Equivalence Between Belief Theories and Naïve Bayesian Fusion for Systems with Independent Evidential Data: Part II, The Example

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1 Equivalence Between Belief Theories and Naïve Bayesian Fusion for ystems with Independent Evidential Data: Part II, The Example John J. udano Lockheed artin oorestown, NJ, 08057, U john.j.sudano@lmco.com bstract The process of fusing multiple independent sensor measurements, communication link data from other independent systems, and dynamic data base information is essential to support critical decisions in a timely way. any real systems can be mapped to such a process. The independence of the input evidential data with an equal probable uniform prior probability distribution (i.e., Naïve Bayesian fusion greatly simplifies the mathematical techniques used to properly fuse the evidential data. Equivalence between Belief Fusion and Naïve Bayesian is shown for this process. The equivalence comparison is done in probability space. The title of a 2001 colloquium, Data Fusion & Target ID: Dempster-hafer & Probability Theories Holy War, depicts the state of mind of many researchers. The goal of this article is to show that large areas from both mathematical camps are equivalent. This equivalence can be exploited by reducing the computational complexity of the fusion process. The fusion can be done in the linear probability set space rather than the exponential power-set representation of the belief space. For a system with 10 possible hypotheses, The fusion of independent data in belief space would involve the fusion of as many as 1024 members of the power set, while exactly the same results can be obtained by fusion of 10 members in probability space. This implies a non-trivial saving in computation complexity for the implementation of many real systems, such as medical diagnostic systems, automated cognitive performance evaluation, oil exploration systems, combat identification, ballistic missile component discrimination, and semi-automated homeland security systems. The numerical examples in this article help clarify the mathematical techniques and confirm the equivalence results. 1 Introduction In the article Equivalence Between Belief Theories and Naïve Bayesian Fusion for ystems with Independent Evidential Data: Part I, The Theory the author describes the equivalence between two mathematical techniques. In this article, numerical examples are calculated showing the equivalence. Let Ω {, B, C} be the set of 3 possible outcomes with the following notation for the power set. ~[1,0,0] B~[0,1,0] C~[0,0,1] B~[1,1,0] C~[1,0,1] BC~[0,1,1] BC~[1,1,1] Four independent probability distributions are calculated from a weighted random process and sorted. PD , , < PD , , < PD , , < PD , , < (1 From the above probability distributions, calculate the BBs with the Inverse Pignistic Probability Transform (IPPT using the Generalized um ean [11] for the values of st1. The mapping has the property that the pignistic probability proportional to belief is equal to the original probability distribution; i.e., PrBl(Pd(. Generally this is not the case. The BBs generated thus can be used by all pignistic probability transforms to estimate probabilities. Therefore, proper comparisons can be made for all combinations of belief theories and Naive Bayesian Fusion. pecial care must be exercised in using PrBl since it can give erroneous results for non-mature data sets. For a given probability set P( i with IPPT values of s1 and t1, the basic belief assignments are calculated as: mm( i 1,1 P( i / D mm( i, j 1,1 ( P( i + P( j /( 2D (2 mm(i, j,k 1,1 ( P( i + P( j + P( k /( 3D mm(1,2,...n 1,1 ( P( 1 + P( P( N /( N * D D is calculated analytically as N D (2-1/N. For each PDi the calculated the BBs are: mm1@1, 0, 0D mm1@0, 1, 0D mm1@0, 0, 1D mm1@1, 1, 0D mm1@1, 0, 1D mm1@1, 1, 1D mm1@0, 1, 1D (3 mm2@1, 0, 0D mm2@0, 1, 0D mm2@0, 0, 1D mm2@1, 1, 0D mm2@1, 0, 1D mm2@1, 1, 1D mm2@0, 1, 1D (4

2 0, 0D , 0D , 1D , 0D , 1D , 1D , 1D (5 0, 0D , 0D , 1D , 0D , 1D , 1D , 1D (6 The singleton Plausibilities [9] for each set of BBs are calculated as: Plmmk ( J mmk ( K K J 0 (7 Plmm1 { , , } Plmm2 { , , } Plmm3 { , , } Plmm4 { , , } (8 The pignistic probability proportional to normalized plausibility (PrNPl is computed for each singleton element of C Ω with C Ω for all 2. 1 PrNPl k ( C mmk ( C 0 (9 PrNPl , , < PrNPl , , < PrNPl , , < PrNPl , , < (10 mets Pignistic probability for each set of BBs is calculated as: BetPmmk ( i i mmk ( (11 BetPmm1 { , , } BetPmm2 { , , } BetPmm3 { , , } BetPmm4 { , , } (12 The Pignistic Probability transforms proportional to Plausibilities (PrPl are calculated for each set of BBs PrPlk ( i C i Pl ( i [ Pl ( mmk ( ] (13 PrPl1@1, 0, 0D PrPl2@1, 0, 0D PrPl1@0, 1, 0D PrPl2@0, 1, 0D PrPl1@0, 0, 1D PrPl2@0, 0, 1D (14 PrPl3@1, 0, 0D PrPl4@1, 0, 0D PrPl3@0, 1, 0D PrPl4@0, 1, 0D PrPl3@0, 0, 1D PrPl4@0, 0, 1D (15 The pignistic probability proportional to all Plausibilities (PraPl is equal to the belief and a component proportional to the sum of all the plausibilities PraPl ( i Bel( i + ε Pl( i with 1- Bel( i i Ω ε Pl( i i Ω (16 PraPl1 { , , } PraPl2 { , , } PraPl3 { , , } PraPl4 { , , } (17 The Hybrid Pignistic Probability (PrHyb [10] distributes the BBs proportionally to PraPl. PraPl ( i PrHybk ( i mmk ( C [PraPl ( ] i (18 PrHyb1@1, 0, 0D PrHyb2@1, 0, 0D PrHyb1@0, 1, 0D PrHyb2@0, 1, 0D PrHyb1@0, 0, 1D PrHyb2@0, 0, 1D (19 PrHyb3@1, 0, 0D PrHyb4@1, 0, 0D PrHyb3@0, 1, 0D PrHyb4@0, 1, 0D PrHyb3@0, 0, 1D PrHyb4@0, 0, 1D (20 The Pignistic Probability transforms proportional to Beliefs (PrBl are calculated for each set of BBs. Note PrBl is equal to the original probability distribution [9]. PrBlmmk ( i C i mmk ( i [ mmk ( mmk ( ] (21 PrBlmm1@1, 0, 0D PrBlmm2@1, 0, 0D PrBlmm1@0, 1, 0D PrBlmm2@0, 1, 0D PrBlmm1@0, 0, 1D PrBlmm2@0, 0, 1D (22 PrBlmm3@1, 0, 0D PrBlmm4@1, 0, 0D PrBlmm3@0, 1, 0D PrBlmm4@0, 1, 0D PrBlmm3@0, 0, 1D PrBlmm4@0, 0, 1D (23 2 Dempster-hafer (D Belief Fusion Combining two BBs by using Dempster s rule of combination yields the fused BB. mm1( mm2( C mm12 ( (24 B C mm12@1, 0, 0D mm12@0, 1, 0D mm12@0, 0, 1D mm12@1, 1, 0D mm12@1, 0, 1D mm12@0, 1, 1D mm12@1, 1, 1D (25 From above BBs, compute the pignistic probability PrNPl. PrNPl12 { , , } (26 Combining the fused BBs with the third belief data input by using Dempster s rule of combination yields the fused BB. mm3( C mm123 ( (27 B C

3 0, 0DD , 0DD , 1DD , 0DD , 1DD , 1DD , 1DD (28 From above BBs, compute PrNPl. PrNPl , , < (29 Combining the fused BBs with the fourth belief data input by using Dempster s rule of combination yields the fused BB. mm1234@1, 0, 0D mm1234@0, 1, 0D mm1234@0, 0, 1D mm1234@1, 1, 0D mm1234@1, 0, 1D mm1234@0, 1, 1D mm1234@1, 1, 1D (30 From above BBs, compute PrNPl. PrNPl , , < (31 3 Naïve Bayesian Fusion of the Four PrNPl Probability Distributions The Naïve Bayesian fusions of the four Pignistic Probability proportional to Plausibilities (PrNPl are calculated as: PrNPl1( DPrNPl2( D...PrNPlN( D BPrNPl12...N(D PrNPl1(...PrNPlN( (32 BPrNPl , , < BPrNPl , , < BPrNPl , , < BPrNPl , , < (33 Note the equivalence between the Pignistic Probability estimate PrNPl of the D fusion (26, (29, (31 and the probability of the Naïve Bayesian fusions as computed using BPrNPl (33 of the original BBs. 4 The Fixsen-ahler odified Dempster- hafer (D Belief Fusion Combining two BBs by using the D combination rule gives the fused BBs. mm1( mm2( C B C B C (34 mm12@1, 0, 0D mm12@0, 1, 0D mm12@0, 0, 1D mm12@1, 1, 0D mm12@1, 0, 1D mm12@0, 1, 1D mm12@1, 1, 1D (35 From above BBs, compute mets Pignistic probabilities: BetP12@1, 0, 0D BetP12@0, 1, 0D BetP12@0, 0, 1D (36 Combining the next input with the fused BBs by using D combination rule gives the fused BBs. mm123( mm3( C B C B C (37 mm123@1, 0, 0D mm123@0, 1, 0D mm123@0, 0, 1D mm123@1, 1, 0D mm123@1, 0, 1D mm123@0, 1, 1D mm123@1, 1, 1D (38 From above BBs compute mets Pignistic probabilities: BetP123@1, 0, 0D BetP123@0, 1, 0D BetP123@0, 0, 1D (39 Combining all four inputs by using the D combination rule gives the fused BBs. mm1234@1, 0, 0D mm1234@0, 1, 0D mm1234@0, 0, 1D mm1234@1, 1, 0D mm1234@1, 0, 1D mm1234@0, 1, 1D mm1234@1, 1, 1D (40 From above BBs, compute mets Pignistic probabilities: BetP1234@1, 0, 0D BetP1234@0, 1, 0D BetP1234@0, 0, 1D (41 5 The udano Generalized Belief Fusion with Cardinality Weighting The Generalized Belief Fusion is calculated with cardinality weighting: B C mm1( mm2( C B C (42 mm12@1, 0, 0D mm12@0, 1, 0D mm12@0, 0, 1D mm12@1, 1, 0D mm12@1, 0, 1D mm12@0, 1, 1D mm12@1, 1, 1D (43 From above BBs compute the pedigree Pignistic probabilities with cardinality weighting: PrPed12@1, 0, 0D PrPed12@0, 1, 0D PrPed12@0, 0, 1D (44 Fusing the first three inputs: mm123@@1, 0, 0DD mm123@@0, 1, 0DD mm123@@0, 0, 1DD mm123@@1, 1, 0DD mm123@@1, 0, 1DD mm123@@0, 1, 1DD mm123@@1, 1, 1DD (45 From above BBs, compute the pedigree Pignistic probabilities with cardinality weighting: PrPed123@1, 0, 0D PrPed123@0, 1, 0D PrPed123@0, 0, 1D (46

4 Fusing all four inputs: 0, 0D , 0D , 1D , 0D , 1D , 1D , 1D (47 From above BBs, compute the pedigree Pignistic probabilities: 0, 0D , 0D , 1D (48 6 Naïve Bayesian Fusion of the mets Pignistic and udano Pedigree Pignistic Probabilities with Cardinality weighting The Naïve Bayesian fusions of the mets Pignistic Probability BetP (12 and Pedigree Pignistic Probabilities with Cardinality weighting are calculated as: BetP1( D mm1( D 0 mm1( D (49 BetP1( DBetP2( D...BetPN( D BBetP12...N(D BetP1(...BetPN( (50 BBetP , , < BBetP , , < BBetP , , < (51 The Pedigree Pignistic Probability for a single set of BBs with cardinality weighting is calculated as: PrPed1( C PrPed1( C C C mm1( C [ ρ 1( C] C [ ρ1( ] mm1( C [ C ] C [ ] ince C is a singleton its cardinality value is one. PrPed1( C C mm1( (52 (53 (54 Note that the Pedigree Pignistic probability with cardinality weighting (54 is the same as mets Pignistic Probability (49 The Naïve Bayesian fusions of the Pedigree Pignistic Probability with cardinality weighting are calculated as: PrPed1( DPrPed2( D...PrPedN( D BPrPed12...N(D PrPed1(...PrPedN( BPrPed , , < BPrPed , , < BPrPed , , < (56 (55 Note the equivalence between the mets Pignistic Probability estimate BetP of the D fusion (36, (39, (41 and the udano Pedigree Pignistic probabilities with cardinality weighting of the GBF fusion (44, (46, and (48, and the probability of the Naïve Bayesian fusions as computed using BBetP (51 and BPrPed (56. 7 The udano Generalized Belief Fusion with Pl Weighting Compute the Generalized Belief Fusion lgorithm with the Pl weighting function. (57 mm1( mm2( C C [Pl( ] C [Pl( ] B C C [Pl( ] C [Pl( C] mm12@1, 0, 0D mm12@0, 1, 0D mm12@0, 0, 1D mm12@1, 1, 0D mm12@1, 0, 1D mm12@0, 1, 1D mm12@1, 1, 1D (58 Compute the Pedigree Pignistic Probability with Pl weighting: PrPed ( C C C [Pl( C] C [Pl( C] C [Pl( ] C [Pl( ] (59 PrPed12@1, 0, 0D PrPed12@0, 1, 0D PrPed12@0, 0, 1D (60 Compute the Generalized Belief Fusion lgorithm with the Pl weighting function for the three inputs. mm123@1, 0, 0D mm123@0, 1, 0D mm123@0, 0, 1D mm123@1, 1, 0D mm123@1, 0, 1D mm123@0, 1, 1D mm123@1, 1, 1D (61 Compute the Pedigree Pignistic Probability with Pl weighting for the above BBs: PrPed123@1, 0, 0D PrPed123@0, 1, 0D PrPed123@0, 0, 1D (62 Compute the Generalized Belief Fusion lgorithm with the Pl weighting function for all four inputs. mm1234@1, 0, 0D mm1234@0, 1, 0D mm1234@0, 0, 1D mm1234@1, 1, 0D mm1234@1, 0, 1D mm1234@0, 1, 1D mm1234@1, 1, 1D (63 From the above BBs, compute the Pedigree Pignistic probabilities: PrPed1234@1, 0, 0D PrPed1234@0, 1, 0D PrPed1234@0, 0, 1D (64

5 8 Naïve Bayesian Fusion of the Pignistic Probability Proportional to Plausibility The Naïve Bayesian fusions of the Pignistic Probability proportional to Plausibilities (PrPl are calculated as: PrPl1( DPrPl2( D...PrPlN( D BPrPl12...N(D PrPl1(...PrPlN( B Ω BPrPl1 { , , } BPrPl12 { , , } BPrPl123 { , , } BPrPl1234 { , , } (66 Note the equivalence between the Pedigree Pignistic Probability with Pl weighting of the Generalized Belief Fusion lgorithm with the Pl weighting function (60, (62, (64 and the probability of the Naïve Bayesian fusions as computed using BPrPl (66. (65 9 The udano Generalized Belief Fusion with PraPl Weighting The probability proportionality function ρ has the PraPl weighting function in the Generalized Belief Fusion lgorithm. PraPl _ B C... Z mm 1( mm2 ( C C [PraPl( ] C [PraPl( ] C [PraPl( ] C [PraPl( C] (67 mm12@1, 0, 0D mm12@0, 1, 0D mm12@0, 0, 1D mm12@1, 1, 0D mm12@1, 0, 1D mm12@0, 1, 1D mm12@1, 1, 1D (68 From above BBs, compute the Pedigree Pignistic probabilities with the PraPl weighting function: PrPed12@1, 0, 0D PrPed12@0, 1, 0D PrPed12@0, 0, 1D (69 Fusing the three BBs inputs: mm123@1, 0, 0D mm123@0, 1, 0D mm123@0, 0, 1D mm123@1, 1, 0D mm123@1, 0, 1D mm123@0, 1, 1D mm123@1, 1, 1D (70 From above BBs, compute the Pedigree Pignistic probabilities with the PraPl weighting function: PrPed123@1, 0, 0D PrPed123@0, 1, 0D PrPed123@0, 0, 1D (71 Fusing all four inputs: mm1234@1, 0, 0D mm1234@0, 1, 0D mm1234@0, 0, 1D mm1234@1, 1, 0D mm1234@1, 0, 1D mm1234@0, 1, 1D mm1234@1, 1, 1D (72 From above BBs compute the Pedigree Pignistic probabilities with the PraPl weighting function: PrPed1234@1, 0, 0D PrPed1234@0, 1, 0D PrPed1234@0, 0, 1D (73 10 Naïve Bayesian Fusion of the Hybrid Pignistic Probability The Naïve Bayesian fusions of the Hybrid Pignistic Probability are calculated as: PrHyb1( DPrHyb2( D...PrHybN( D BPrHyb12...N(D PrHyb1(...PrHybN( (74 BPrHyb1 { , , } BPrHyb12 { , , } BPrHyb123 { , , } BPrHyb1234 { , , } (75 i y k { Note the equivalence between the udano Pedigree Pignistic Probability of the Generalized Belief Fusion lgorithm with PraPl weighting function (69, (71, (73 and the probability of Naïve Bayesian Fusion as computed using PrHyb ( The udano Generalized Belief Fusion with Belief Weighting The Generalized Belief Fusion with Belief as the weighting functions: B C mm ( mm ( C C [ Bel( ] C [ Bel( ] 1 2 C [ Bel( ] C [ Bel( C] (76 mm12@1, 0, 0D mm12@0, 1, 0D mm12@0, 0, 1D mm12@1, 1, 0D mm12@1, 0, 1D mm12@0, 1, 1D mm12@1, 1, 1D (77 From above BBs, compute the Pedigree Pignistic probabilities with the Bel weighting function: PrPed12@1, 0, 0D PrPed12@0, 1, 0D PrPed12@0, 0, 1D (78 Fusing the first three inputs: mm123@@1, 0, 0DD mm123@@0, 1, 0DD mm123@@0, 0, 1DD mm123@@1, 1, 0DD mm123@@1, 0, 1DD mm123@@0, 1, 1DD mm123@@1, 1, 1DD (79

6 From above BBs compute the Pedigree Pignistic probabilities with the Bel weighting function: 0, 0D , 0D , 1D (80 Fusing all four inputs: 0, 0D , 0D , 1D , 0D , 1D , 1D , 1D (81 Probability of Naïve Bayesian Fusion as computed using the original Probability Distributions PD ( Results Figure 1: hows the comparison of thirteen methods of fusing the same four independent belief measurements; the probability increase of the most probable state is shown. From above BBs compute the Pedigree Pignistic probabilities: 0, 0D , 0D , 1D ( Naïve Bayesian Fusion of the Pignistic Probability Proportional to Beliefs The Naïve Bayesian fusions of the Pedigree Pignistic probabilities with the Bel weighting function or the Pignistic Probability proportional to Belief (22,23 are calculated as: PrBlmm1( DPrBlmm2( D...PrBlmmN( D BPrBl12...N(D PrBlmm1(...PrBlmmN( i y k { (83 BPrBl , , < BPrBl , , < BPrBl , , < BPrBl , , < (84 13 Naïve Bayesian Fusion of the Original Four Probability Distributions The Naïve Bayesian fusions of the four original probability distributions (1 are calculated as: BPD12.. N( D PD1( D... PDN( D PD1(... PDN( B Ω Figure 1: GBF (Bel + PrPed(Bl BNF(PrBl BNF(PD GBF(PraPl + PrPed BNF(~PrHyb GBF(Pl + PrPed BNF(PrPl GBF(Cardinality + PrPed D+BetP BNF(PrPed(Cardinality BNF(BetP D-BF +PrNPl GBF(1 + PrPed BNF(PrNPl n example has been generated for fusing four independent belief data sources. Thirteen fusion comparisons for the same four input measurements have been computed. The thirteen fusion results fall into five unique classes. Figure 1 shows the thirteen fusion comparisons of the most probable hypothesis for the same four independent input measurements. The results fall into five unique curves: 2 1 BPD , , < BPD , , < BPD , , < BPD , , < (85 Note the equivalence between the udano Pedigree Pignistic Probability of the Generalized Belief Fusion lgorithm with Belief weighting function (78, (80, (82 and the Probability of Naïve Bayesian Fusion as computed using PrBl (84 and the Curve 1. hows the numerical equivalence between the Dempster-hafer belief fusion mapped via the pignistic probability estimate proportional to the normalized plausibility (PrNPl and the Naïve Bayesian fusion of the PrNPl computed for each input BB set. Curve 2. hows the numerical equivalence between (a the Fixsen-ahler odified Dempster-hafer (D Belief Fusion mapped via mets pignistic probability estimate

7 (BetP, (b the udano Generalized Belief Fusion with Cardinality Weighting mapped via the Pedigree Pignistic probabilities with cardinality weighting, and (c Naïve Bayesian fusion of the BetP computed for each input BB set. Curve 3. hows the numerical equivalence between the udano Generalized Belief Fusion with the Plausibility weighting mapped via the Pedigree Pignistic probabilities with Plausibility weighting, and the Naïve Bayesian fusion of the pignistic probability estimate proportional to the plausibility (PrPl computed for each input BB set. Curve 4. hows the numerical equivalence between the udano Generalized Belief Fusion with the pignistic probability proportional to all Plausibilities (PraPl weighting mapped via the Pedigree Pignistic probabilities with PraPl weighting, and the Naïve Bayesian fusion of the Hybrid Pignistic Probability computed for each input BB set. Curve 5. hows the numerical equivalence between the udano Generalized Belief Fusion with the pignistic probability proportional to Belief (Bel weighting mapped via the Pedigree Pignistic probabilities with Bel weighting, and the Naïve Bayesian fusion of the Pignistic Probability proportional to Belief computed for each input BB set. ince the BBs are computed by the Inverse Pignistic Probability Transform (IPPT using the Generalized um ean, so that PrBl(Pd(, then the of Naïve Bayesian Fusion as computed using the original Probability Distributions is also equivalent. 15 Conclusion The process of fusing multiple independent sensor measurements, communication link data from other independent systems, and dynamic data base information is essential to support critical decisions in a timely way. any real systems can be mapped to such a process. The independence of the input evidential data with an equal probable uniform prior probability distribution (i.e., Naïve Bayesian fusion greatly simplifies the mathematical techniques used to properly fuse the evidential data. Equivalence between the Pignistic Probability Estimates of the Belief Fusion of the BBs and the Naïve Bayesian fusion of the Pignistic Probability Estimates of the individual BBs has been shown for this process. The equivalence comparison is done in probability space. The practical implications are notable for information fusion processes in many real systems. For many such systems, some inputs to the information fusion process are better represented by Ω the exponential belief, Power - set ( Ω 2, representation of the incomplete information set. Via an appropriate pignistic probability transform, all these inputs are mapped into the linear probability Ω set representations and fused. This greatly simplifies the computation complexity since the equivalent fusion results are obtained in linear probability space rather than exponential belief space. References [1] hafer, G., athematical Theory of Evidence, Princeton University Press, [2] D. Fixsen and R. ahler (1992 " Dempster-hafer pproach to Bayesian Classification," Proc. 5th Int'l ymp. on ensor Fusion, Vol. I (Unclassified, Naval Training Center, Orlando FL, pril , pp [3] mets P., Kennes, R., The Transferable Belief odel, rtificial Intelligence, vol. 66, pages , [4] Fister, T., itchell, R., odified Dempster-hafer with Entropy Based Belief Body Compression, Proc Joint ervice Combat Identification ystems Conference (CIC, Naval Postgraduate chool, C, ugust 1994, pp [5] Fixsen, D.; ahler, R.P.. The odified Dempster- hafer pproach to Classification IEEE Transactions on ystems, an and Cybernetics, Part, Volume: 27 Issue: 1, Pages: , Jan [6] David Lewis, Naive (Bayes at Forty: The Independence ssumption in Information Retrieval, Proceedings of the 10th European Conference on achine Learning, ECL-98. [7] I. ndroutsopoulos, J. Koutsias, K. V. Chandrinos, G. Paliouras, and C. D. pyropoulos., n Evaluation of Naïve Bayesian nti-pam Filtering, In Proc. of the workshop on achine Learning in the New Information ge, [8] Peri, Joseph.., Data Fusion & Target ID: Dempster- hafer & Probability Theories Holy War, ay 18, 2001, Parsons uditorium, JHU pplied Physics Lab Colloquium [9] udano, John, J., Pignistic Probability Transforms for ixes of Low- and High- Probability Events, Fourth International Conference on Information Fusion 2001, ontreal, QC, Canada, ugust 2001, pages TUB [10] udano, John, J., The ystem Probability Information Content (PIC Relationship to Contributing Components, Combining Independent ulti-ource Beliefs, Hybrid and Pedigree Pignistic Probabilities, Proceedings of the Fifth International Conference on Information Fusion, 2002 Volume: 2, 2002 Pages: [11] udano, John, J., Inverse pignistic probability transforms, Proceedings of the Fifth International Conference on Information Fusion, Volume: 2, 2002 Pages:

8 [12] Dezert, Jean, "Foundations for New Theory of Plausible and Paradoxical Reasoning", to appear in "Information and ecurity Journal", n International Journal, Edited by Tzvetan emerdjiev, CLPP, Bulgarian cademy of ciences, ophia, Nov [13] udano, John, J., Generalized Belief Fusion lgorithm Proceedings of the ixth International Conference on Information Fusion, 2003 [14] udano, John, J., Equivalence Between Belief Theories and Naïve Bayesian Fusion for ystems with Independent Evidential Data: Part I, The Theory, Proceedings of the ixth International Conference on Information Fusion, 2003