belief Assignment Bel(A) = B B A m(b) means the following: m(a) characterizes degree of evidence that x belongs only to A Bel(A) represents evidence that x belongs to A and subsets of A Pl(A) = B A B m(b) represents both kinds of evidence: total evidence that x belongs to A or to any subset of A evidence associated with sets that overlap with A for all A P(X), Pl(A) Bel(A) R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 15 / 61

Focal Element every A P(X) for which is called focal element of m m(a) > 0 they are subsets of X on which available evidence focuses X is finite m can be fully described by list F of its focal elements pair F, m is called body of evidence R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 16 / 61

Total Ignorance total ignorance is expressed by m(x) = 1 and m(a) = 0 A X i.e., we know that element is in universal set but we have no evidence about is location in any A X in terms of belief it is exactly the same: Bel(X) = 1 and Bel(A) = 0 A X in terms of plausibility it is different: Pl( ) = 0 and Pl(A) = 1 A R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 17 / 61

Dempster s Rule of Combination now, consider two independent sources of evidence (e.g., from two experts) expressed by m 1 and m 2 on some P(X) they must be appropriately combined to obtain joint basic assignment m 1,2 generally, evidence can be combined in many ways standard way is expressed by Dempster s rule of combination m 1,2 (A) = for all A and m 1,2 ( ) = 0 where K = B C=A m 1(B) m 2 (C) 1 K B C= m 1 (B) m 2 (C) R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 18 / 61

Dempster s Rule of Combination m 1 (B) (focusing on B P(X)) and m 2 (C) (focusing on C P(X)) are combined by m 1 (B) m 2 (C) (focusing on B C) exactly the same for joint probability distribution which is calculated from two independent marginal distributions since some intersections of focal elements may result in A, we must add corresponding product to obtain m 1,2 also, some intersections may be empty since, m 1,2 ( ) = 0, K is thus not included in definition of m 1,2 to ensure A P(X) m(a) = 1, each product is divided by 1 K R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 19 / 61

Example assume that an old code fragment was discovered by Rudolf Kruse it strongly resembles code fragments of Christian Moewes several questions can come up: 1. Is the discovered code fragment a real fragment of Christian? 2. Is the discovered code fragment a product of one of Christian s students? 3. Is the discovered code fragment a fake? R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 20 / 61

Example let C, S, F denote subsets of universe X of all code fragments C denotes set of all code fragments by Christian S represents set of all code fragments by Christian s students F represents set of all fakes of Christian s code fragments two experts (named Georg and Matthias) performed careful examinations of the code fragment afterwards, they provided Rudolf Kruse with basic assignments m 1 and m 2 as follows R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 21 / 61

Two Independent Sources experts Georg Matthias focal elements m 1 Bel 1 m 2 Bel 2 C.05.05.15.15 S 0 0 0 0 F.05.05.05.05 C S.15.2.05.2 C F.1.2.2.4 S F.05.1.05.1 C S F.6 1.5 1 degrees of evidence obtained by examination and supported by different claims that code fragment belongs to one of the sets given m 1 and m 2, we can easily compute Bel 1 and Bel 2, resp. R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 22 / 61

Calculation of Normalization Factor K = m 1 (C) m 2 (S)+m 1 (C) m 2 (F)+m 1 (C) m 2 (S F) + m 1 (S) m 2 (C)+m 1 (S) m 2 (F)+m 1 (S) m 2 (C F) + m 1 (F) m 2 (C)+m 1 (F) m 2 (S)+m 1 (F) m 2 (C S) + m 1 (C S) m 2 (F)+m 1 (C F) m 2 (S)+m 1 (S F) m 2 (C) =.03 so, the normalization factor is then 1 K =.97 R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 23 / 61

Calculation of Joint Basic Assignment m 1,2 (C) = [m 1 (C) m 2 (C)+m 1 (C) m 2 (C S) + m 1 (C) m 2 (C F)+m 1 (C) m 2 (C S F) + m 1 (C S) m 2 (C)+m 1 (C S) m 2 (C F) + m 1 (C F) m 2 (C)+m 1 (C F) m 2 (C S) + m 1 (C S F) m 2 (C)]/.97 =.21 m 1,2 (S) = [m 1 (S) m 2 (S)+m 1 (S) m 2 (C S)+m 1 (S) m 2 (S F) + m 1 (S) m 2 (C S F)+m 1 (C S) m 2 (S) + m 1 (C S) m 2 (S F)+m 1 (S F) m 2 (S) + m 1 (S F) m 2 (C S)+m 1 (C S F) m 2 (S)]/.97 =.01 R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 24 / 61

Calculation of Joint Basic Assignment m 1,2 (C F) = [m 1 (C F) m 2 (C F) + m 1 (C F) m 2 (C S F) + m 1 (C S F) m 2 (C F)]/.97 =.2 m 1,2 (C S F) = [m 1 (C S F) m 2 (C S F)]/.97 =.31 R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 25 / 61

Joint Basic Assignment combined Georg Matthias evidence focal elements m 1 Bel 1 m 2 Bel 2 m 1,2 Bel 1,2 C.05.05.15.15.21.21 S 0 0 0 0.01.01 F.05.05.05.05.09.09 C S.15.2.05.2.12.34 C F.1.2.2.4.2.5 S F.05.1.05.1.06.16 C S F.6 1.5 1.31 1 R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 26 / 61

Projection consider now basic assignment m : P(X Y) [0, 1] each focal element of m is binary relation R on X Y let R X denote projection of R on X, then R X = {x X (x, y) R for some y Y } projection m X of m on X is m X (A) = m(r) for all A P(X) R A=R X adding values of m(r) for all focal elements R whose R X is A similarly, computation can be done for R Y and m Y, resp. R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 27 / 61

Marginal Bodies of Evidence m X and m Y are also called marginal basic assignments F X, m X and F Y, m Y are called marginal bodies of evidence F X, m X and F Y, m Y are called noninteractive if and only if for all A F X and B F Y m(a B) = m X (A) m Y (B) and m(r) = 0 for all R A B i.e., joint basic assignment m is uniquely determined by marginal basic assignments R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 28 / 61

Example consider body of evidence given in table focal elements are subsets of X Y = {1, 2, 3} {a, b, c} focal X Y element 1a 1b 1c 2a 2b 2c 3a 3b 3c m(r i ) R 1 = 0 0 0 0 1 1 0 1 1.0625 R 2 = 0 0 0 1 0 0 1 0 0.0625 R 3 = 0 0 0 1 1 1 1 1 1.125 R 4 = 0 1 1 0 0 0 0 1 1.0375 R 5 = 0 1 1 0 1 1 0 0 0.075 R 6 = 0 1 1 0 1 1 0 1 1.075 R 7 = 1 0 0 0 0 0 1 0 0.375 R 8 = 1 0 0 1 0 0 0 0 0.075 R 9 = 1 0 0 1 0 0 1 0 0.075 R 10 = 1 1 1 0 0 0 1 1 1.075 R 11 = 1 1 1 1 1 1 0 0 0.15 R 12 = 1 1 1 1 1 1 1 1 1.15 R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 29 / 61

Example margin bodies of evidence are obtained as follows m X ({2, 3}) = m(r 1 )+m(r 2 )+m(r 3 ) =.25 m X ({1, 2}) = m(r 5 )+m(r 8 )+m(r 11 ) =.3 m Y ({a}) = m(r 2 )+m(r 7 )+m(r 8 )+m(r 9 ) =.25 m Y ({a, b, c}) = m(r 4 )+m(r 10 )+m(r 11 )+m(r 12 ) =.5 X 1 2 3 m X (A) A = 0 1 1.25 1 0 1.15 1 1 0.3 1 1 1.3 Y a b c m Y (B) B = 0 1 1.25 1 0 0.25 1 1 1.5 here, marginal bodies of evidence are noninteractive R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 30 / 61

Conversion Formulas given m(a) = Bel(A) = Pl(A) = m m(a) m(b) m(b) B A B A Bel ( 1) A\B Bel(B) Bel(A) 1 Bel(A C ) B A Pl ( 1) A\B (1 Pl(B C )) 1 Pl(A C ) Pl(A) B A R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 31 / 61

Outline 1. Introduction 2. Dempster-Shafer Theory of Evidence 3. Possibility Theory Necessity and Possibility Possibility Distribution Basic Distribution Mathematical Properties Fuzzy Sets and Possibility Theory Possibility versus Probability Comparison of both Theories Interpretations R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 32 / 61

Possibility Theory possibility theory is special kind of evidence theory bodies of evidence contain nested focal elements belief and plausibility are called necessity and possibility, resp. Theorem Let a given finite body of evidence F, m be nested. Then, the associated belief and plausibility measures have the following properties for all A, B P(X) Bel(A B) = min{bel(a), Bel(B)} Pl(A B) = max{pl(a), Pl(B)} R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 33 / 61

Necessity and Possibility let necessity and plausibility be denoted by Nec and Pos, resp. last theorem provides us with basic equation of possibility theory Nec(A B) = min{nec(a), Nec(B)} Pos(A B) = max{pos(a), Pos(B)} can be also defined for arbitrary universal set (e.g., infinite ones) Nec A k = inf Nec(A k) k K Pos k K A k k K where K is arbitrary index set = sup Pos(A k ) k K R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 34 / 61

Necessity and Possibility since necessity and possibility are special belief and plausibility measures, resp., they satisfy fundamental properties Nec(A)+Nec(A C ) 1 Pos(A)+Pos(A C ) 1 1 Pos(A C ) = Nec(A) from the definitions of Nec and Pos, it follows that { } min Nec(A), Nec(A C ) = 0 { } max Pos(A), Pos(A C ) = 1 R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 35 / 61

Necessity and Possibility necessity and possibility measures constrain each other Theorem For every A P(X), any necessity measure Nec on P(X) and the associated possibility measure Pos satisfy the following implications: Nec(A) > 0 Pos(A) = 1, Pos(A) < 1 Nec(A) = 0. R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 36 / 61

Possibility Distribution Function given possibility measure Pos on P(X) let function r : X [0, 1] s.t. x X : r(x) = Pos({x}) be called possibility distribution function associated with Pos Theorem Every possibility measure Pos on a finite power set P(X) is uniquely determined by a possibility distribution function r : X [0, 1] via the formula for each A P(X). Pos(A) = max x A r(x) when X is infinite, then Pos(A) = sup x A r(x) R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 37 / 61

Possibility Distribution let possibility distribution function r be defined on X = {x 1,..., x n } possibility distribution associated with r is n-tuple where r i = r(x i ) for all x i X r = (r 1,..., r n ) it is useful to order r s.t. r i r j when i < j R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 38 / 61

Example now, let Pos be defined on P(X) by m w.l.o.g., focal elements are some or all of subsets in complete sequence of nested subsets A 1 A 2... A n = X where A i = {x 1,..., x i } i IN n i.e., m(a) = 0 for each A A i (i IN) and n i=1 m(a i ) = 1 R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 39 / 61

Example complete sequence of nested focal elements of Pos on P(X) where X = {x 1,..., x n } m(a 3 ) m(a 4 )... m(a n ) m(a 2 ) m(a 1 ) x 1 x 2 x 3 x 4... x n r(x 1 ) r(x 2 ) r(x3 ) r(x 4 )... r(x n ) R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 40 / 61

Basic Distribution every Pos on finite set can be represented by n-tuple m = (m 1,..., m n ) for some finite n IN where m i = m(a i ) for all i IN n clearly, n i=1 m i = 1 and m i [0, 1] for all i IN n m is called basic distribution each m represents exactly one possibility distribution r x i X : r i = r(x i ) = Pos({x i }) = Pl({x i }) n n i IN n : r i = Pl({x i }) = m(a k ) = k=i solving all n equations for m i (i IN n ) leads to m i = r i r i+1 one-two-one correspondence between r and m R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 41 / 61 k=i m k

Example one possibility measure defined on X with A 1 A 2... A 7 m(a 6 ) =.1 m(a 3 ) =.4 m(a 7 ) =.2 m(a 2 ) =.3 x 1 x 2 x 3 x 4 x 5 x 6 x 7 r(x 1 ) = 1 r(x 7 ) =.2 r(x 2 ) = 1 r(x 6 ) =.3 r(x 3 ) =.7 r(x 5 ) =.3 r(x 4 ) =.3 n i=1 m(a i ) = 1 is satisfied, m(a 1 ) = m(a 4 ) = m(a 5 ) = 0 R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 42 / 61

Perfect Evidence and Total Ignorance smallest possibility distribution ř has form ř = (1, 0, 0,..., 0) ř represents perfect evidence with no uncertainty involved largest possibility distribution ˆr can be written as ˆr = (0, 0,..., 0, 1) ˆr denotes total ignorance, i.e., no relevant evidence is available the larger r, the less specific the evidence is (the more ignorant) R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 43 / 61

Joint Possibility Distribution consider joint possibility distributions r defined on X Y projections r X, r Y are called marginal possibility distribution formula for r X (x) follows from x X : r X (x) = max r(x, y) y Y y Y : r Y (y) = max r(x, y) x X Pos X ({x}) = Pos({(x, y) x Y }) Pos X ({x}) = r X (x) Pos({(x, y) x Y }) = max r(x, y) y Y same argumentation can be done for r Y R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 44 / 61

Possibilistic Noninteraction Definition Nested bodies of evidence X and Y represented by r X and r Y, resp., are called noninteractive if and only if for all x X and all y Y r(x, y) = min{r X (x), r Y (y)} recall product rule m(a B) = m X (A) m Y (B) this definition is not based upon product rule of evidence theory does not conform to general noninteractive bodies of evidence here, product rule would not preserve nested structure R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 45 / 61

Possibilistic Noninteraction let r X and r Y be marginal poss. dist. functions on X and Y, resp. let r be joint poss. dist. function on X Y defined by r X and r Y assume Pos X, Pos Y and Pos correspond to r X, r Y and r, then Pos(A B) = min{pos X (A), Pos Y (B)} for all A P(X) and all B P(Y) where Pos X (A) = max r X(x), x A Pos Y (B) = max r Y(y), y B Pos(A B) = max r(x, y) x A,y B R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 46 / 61

Possibilistic Independence two marginal possibilistic bodies of evidence are independent if and only if conditional possibilities equal marginal possibilities, i.e., for all x X and all y Y r X Y (x y) = r X (x), r Y X (y x) = r Y (y) r X Y (x y) and r Y X (y x) are conditional possibilities on X Y possibilistic independence is stronger than poss. noninteraction possibilistic independence possibilistic noninteraction possibilistic noninteraction possibilistic independence R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 47 / 61

Fuzzy Sets and Possibility Theory possibility theory can be also formulated using fuzzy sets instead of nested bodies of evidence [Dubois and Prade, 1993] fuzzy sets (similar to bodies of evidence) are based on families of nested sets, i.e., α-cuts Pos measures are connected with fuzzy sets via possibility distribution function e.g., let X denote variable taking values in set X X = x describes X is x where x X fuzzy set F on X expresses constraint on values assigned to X given value x X, F(x) is degree of compatibility of x described by F R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 48 / 61

Fuzzy Sets and Possibility Theory given X is F, F(x) is degree of possibility that X = x i.e., given F on X and X is F, x X : r F (x) = F(x) given r F, associated possibility measure is A P(X) : Pos F (A) = sup r F (x) x A for normal fuzzy sets, Nec F can be calculated by A P(X) : Nec F (A) = 1 Pos F (A C ) R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 49 / 61

Example let variable T be temperature in C (only integers) actual value is given in terms of T is around 21 C 1.00 0.75 0.50 0.25 0 F = r F 17 18 19 20 21 22 23 24 25 19 20 21 22 23 r F : 1/3 2/3 1 2/3 1/3 incomplete information induces possibility distribution function r F r F is numerically identical with membership function F nested α-cuts play same role as focal elements Question: Which values do Nec and m take here? R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 50 / 61

Summary if r F is derived from normal fuzzy set F, then both formulations of possibility theory are equivalent full equivalence breaks when F is not normal basic probability assignment function m is not directly applicable all other properties remain equivalent in both formulations possibility theory is measure-theoretic counterpart of fuzzy set theory based upon standard fuzzy operations it provides tools for processing incomplete information expressed by fuzzy propositions it plays major role in fuzzy logic and approximate reasoning R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 51 / 61

Possibility versus Probability probability theory and possibility theory are distinct to show that, let us examine both theories by evidence theory probability theory and possibility theory are special branches of it introduce probabilistic counterparts of possibilistic characteristics probability measure Pro must satisfy Pro(A B) = Pro(A)+Pro(B) for all A, B P(X) s.t. A B = Pro is special type of belief measure R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 52 / 61

Probability Distribution Function Theorem A belief measure Bel on a finite power set P(X) is a probability measure if and only if the associated basic probability assignment function m is given by m({x}) = Bel({x}) and m(a) = 0 for all subsets of X that are no singletons. probability measures on finite sets are thus fully represented by p : X [0, 1] s.t. p(x) = m({x}) p is called probability distribution function let p = (p(x) x X) be probability distribution on X R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 53 / 61

Probability Measure recall definition of belief and plausibility Bel(A) = m(b) and Pl(A) = m(b) B B A B A B if basic probability assignment focuses only on singletons, then A P(X) : Bel(A) = Pl(A) = m({x}) x A Bel and Pos are equal it is convenient to use one symbol Pro A P(X) : Pro(A) = x A p(x) Pro is called probability measure R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 54 / 61

Comparison of both Theories Property Probability Theory Possibility Theory measures probability Pro possibility Pos, necessity Nec body of evidence singletons family of nested subsets representation p : X [0, 1] via r : X [0, 1] via Pro(A) = x A p(x) Pos(A) = max x A r(x) normalization x X p(x) = 1 max x X r(x) = 1 total ignorance x X : p(x) = 1 X x X : r(x) = 1 noninteraction p(x, y) = p X (x) p Y (y) r(x, y) = min{r X (x), r Y (y)} independence p X Y (x y) = p X (x) r X Y (x y) = r X (x) p Y X (y x) = p Y (y) r Y X (y x) = r Y (y) independence noninteraction independence noninteraction R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 55 / 61

Comparison of both Theories Probability Theory Possibility Theory Pro(A B) = Pro(A)+Pro(B) Pro(A B) Pos(A B) = max{pos(a), Pos(B)} Nec(A B) = min{nec(a), Nec(B)} not applicable Nec(A) = 1 Pos(A C ) Pos(A) < 1 Nec(A) = 0 Nec(A) > 0 Pos(A) = 1 Pro(A)+Pro(A C ) = 1 Pos(A)+Pos(A C ) 1 Nec(A)+Nec(A C ) 1 max{pos(a), Pos(A C )} = 1 min{nec(a), Nec(A C )} = 0 R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 56 / 61

FUZZY MEASURES: monotonic and continuous or semi continuous PLAUSIBILITY MEASURES: subadditive and continuous from below PROBABILITY MEASURES: additive POS. MEASURES BELIEF MEASURES: superadditive and continuous from above NEC. MEASURES R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 57 / 61

Discussion possibility, necessity and probability measures do not overlap except for one special measure characterized by only one focal element (i.e., singleton) distribution functions of possibility and probability become equal one element of universal set is assigned 1, all others assigned 0 this measure represents perfect evidence R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 58 / 61

Discussion differences in mathematical properties make each theory suitable for modeling certain types of uncertainty probability theory is ideal for formalizing uncertainty in situations where class frequencies are known or evidence is based on outcomes of many independent random experiments possibility theory is ideal for formalizing incomplete information expressed by fuzzy propositions necessity and possibility can be seen as lower and upper probabilities [Dempster, 1967] R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 59 / 61

Interpretations of Possibility Theory based on similarity possibility r(x) reflects degree of similarity between x and prototype x i having r(x i ) = 1 i.e., r(x) is expressed by distance between x and x i the closer x to x i, the more possible its interpretation closeness comes from measurement or subjective judgement based on preference due to ordering Pos defined on P(X) A, B P(X), A Pos B means B is at least as possible as A A Pos B A C Nec B C R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 60 / 61

Literature for the Lecture Choquet, G. (1954). Theory of capacities. Annales de l Institut Fourier, 5:131 295. Dempster, A. P. (1967). Upper and lower probabilities induced by a multivalued mapping. The Annals of Mathematical Statistics, 38(2):325 339. Dubois, D. and Prade, H. (1993). Fuzzy sets and probability: Misunderstanding, bridges and gaps. In Proceedings of the Second IEEE International Conference on Fuzzy Systems, pages 1059 1068, San Francisco, CA, USA. Shafer, G. (1976). A Mathematical Theory of Evidence. Princeton University Press. R. Kruse, C. Moewes Fuzzy Systems Possibility Theory 2009/12/16 61 / 61