The Internal Conflict of a Belief Function

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1 The Internal Conflict of a Belief Function Johan Schubert Abstract In this paper we define and derive an internal conflict of a belief function We decopose the belief function in question into a set of generalized siple support functions (GSSFs) Reoving the single GSSF supporting the epty set we obtain the base of the belief function as the reaining GSSFs Cobining all GSSFs of the base set, we obtain a base belief function by definition We define the conflict in Depster s rule of the cobination of the base set as the internal conflict of the belief function Previously the conflict of Depster s rule has been used as a distance easure only between consonant belief functions on a conceptual level odeling the disagreeent between two sources Using the internal conflict of a belief function we are able to extend this also to non-consonant belief functions 1 Introduction In this paper we define and derive an internal conflict of a belief function within Depster-Shafer theory [1 3, 14] We decopose the belief function in question into a set of generalized siple support functions (GSSFs) Reoving the single GSSF supporting the epty set we obtain the base of the belief function as the reaining GSSFs Cobining all GSSFs of the base set, we obtain a base belief function by definition We define the conflict in Depster s rule of this cobination as the internal conflict of the belief function We propose that the base belief function is a better easure than the original belief function which can be obtained by cobining the base belief function with pure conflict, ie, {[ 1, ], [ 1( Θ), Θ ]} There are several different ways to anage a high conflict in cobination of belief functions within Depster-Shafer theory For an overview of different Johan Schubert Departent of Decision Support Systes, Division of Inforation and Aeronautical Systes, Swedish Defence Research Agency, SE Stockhol, Sweden e-ail: schubert@foise, This work was supported by the FOI research project Real-tie Siulation Supporting Effects-based Planning, which is funded by the R&D prograe of the Swedish Ared Forces T Denœux & M-H Masson (Eds): Belief Functions: Theory & Appl, AISC 164, pp springerlinkco Springer-Verlag Berlin Heidelberg 01

2 170 J Schubert alternatives to anage the cobination of conflicting belief functions, see articles by Sets [16] and Liu [7] For a recent survey of alternative distance between belief functions, see Joussele and Maupin [5] In section we review a ethod for decoposing a belief function into a set of GSSFs [15] In section 3 we derive the base set of a belief function and construct a base belief function fro the base set corresponding to the belief function under decoposition In section 4 we derive the internal conflict of the belief function and show how this extends the conflict fro being a distance easure only for consonant belief functions to a functioning distance easure also between non-consonant belief functions In section 5 we provide an exaple Finally, conclusions are drawn (section 6) Decoposing a Belief Function All belief functions can be decoposed into a set of GSSFs on a frae of discernent Θ using the ethod developed by Sets [15] A GSSF is either a traditional siple support function (SSF) [14] or an inverse siple support function (ISSF) [15] Let us begin by defining an ISSF: Definition 1 An inverse siple support function on a frae of discernent Θ is a function : Θ (, ) characterized by a weight w ( 1, ) and a focal eleent A Θ, such that ( Θ) w, A ( ) 1 w and X ( ) 0 when X { A, Θ } Let us recall the eaning of SSFs and ISSFs [15]: An SSF 1 ( A) [ 0, 1] represents a state of belief that You have soe reason to believe that the actual world is in A (and nothing ore) An ISSF ( ) (,0) A on the other hand, represents a state of belief that You have soe reason not to believe that the actual world is in A Note that not believing in A is different than believing in A c A siple exaple is one SSF 1 (A) 1/ 4 and ( ) 3/4 1 Θ, and one ISSF (A) 1/ 3 and ( ) 4/3 Θ Cobining these two functions yields a vacuous belief function 1 ( Θ) 1 The decoposition ethod is perfored in two steps eqs (1) and () First, for any non-dogatic belief function Bel 0, ie, where 0 ( Θ) > 0, calculate the coonality nuber for all focal eleents A by eq (1) We have Q0( A) 0( B) (1) B A For dogatic belief functions assign 0 ( Θ) ε > 0 and discount all other focal eleents proportionally Secondly, calculate i ( C) for all decoposed GSSFs, where C Θ including C, and i is the ith GSSF There will be one GSSF for each subset C

3 The Internal Conflict of a Belief Function 171 of the frae unless i ( C) happens to be zero In the general case we will have Θ GSSFs We get for all C Θ including C i ( C) 1 Q 0 ( A) 1 A C i ( Θ) 1 i ( C) ( ) A C + 1 () Θ where i [1, 1] Here, C Θ of is the ith subset of Θ in nuerical order i ( C) which also includes C, ie, {[ 1( ), ], [ 1( ), ]} of eq () Θ Θ is the first decoposed GSSF 3 Transforing a Belief Function into a Base Belief Function Using eqs (1) and () we ay decopose a belief function 0 into a set of GSSFs We call the non-conflict GSSFs of the decoposition the base of 0 Definition The base of a belief function 0 is the set of decoposed siple support and inverse siple support function { } Θ 1 i i (3) deliberately excluding 1 that supports only {, Θ }, where { } Θ 1 i i 1 (4) is the full set of Θ 1 siple support and inverse siple support function fro the decoposition of 0 by eqs (1) and () Definition 3 A base belief function of a belief function 0 is the belief function resulting fro the unnoralized cobination of the base of 0, ie, Definition 4 A base conflict of a belief function 0 is the obtained conflict of the first GSSF that supports {[ 1( ), ], [ 1( ), ]} { } i Θ 1 i Θ Θ of the decoposition of a belief function of 0 by eqs (1) and () (5) 1 With Θ {, abc, } the nuerical order of all subsets in Θ including is Θ {, {a}, {b}, {a, b}, {c}, {a, c}, {b, c}, {a, b, c}}

4 17 J Schubert Theore 1 A belief function 0 can be recovered by cobination of its corresponding base belief function with the base conflict 1, ie, 0 1 (6) Proof Iediate by definition and 3 Ñ 4 The Internal Conflict of a Belief Function The conflict received fro a cobination of belief functions by Depster s rule is not a easure of dissiilarity between the cobined belief functions Indeed, belief functions can be quite different and yet have zero conflict as intersection of their focal eleents are non-epty Instead the conflict of Depster s rule is best viewed as a different kind of distance easure; a easure of conceptual disagreeent between sources When they disagree highly it is a sign that soething is wrong It should be noted that there is at least two possible sources of conflict other than easureent errors We ay have odeling errors or faulty sources [4] Faulty sources are corrected by appropriate discounting (eg, [6, 1, 16]) while odeling errors are corrected by adopting an appropriate frae of discernent [13, 14] In this section we define and investigate an internal conflict of a belief function We further devise a way to obtain the internal conflict Definition 5 The internal conflict of a belief function 0 is the conflict received in the unnoralized cobination of the base of 0 to obtain the base belief function, ie, where { i } Θ 1 i (7) For siplicity, view the intersection of eq (7) as taking place in a Θ hyper cube of all Θ GSSFs Note that can take both positive and negative values Theore The internal conflict within a base belief function is strictly a function of conflicts between different GSSFs supporting subsets of the frae Proof Iediate by observation of the cobination in eq (7) as 1 with body of,, ( Θ), Θ is not included in the cobination Ñ evidence {[ 1 ] [ 1 ]} Theore 3 There exist an infinite size faily of unnoralized belief functions p { 0 p 1 (, ), p 1 } with an identical base belief function and identical internal conflict

5 The Internal Conflict of a Belief Function 173 Proof Let us generate a faily fro the base: Take any belief function 0 on a frae Θ of size n Θ Decopose 0 into its GSSFs Cobine { } n n i i using eq (7) into the base belief function Let us ignore the value obtained for 1 in the decoposition Instead, let 1 (, ), 1 1 The p faily of belief functions { 0 } is generated by cobining the base belief function with each of the { } The faily is of infinite size Ñ 1 When going in the other direction fro faily to base: Note, that in the special case of a noralized base belief function it can be recovered fro any faily eber by noralization If we cobine a non-consonant belief function with itself we should not be surprised that we receive a conflict A non-consonant belief function can be expressed as a construct fro the base set of that belief function If the belief function cobined with itself is constructed in two steps by first cobining all GSSFs pairwise with theselves, these Θ cobinations of GSSFs with identical focal sets are conflict free (excluding and Θ) Cobining the resulting Θ GSSFs in the second step obviously has epty intersections aong their focal eleents resulting in conflict Thus, the internal conflict received is a function of conflicts fro different GSSFs (excluding 1 ) that are used to construct the non-consonant belief function Thus, the conflict noticed in the cobination exists internally within non-consonant belief functions before cobination and is a consequence of the scattering of ass within the distribution This akes the internal conflict appropriate as a conceptual distance easure also between non-consonant belief functions as it easures the internal conflict in the cobination of GSSFs fro two different base sets corresponding to the two different base belief functions without the added pure conflict of 1 (supporting only and Θ) that is always included in the conflict obtained by Depster s rule Thus, fro theore and 3 follows that the internal conflict is an appropriate distance easure for all belief functions as it excludes the pure conflict of 1 (ie, also for non-consonant belief functions), where this distance easure sought after is a easure of conceptual disagreeent between sources When calculating the conceptual distance easure based on internal conflict between two belief functions we first transfor the two belief functions 0 and 0 to their base belief function for using eqs (1) and () to find the base set, this is followed by eq (5) to construct the base belief function, and We perfor a conjunctive cobination and find the internal conflict of the resulting base belief function using eq (7)

6 174 J Schubert This easure of internal conflict is the ost objective conflict easure since it excludes pure conflict and is iune to noralizations and incoing belief functions fro sources without any inforation on conflicts in earlier cobinations In the proble of partitioning ixed-up belief functions into subsets that correspond to different subprobles [8 11] we ay use the distance easure of internal conflict for all belief functions (ie, also for non-consonant belief functions) 5 An Exaple Let us study a siple exaple In this exaple we will represent all belief functions using nuerical ordering of focal eleents We assue a frae of discernent Θ { a, b} and a belief function 0 that is build up by cobination of two SSFs and 3 that are yet unknown to us, where We have, [ 0, 04, 0, 06 ], [ 0, 0, 04, 06] (8) [ 016, 04, 04, 036] (9) Using eqs (1) and (), 0 can be decoposed into the base SSFs and 3 Here, the base conflict is 0, ie, 1 in the decoposition of 0 is a vacuous SSF; 1 (Θ) 1 Fro the base set eq (8) we can construct the base belief function which in this case is identical to the belief function 0 If 0 is noralized then the situation is different Let us call this noralization We have, [ ] 0, 0857, 0857, 0486 (10) Decoposing we get [ ] n [ ] [ 0, 0, 04, 06] 1n 3n 01905, 0, 0, 11905, 0, 04, 0, 06, (11) which is the sae base for as in the decoposition of 0 Thus, 0 and has the sae base belief function, which is 0 However, we obtain an inverse base conflict of 1n when decoposing copared to 1 (Θ) 1 in the decoposition of 0 Furtherore, let us assue that we have a second belief function which is build up by cobination of two SSFs and 3 that are also unknown to us, where [ ] [ ] 0, 03, 0, 07, 0, 0, 03, 07 (1) 3 0

7 The Internal Conflict of a Belief Function 175 We have 0 3 [ 9, 01, 01, 049] (13) As above, using eqs (1) and () 0 can be decoposed into the base and 3 ( 1 is vacuous) Using eq (5) we construct the base belief function Finally, given both and we cobine the to obtain [ 03364, 0436, 0436, 01764] (14) We notice a conflict of in the cobination of and Assuing instead that we receive fro a source (let us then call it ) we can decopose it to obtain a pure base without any base conflict; [ ] [ ] [ 0, 0 ] 0, 0, 0, 1, 0, 058, 0, 04, 1 3 3,058,04 We should notice that the two base SSFs and 3 are theselves conflict free cobinations and 3 3 resulting in and 3, respectively Recobining and 3 yields a base belief function identical to 0 Thus, the conflict of 0 and the internal conflict of are identical in this case Had 0 been noralized the situation is soewhat different Let us call the noralization We have, It can be decoposed into [ ] (15) 0, 03671, 03671, 0658 (16) [ ] n [ ] [ 0, 0, 058, 04 ] 1n 3n , 0, 0, 1308, 0, 058, 0, 04, We observe that 0 and have the sae base set in that n and 3 3n Finally, let us study a cobination of a belief function with itself We cobine with itself We have, (17) [ 0695, 0399, 0399, 706] (18)

8 176 J Schubert where belief function Decoposing 0695 we get, is the internal conflict distance easure between the two [ ] n [ ] [ 0, 0, 0836, ] 1n 3n 1711, 0, 0, 711 0, 0836, 0, 01764, (19) As before we have a base set of two SSFs Here the base set of is and 3n, where n n n n, 3n 3n 3n (0) 6 Conclusions We conclude that the internal conflict of a non-consonant belief function is actually a function of conflicts between different GSSFs in the base set of that belief function Here all GSSFs that have identical focal eleents have zero conflict Thus, the internal conflict between two belief functions is an appropriate distance easure on a conceptual level that easures disagreeent between sources References 1 Depster, AP: Upper and lower probabilities induced by a ultiple valued apping The Annals of Matheatical Statistics 38, (1967) Depster, AP: A generalization of Bayesian inference Journal of the Royal Statistical Society B 30, (1968) 3 Depster, AP: The Depster-Shafer calculus for statisticians International Journal of Approxiate Reasoning 48, (8) 4 Haenni, R: Shedding new light on Zadeh s criticis of Depster s rule of cobination In: Proceedings of the Seventh International Conference on Inforation Fusion, pp (5) 5 Joussele, A-L, Maupin, P: Distances in evidence theory: Coprehensive survey and generalizations International Journal of Approxiate Reasoning 53, (01) 6 Klein, J, Colot, O: Autoatic discounting rate coputation using a dissent criterion In: Proceedings of the Workshop on the Theory of Belief Functions, pp 1 6 (paper 14) (010) 7 Liu, W: Analyzing the degree of conflict aong belief functions Artificial Intelligence 170, (6) 8 Schubert, J: On nonspecific evidence International Journal of Intelligent Systes 8, (1993) 9 Schubert, J: Clustering belief functions based on attracting and conflicting etalevel evidence using Potts spin ean field theory Inforation Fusion 5, (4)

9 The Internal Conflict of a Belief Function Schubert, J: Managing decoposed belief functions In: Bouchon-Meunier, B, Marsala, C, Rifqi, M, Yager, RR (eds) Uncertainty and Intelligent Inforation Systes, pp World Scientific Publishing Copany, Singapore (8) 11 Schubert, J: Clustering decoposed belief functions using generalized weights of conflict International Journal of Approxiate Reasoning 48, (8) 1 Schubert, J: Conflict anageent in Depster-Shafer theory using the degree of falsity International Journal of Approxiate Reasoning 5, (011) 13 Schubert, J: Constructing and evaluating alternative fraes of discernent International Journal of Approxiate Reasoning 53, (01) 14 Shafer, G: A Matheatical Theory of Evidence Princeton University Press, Princeton (1976) 15 Sets, P: The canonical decoposition of a weighted belief In: Proceedings of the 14th International Joint Conference on Artificial Intelligence, pp (1995) 16 Sets, P: Analyzing the cobination of conflicting belief functions Inforation Fusion 8, (7)

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