A Supporting Uncertainty Decision Making Processes based on Fusion of Audio – Vision Evidence

A Supporting Uncertainty Decision Making Processes based on Fusion of Audio – Vision Evidence

A. E. Amin

Department of Computer Science, Mansoura University, Mansoura 35516, Egypt

Abstract:-This paper presents a new method for decision making based on the fusion of audio-visual evidences. Evidences fusion is characterized that the decision will be more accurate and specific because it does not depend on one evidence's alone as in the probabilistic approach.

The decision-making process depends on decisional separation of conflict conditionally between contributions of several independent sources of information represented by audio and images.

In order to provide effective of evidence fusion, one must employ an analytical framework that can capture the uncertainty inherent in audio and visual data. In particular, feature extraction of audio and visual data results in propositions that inherently possess significant semantic ambiguity. An evidence fusion must be able to exploit the respective advantages of audio and visual data while mitigating their particular weaknesses.

Keywords: Evidence fusion, Evidence measures, decision making.

1. INTRODUCTION

   Decision making DM is the study that identifying and choosing from a nonempty set of alternatives A

   and decision making under pure uncertainty[6]. In decision making under pure uncertainty, the decision-maker has no knowledge regarding any of the states of nature outcomes, and/or it is costly to obtain the needed information.

   In the environment of uncertainty, more than one type of event can take place and the decision maker is completely in dark regarding the event that is likely to take place. The decision maker is not in a position, even to assign the probabilities of happening of the events as shown in figure 1. In most decision whose outcomes depend on uncertain events are contain implicitly degree of belief. Belief function is used to determine degree of belief which can be defined as a function satisfying three axioms which can be viewed as a weakening of the Kolmogorov axioms that characterize probability functions [7]. The view of belief function as a generalization probability theory is quit different from a representation of a body of evidence [8].

   Evidence theory [9] has often been promoted as an alternative approach for fusion information when the hypotheses for Bayesian approach cannot be precisely stated.

   probabilities based on a given set of criteria C and

   preferences of the decision maker i.e.:

   DM

   f A, C

   f : A C A, A U ,A

   The outcomes of a DM process are determined by the decision making strategies selected by decision makers when a set of alternative decisions has been identified. There is a great variation of DM strategies developed in traditional decision as well as cognitive science, system science, management science, and economics. Decision making is one of the fundamental cognitive processes modeled in the layered reference model of the brain (LRMB) [1, 2]. Modeling for decision making involves two distinct parties one is the decision maker and the other is the model builder known as the analyst [3]. There are three most widely types of decision models that help to analyze depending on the amount and degree of knowledge namely decision making by buying information [4], decision making under risk [5]

   Contextual Evidence

   Best available research evidence

   Decision making

   Experiential evidence

   Environment and organizational context

   Fig. 1: Evidence based decision making environment

   The evidence alone is not enough to make the decision because its calculation method depends on randomized trials and other quantifiable methods. So evidence alone is considered an only one key component in the decision- making process but its have high probability of being affected by noisy data and lack of distinctiveness which contributes to the inability of decision maker in making the right decision.

   Fusion of evidence should actually be used to contribute to the decision by predicting the performance of the fused evidence and comparing it with the corresponding belief function of the best expert. In the recent literature [13] there has been a large amount of work devoted to the definition of new rules. For example, Dempster-Shafer

   f and Rare associated with some supporting evidence as f h, E and R , E respectively. An evidence argument is a pair h, E , where his a formula in and E e1 , e2 ,…, en is a set of formula in denoted by Eh. An element ei Ehrepresents

   an indivisible chunk of information serving as evidence called a focal element of the evidence for h. A focal

   element is an element of the power set to which a non-zero belief is assigned. It is possible

   1

   2 1 2

   that h, E , h, E f , such that E E . For

   theory [14] which based on belief functions [15] and combines different pieces of evidences into a single value

   every pair

   h, E :

   that approximates the probability of an event. And there are theoretical framework [16] is developed for combining multiple experts and the most usual classifiers combinations schemes, such as the product, sum, min, max and median rules.

   Subsequently, this paper is illustrated by implementing two well known audio and visual evidences. Typically, these evidences take into account a consensual evaluation of the sources by invalidating irrelevant sources of information on the basis of a majority decision. The remainder of this paper is organized as follows: Section 2 basic notation. Section 3 proposed method. Section 4 experimental and results Finally, Section 5 concludes.

2. BASIC NOTATION

General formula which represents the knowledge base is

1. h K or h ; and

2. E e1 , e2 ,…, en is a set of evidence for hsuch that ei e j for any i j .

   For every set of evidence E there are constituent members have a probability mass function denoted by

   mE,.: E 0,1 and satisfies the constraint:

   mE, e1 m(E, e2 ) … mE, en 1

   mE, 0 for all E

   K f , R where f represents facts and Ris

   inference engine that can reason about those facts. Each of

   1. Evidence Types:

      Evidence can be divided into four type's [17] namely consonant evidence, consistent evidence, arbitrary evidence, and disjoint evidence. As shown in figure 2 evidence types are represents as sets of elements of the

      frame of discernment for where there are non-zero basic probability assignments.

      B A C

      D C B A D

      1. Consonant evidence (b) Consistent evidence

         B A C A C

         D

         B D

         1. Disjoint evidence

         2. Arbitrary evidence

            Fig. 2. Four types of evidence

            Consonant evidence can be represented as nested structure of subsets, where smallest subset elements are included in the next larger subset. This can correspond to the situation where information is obtained over time that increasingly narrows or refines the size of the evidentiary set. While, consistent evidence means that there is at least one element that is common to all subsets. But arbitrary evidence corresponds to the situation where there is no element common to all subsets, though some subsets may have elements in common. Whereas disjoint evidence implies that any two subsets have no elements in common with any other subset.

   2. Evidenc Combining:

      Evidence combining can be stated in the context of information fusion. Depending on the type of information that is fused, the fusion scheme can be classified as sensor level, feature level, score level and decision level fusion.

      Feature level fusion refers to combining deferent feature

   3. Evidence measures:

      By applying evidence combination rules there are several evidence measures (EM) can be created. An evidence belief function (EBF) is a numerical reasoning method represents the evidence in the form of generalized probabilities [20]. EBF a problem is described all possible values of element in an environment and provides a way to represents the hesitation and ignorance. The elements of the environment are mutually exclusive and exhaustive. The exhaustive set of mutually exclusive elements is referred to as a frame of

      discernment denoted as . Let Bel be such EBF that

      represents belief in the propositions that correspond to the elements of 2 :

      Bel : 2 0,1

      A Bel Awith A 1

      A

      where, Bel

      A is the total belief committed to A.

      sets that are extracted from multiple biometric sources. When the feature sets are dependences a single resultant

      The counterpart of Bel is the plausibility measure pl :

      feature set can be calculated as a weighted average of the individual feature sets [18]. Whereas the feature sets are

      2 0,1

      with

      plA

      mB

      B A

      independents a single feature set form can be concatenated [19]. Concatenated feature set is demonstrating different properties of uncertainty about the evidence, generate different characterizations of the evidence as observed through the evidence they can obtain. The obtained evidence can be characterized by the basic probability assignments to the frame of discernment of evidence.

      The measure plAshall not be understood as a complement of Bel A. Only

      A | mA 0 Bel A plA

      has to be fulfilled.

      In addition to Bel and pl third evidence measure can be defined as commonality measure cmn[21]. With cmn : 2 0,1 and

      cnmA m(B)

      B A

      The complements to the measures Bel and pl are doubt and disbelief respectively. Doubt [22] can defined as complements to the plausibility measure it seems to make

      more sense to distinguish between doubt and disbelief. Lack of belief does not imply disbelief [23]. The disbelief of set A is the belief in the complement. There is

      Z

      Bel (A) 1 pl( A) plA 1 Bel A with

      Bel A plA

      The difference plA Bel A describes the uncertainty

      concerning the hypothesis A represented by the evidential interval as shown in figure 3.

      Y

      Bel(A) dout(A)

      B A A

      P(X,Y)

      pl(A) disBel(A)

      B

      0 1 X

      Uncertainty P(X,Y)

      0 X

      1

      Fig 3. measures of belief and plausibility and its complements

3. PROPOSED METHOD

   The proposed method presents new method for improving decision making based on the decision function, which adjudicate in a decisional dispute conditionally to basic decisions provided by the several sources of information. As shown in figure 4, there are three subsections which introduce evidence representation, evidence fusion and decision making.

   1. Evidence representation:

      We consider there are two different sources for evidence

      Let 1 ,2 ,…,n be a frame of discernment of a decision making problem under consideration ndistinct elementsi ,i 1,2,…, n . Evidence belief mass function defined as a mapping from the power set of denoted by

      2 that must satisfy the two conditions. The first is mass of empty set which represent the impossible event is zero and the other is the mass of belief is normalized to one. An element is called focal element if and only if m 0 . Focal element represents a degree of belief attached to the proposition and to uncertainly

      namely image evidence EvIm and sound evidence Ev . Each of evidence detects a set of objects

      proposition, based on some evidence.

      Each normalized mass function mN .can represent by

      denoted

      by EvIm

      So

      im im im Im

      ev , ev ,…, ev for Ev

      and

      several function associated with belief function known as

      plausibility, commonality and disbelief. Plausibility

      function pl is the most important which represents the

      1 2

      n

      upper limits of uncertainty whereas the belief function

      1

      2

      n

      EvSo evso , evso ,…, evso for EvSo . All evidence

      bel

      that has been obtained from the classifier that gives information on the actual class of a test pattern. This information can be represented by a belief mass function

      m. after the presentation on the expert.

      represents the lower limits. Each of

      pl and bel functions are in one to one correspondence, they may be obtained from each other through linear transformation.

      Image Database

      Sound Database

      Feature Extraction

      Classifier

      Classifier

      Feature Extraction

      Query Image

      Image Evidence

      Sound Evidence

      Query Image

      Image Belief Function

      Expert

      Sound Belief Function

      Evidence Fusion

      Frame of discernment

      Decision Making

      Fig 4: proposed method to improving decision making.

   2. Evidence representing algorithm:

      , Ev

      Let there are probability (p), p 0,1of two evidences

      Then, evidences of the fused belief function

      m1:s | E

      EvIm , EvSo

      and a partition

      Ev

      Im1:n So1:m

      of

      are generated by means of the sub arbitraments related to

      1, ssuch as:

      EX | Y1:s ; m1:s pEvIm X | YEv

      i

      m , considered as probabilistic distribution over the set 2 .

      For each i

      Im1:n

      ; mEv

      1- Entries generation:

      Im1:n

      1 pEvSo X | YEv

      1:m

      , mEv

      So1:m

      EvIm and EvSo :

      Evidence representing algorithm:

      1, n, generates 2 according to the basic belief function

      considered as a probabilistic distribution over the set 2 .

      • In case empty set means impossible event. Otherwise event is focal element.

      ; m

      1:n 1:n

      • Generate 2 according to experts E |

      2- Conditional arbitrament:

   3. Evidence Fusion:

      The purpose of aggregation of information is to meaningfully summarize and simplify information

      of two basic probability assignment of m1 and m2 in the following manner:

      m1 Bm2 C

      rationally obtained from an independent source or multiple sources. Hence the evidence fusion algorithm can be done

      m12

      A BC A

      1 K

      where

      A

      (I)

      by algorithm 1. Combination rules are the special types of aggregation methods for data obtained from multiple

      m12

      0

      sources. From a set theoretic standpoint, the combination and disjunction of evidence is employed by AND (set intersection) and OR (set union) operation respectively. The combination rule is determined from the aggregation

      K m1 Bm2 C

      BC

      The normalization factor (1-K) has the effect of completely ignoring conflict and attributing any probability mass associated with conflict to the null set [24]. The

      imk Sok

      n Experts ex:ex1 , ex2 …, exn

      mev mev 0

      Im

      Im

      imn

      im1 im2

      Im

      So

      So

      son

      ,…, ev , Ev .bel, Ev . focal

      so1 so2

      So

      Visual evidence: Ev ev , ev

      combiation rule results which based on conjunctive pooled evidence can be measured by evidence measures.

      Algorithm 1: Evidence Fusion

      Data: Audio evidence: Ev ev , ev

      ,…, ev , Ev .bel, Ev . focal

      Results: Fusion of EvIm and EvIm : Fev

      for i 1:n do

      fusion

      for EvSo . focal in exi do for EvIm . focal in exi do

      K EvSo. focal EvIm . focal

      fusion. focal K

      fusion.bel EvSo.bel EvIm .bel

      Concatenate same focal in fusion

      Fev fusion

   4. Decision Making:

   A belief function has to be transformed into a probability function for decision making. The belief function that quantifies knowledge of the actual class of xis

   transformed into a pignistic probability distribution [25].

   Each mass of belief mA is divided equally between the

4. EXPERIMENTAL AND RESULTS

   The most important and immediate application of this proposed method is helping decision makers decide most appropriate in given situation as shown in figure 5. Such decisions are based on information from set of

   A

   A

   hypothesis consisting of basic

   elements of

   for all

   . This leads to pignistic

   hypotheses c , c …, c , pieces of evidence that get from

   probability distribution of class w defined as [26]:

   1 2 m

   BetP (wk

   ) m( A),

   w AA

   k

   wk

   (II)

   two sources audio-visual evidences and decision maker opinion. Audio and visual sources, provided by the sound signal sensor and image processing specifying the sets of features and the probabilities conditional on the features

   and corresponding cases characteristics. Expert's opinion is provided by the case concerning his characteristics and preferences on the basis of which relevant utility functions are to be chosen.

   Input Experts No. n

   Input Image

   Input Sound

   Image Feature Extraction

   Sound Feature Extraction

   EvIm

   EvSo

   Sound Classifier

   Image Classifier

   FX So

   FX Im

   FX Im Databas

   FX So Databas

   Image Evidence Focal

   Sound Evidence Focal

   for i 1:n

   EvIm Belief Function

   EvSo Belief Function

   n n 1

   Y

   N

   Any

   Select decision (d)

   EvIm

   Any

   n n 1

   N

   Y

   EvSo

   Evidence Fusion Applied Eq. (I)

   d f Ev, R

   Decision Making by applied Eq. (II)

   Fig. 5: Decision making based on audio-visual evidences.

   For illustrates, the decisions can be determined to three precisely defined hypothesis represented by:

   c1 , c2 , c3

   The corresponding power set of is:

   depends on the point of view of decision-makers, where his opinion depending on the evidence probability that affect in the hypothesis.

   One of the decision makers mainly states that the

   hypothesis c1 or c2 are the reason for the problem. In

   2 ,c ,c ,c ,c , c ,c , c ,c , c ,c , c , c

   ev

   1 2 3 1 2 1 3

   2 3 1 2 3

   other words, the piece of evidence three Im

   might

   3

   Each case can be described by two major symptoms called

   audio So

   and visual Im

   evidences. The decision

   have occurred and resulted in the consequences c1

   or c2 . Whereas, the second decision maker was focused

   qualitative evidences-hypothesis is given in Table 1.

   on hypothesis c1 and c3 . The complete survey of the

   Table 1: qualitative evidences-hypothesis

   		Evidence	Degree of belief	Hypothesis
   Decision makers	1st	evIm 1	Pev Im1	c1
   		evIm 2	Pev	c2
   		evIm 3	Im 2 P evIm	c1 , c2
   		evIm 4	3 P evIm	c1 , c2 , c3
   	2nd	evSo 1	4 P evSo	c1
   		evSo 2	1 P evSo	c2
   		evSo 3	2 P evSo	c1 , c3
   		evSo 4	3 P evSo 4	c1 , c2 , c3

   As shown from table 1, there are different evidences Degree of belief that effected on the hypothesis is

   1

   3

   4

   as evIm , evIm and evIm leads to a different set of

   consequences that contain same hypotheses as elements c1 . So, the possibility of decision making be very difficult because presence uncertainly area containing on degree of beliefs of evidences leads to which hypothesis is selected by the decision makers.

   determined for each evidence by experts and that is used by decision maker to take the appropriate decision as shown in table 2. Based on the degree of belief can be calculated belief and doubt, commonality, plausibility and disbelief measures that helping to define the uncertainty evidences area to decision-making as shown in table 3. From the lower boundary (belief) and higher boundary (plausibility) can fuses each of audio evidence with visual evidence to builds the fusion evidence as shown in table 4.

   Table 2: relation of degree of belief for decision maker by power set

   1st decision maker	2	2nd decision maker
   mev 0.2 Im1	c1	mev 0.2 So1
   mev 0.1 Im2	c2	mev 0 So2
   mev 0 Im3	c3	mev 0.2 So3
   mev 0.6 Im4	c1 c2	mev 0 So4
   mev 0 Im5	c1 c3	mev 0.4 So5
   mev 0 Im6	c2 c3	mev 0 So6
   mev 0.1 Im7	c1 c2 c3	mev 0.2 So7

   Table 3: belief and plausibility measures for evidences

   mev Im K	belev Im K	plev Im K	2	mev SoK	belev SoK	plev SoK
   0.2	0.2	0.9	c1	0.2	0.2	0.8
   0.1	0.1	0.8	c2	0	0	0.2
   0	0	0.1	c3	0.2	0.2	0.8
   0.6	0.9	1	c1 c2	0	0.2	0.8
   0	0.2	0.9	c1 c3	0.4	0.8	1
   0	0.1	0.8	c2 c3	0	0.2	0.8
   0.1	1	1	c1 c2 c3	0.2	1	1

   Table 4: the fusion table contains audio and visual evidences cut set.

   td>

   evIm 1	evIm 2	evIm 3	evIm 4	evIm 5	evIm 6	evIm 7
   evSo 1	c1			c1	c1		c1
   evSo 2		c2		c2		c2	c2
   evSo 3			c3		c3	c3	c3
   evSo 4	c1	c2		c1 c2	c1	c2	c1 c2
   evSo 5	c1		c3	c1	c1 c3	c3	c1 c3
   evSo 6		c2	c3	c2	c3	c2 c3	c2 c3
   evSo 7	c1	c2	c3	c1 c2	c1 c3	c2 c3

   Information Lake represents a significant problem and influential in the decision. So reducing the size of information and reduce the time of the decision distinguishes the proposed system where the negligence of

   ( mev 0, mev 0). In our example, columns Im3 , Im5 and Im6 , and rows So2 ,

   ImK SoK

   So and So are not applicable as shown in table 5:

   rows and columns relating to non focal elements 4 6

   Table 5: The reduced fusion evidences.

   	evIm 1	evIm 2	evIm 4	evIm 7
   evSo 1	c1		c1	c1
   evSo 3				c3
   evSo 5	c1		c1	c1 c3
   evSo 7	c1	c2	c1 c2

   After reduction of information dimensionality the effect of both audio and visual evidences on the hypothesis available to the decision-making is calculated. Table 6 is illustrate the formal procedure by applied the equation (I). For each

   hypothesis can calculate the effect of both audio and visual evidence in them. While the sum over all calculated combination in table 6 is identical with the denominator of equation (I) to calculate the evidence measures of combined hypotheses as shown in table 7.

   Table 6: effect o audio-visual evidences on decision hypothesis

   	evIm 1	evIm 2	evIm 4	evIm 7
   evSo 1	0.04		0.12	0.02
   evSo 3				0.02
   evSo 5	0.08		0.24	0.04
   evSo 7	0.04	0.02	0.12	0.02

   Table 7: the evidence measures for fuses hypotheses.

   2	m	bel	cmn	pl
   	0.0263	1	0.0263	1
   c1 c2	0.1579	0.8947	0.1842	0.9737
   c1 c3	0.0526	0.7895	0.0789	0.9737
   c1	0.7105	0.7105	0.9474	0.9471
   c2	0.0263	0.0263	0.2105	0.2105
   c3	0. 0263	0.0263	0.1053	0.1053

   From the results shown in table 7, the decision maker A decision function represents an arbitrament process

   should avoid hypotheses two c2 and three c3 due to

   conditionally to the contributions of several independent

   the same low values of belief and c3

   takes roughly half

   evidences. It has been shown that evidence fusions based on the concept of decision functions have a straightforward

   the range of uncertainty region of audio and visual evidences and plausibility that c2 . Also, the first hypothesis c1 is excludes due to the wide range of audio and visual evidences uncertainty region 0.24. The

   combination between first and third hypotheses c1 c3 covers a smaller distance between

   lower boundary (belief) and higher boundary (plausibility) than first hypothesis c1 alone 0.18. So the best

   decision is combination first and second hypotheses c1 c2 , where it is smallest range of uncertainty 0.08 with the same (highest) plausibility

   as in case of the combination of first and third hypotheses c1 c3 .

5. CONCLUSION

The decision-making process is difficult and very stressful as it is assumed in the decision-maker should be familiar with the diagnosis knowledge, available procedures, their consequences, and the probabilities of the associated outcomes. The proposed method presents a new method to decision making based on audio and visual evidences which add a new precisely and reliability flavor compared to probabilistic approaches. The fusion of evidences may be responsible for the serious changes of the decision properties.

sampling based implementation.

The proposed method can be used in decision making based on the evidence as audio-visual equipment in the diagnosis of defects in the field of engineering and diagnosis of diseases in the medical field.

REFERENCE

1. Wang, Y., Wang, Y., Patel, S., & Patel, D. (2004). A layered reference model of the brain (LRMB). IEEE Transactions on Systems, Man, and Cybernetics (C), 36(2), 124-133.

2. Wang, Y. (2007b). "The theoretical framework of cognitive informatics". The International Journal of Cognitive Informatics and Natural Intelligence (IJCINI), 1(1), 1-27.

3. Herrera, F., and E. Herrera-Viedma. "Linguistic decision analysis: steps for solving decision problems under linguistic information." Fuzzy Sets and systems 115.1 (2000): 67-82.

4. Verhagen, Tibert, and Willemijn van Dolen. "The influence of online store beliefs on consumer online impulse buying: A model and empirical application." Information & Management 48.8 (2011): 320-327.

5. Heilman, Renata M., et al. "Emotion regulation and decision making under risk and uncertainty." Emotion 10.2 (2010): 257.

6. Rakow, Tim, and Ben R. Newell. "Degrees of uncertainty: An overview and framework for future research on experiencebased choice." Journal of Behavioral Decision Making 23.1 (2010): 1-14.

7. Fagin, Ronald, and Joseph Y. Halpern. "A new approach to updating beliefs." arXiv preprint arXiv:1304.1119 (2013).

8. Cooper, Gregory F. "A method for using belief networks as influence diagrams." arXiv preprint arXiv:1304.2346 (2013).

9. Couso, Inés, and Serafín Moral. "Independence concepts in evidence theory." International Journal of Approximate Reasoning 51.7 (200): 748-758.

10. Schoenfeld, Alan H. How We Think: A Theory of Goal-Oriented Decision Making and Its Educational Applications. Studies in Mathematical Thinking and Learning Series. Routledge, Taylor & Francis Group. 7625 Empire Drive, Florence, KY 41042, 2010.

11. Melnyk, Bernadette Mazurek, and Ellen Fineout-Overholt, eds. Evidence-based practice in nursing & healthcare: A guide to best practice. Lippincott Williams & Wilkins, 2011.

12. Yao, Yiyu. "Three-way decisions with probabilistic rough sets."

    Information Sciences 180.3 (2010): 341-353.

13. Lee, Hyun, et al. "Fusion Techniques for Reliable Information: A Survey." JDCTA 4.2 (2010): 74-88.

14. Yager, Ronald R., and Janusz Kacprzyk. Recent developments in the ordered weighted averaging operators: theory and practice. Vol.

    265. Berlin: Springer, 2011.

15. Fagin, Ronald, and Joseph Y. Halpern. "A new approach to updating beliefs." arXiv preprint arXiv:1304.1119 (2013).

16. Crossan, Mary M., and Marina Apaydin. "A multidimensional framework of organizational innovation: A systematic review of the literature." Journal of Management Studies 47.6 (2010): 1154-1191.

17. Henrekson, Magnus, and Dan Johansson. "Gazelles as job creators: a survey and interpretation of the evidence." Small Business Economics 35.2 (2010): 227-244.

[18]A. Ross, S. Shah, and J. Shah. Image Versus Feature Mosaicing: A Case Study in Fingerprints. In Proceedings of SPIE Conference on Biometric Technology for Human Identi¯cation, volume 6202, pages 1{12, Orlando, USA, April 2006.

1. A. Ross and R. Govindarajan. Feature Level Fusion Using Hand and Face Biometrics. In Proceedings of SPIE Conference on Biometric Technology for Human Identi¯cation II, volume 5779, pages 196{204, Orlando, USA, March 2005.

2. Spohn, Wolfgang. "A general non-probabilistic theory of inductive reasoning." arXiv preprint arXiv:1304.2375 (2013).

3. Karolyi, G. Andrew, Kuan-Hui Lee, and Mathijs A. Van Dijk. "Understanding commonality in liquidity around the world." Journal of Financial Economics 105.1 (2012): 82-112.

4. Hunsley, John, and Eric J. Mash. "Evidence-based assessment." The Oxford handbook of clinical psychology (2011): 76-97.

5. Grosof, Benjamin N. "Evidential confirmation as transformed probability." arXiv preprint arXiv:1304.3439 (2013).

6. Yager, R. R., J. Kacprzyk, et al. (1994). Advances in the Dempster- Shafer Theory of Evidence. New York, John Wiley & Sons, Inc.

7. P. Smets, R. Kennes, The transferable belief model Artificial Intelligence 66(1994) 191 – 879

8. T. Denoeux, A neural network classifier based on Dempster Shafer theory, IEEE transactions on Systems Man and Cybernetics A System and Humans 30(2) (2000) 131 150.