A PROPOSAL FOR A DISTINCTION IN BELIEF FUNCTIONS I

This study proposes a distinction between the belief function and the credibility function in mind design – a section of artificial intelligence. We show the distinction between the credibility and belief measures first. Then we evaluate the possibility of these measures in case studies and consider the credibility and belief values of each case. This is a novel method for the evaluation of belief functions and it is shown that they satisfy the rules of current belief function axioms. Key WordsBelief measure, belief functions, credibility measure, Dempster – Shafer belief functions


INTRODUCTION
Dempster-Shafer theory is put forward to cover several models that use the mathematical object called "belief function" by Shafer [1].The aim of his study was the modeling of someone"s degrees of belief.The Dempster-Shafer theory is a generalization of the Bayesian theory of individual probability.The Bayesian theory requires probabilities for each question of interest; on the other hand, belief functions allow us to base degree of belief for one question on probabilities for a related question, and this is called power set.These degrees of belief may or may not have the mathematical properties of probabilities; how much they differ from probabilities will depend on how closely the two questions are related.

Shafer's Theory of Evidence
Belief functions are defined as follows by Shafer [1].If Θ is a frame of discernment, then a function Bel: 2 Θ → [0, 1] is a belief function if and only if it satisfies following conditions: 1 -Bel(Ø) = 0 2 -Bel(Θ) = 1 3 -For every positive integer n and every collection A 1 , A 2 … A n of subsets of Θ Belief functions are also called Credibility functions by many researchers.Associated with each belief measure, there is a Plausibility measure "Pl" and it is defined by the equation [2] Pl(A) = 1 -Bel(Â) (1) or in the same way [3] Pl Degree of doubt on an evidence A is shown by Dou Therefore we can say Pl The following remarks are made with respect to Plausibility measure [1]: 1) Pl(A) measures the degree to which one fails to doubt A, where Dou(A) = Bel(Â) 2) Pl(A) measures the total belief mass that can be allocated into A, whereas Bel(A) measures the total belief mass that is allocated to A.

4) Bel(A) ≤ Pl(A)
From the super additively property of belief functions above, it is easy to say [2] Pl Bel(A) + Bel(Â) ≤ 1 and Pl(A) + Pl(Â) ≥ 1 Belief functions are generally obtained by support functions.The relation between support functions and belief functions [1] are given in Figure 1.
Figure 1.A general view of belief functions

THE PROPOSED METHOD
In the proposed method, some of the formulations are put forth in a heuristic way.How we think or how we believe is an innate knowledge.Here, we only try to get a formulation of this knowledge and to reveal why it is tackled in such a way.
One can say that belief is much stronger then credibility from the daily use of these words.Belief means allocating your faith to one option and rejecting the other options; whereas, credibility means trustworthiness.Therefore, it is possible to make a distinction between these two words.Credibility functions can be thought as pre-belief functions and is a priori for constructing our belief.Usual belief functions can be called as credibility functions.

Explanation
Belief functions must be much stronger than credibility functions [4].Besides the vocabulary meaning, in our daily life we sometimes say that we don"t believe an idea just by evaluating "how plausible the ideas", but we also look for "how plausible complement of this idea".In other words, we don"t believe an idea just by evaluating only "how credible the ideas", but we also look for "how credible is the complement of this idea".Different ideas about an event are the antithesis of each other.Each idea"s credibility is only as valuable as the other thesis left us to take into account it.
For example, assume there are two ideas about a subject.One of them is highly credible (or plausible) and the other is not.In this case, we take the first idea into account much more.But if both of the ideas are highly credible (or plausible) we won"t be so bold to take the first idea into account.This consideration shows us that each idea is as valuable as its credibility and remaining value of its complement"s credibility.

Table 1. Basic allocations of belief by two possible results (A and B) game [4]
A A B 1 2

B 3 4
As another example, let us consider an experiment with only two possible results in a Bayesian space (head or toss experiment).Assume some degree of credibility is allocated to these two possible results by some support functions.In such circumstances we have four divisions in mind to evaluate this situation and there are certain values for these divisions.These four divisions are represented in Table 1 where division 1 means that both A and B seem to occur.In a Bayesian space it is not possible for both of them to occur simultaneously and it is obvious that this part of division means hesitation.
Similarly division 4 means that both A and B seem not to occur, yet one of them must occur.This division also represents hesitation.Under these conditions belief in A is represented by division 3 in which trusts in A to occur and trust in B not to occur.In a similar way belief in B is represented by division 2.

Approach by an Electronic View
From another point of view, we can think this problem like a xor gate [4].In a xor gate, the result is true (or equal to one) if and only if A is not equal to B. Gate works such that; if both A and B are equal to one, currents blockade each other.So there will be no output (each output will be equal to zero).If A and B are both equal to zero, then there is no blockade but also there is no output current (means outputs are zero).If one of the inputs (e.g.A) is 1,then the other one (B) is zero current."A" won"t be blockaded and output of A is one.
Figure 2.Basic electronic model of belief by two possible results (A and B) game [4] This output can be modeled as In a similar way allocating belief problem can be thought as a xor problem.This is because each idea is the antithesis of the other.

Formulation of the Proposed System
According to all propositions above for a frame of discernment Θ, proposed belief function in the set of 2 Θ is [5] ͞ A: Every items of power set 2 Θ except A Â: Every items of power set 2 Θ which has no intersection with A (A  Â = Ø).
In equation( 13), belief measurement is obtained with plausibility of an option which is weighted with plausibility of the complement of this option.In equation( 14),the terms are replaced with their expressed formulas with basic assignments.In equation( 15 According to such a formulation it is obvious that even if the credibility function is linear belief function will not be linear.

THE FORM OF BELIEF FUNCTION SATISFYING THE DEMPSTER-SHAFER BELIEF FUNCTION AXIOMS
If Θ is a frame of discernment, Dempster-Shafer [1] belief function axioms on power set of Θ are This form of belief function satisfies the Dempster-Shafer belief function axioms [4].
1 -Bel() = 0 This situation can be easily seen that Cr() = 0 and Pl(Ø) = 0. Bel(Ø) = Cr (Ø)*Pl (Ø) so Bel(Ø) = 0 2 -Bel(Θ) = 1 Also this equality can be obtained for Cr (Θ) = 1, Cr (Ø) = 0 it is easily seen that ) This inequality can also be seen easily that for every measure of Cr (A i ) ≤ 1 and Pl(A i ) ≤ 1 multiplication of Cr (A i )* Pl (A i ) is much smaller than any one of them.For simplicity, let us take n = 2 and take a Bayesian credibility function space where A i A j = 0 for i≠j In a Bayesian space where A i A j = 0 for i≠j it is obvious that Cr (A 1 A 2 )*Pl (A 1 A 2 ) part of this equation will be zero.
Consequently, multiplication of the left hand side of these equations will be larger than the right hand side.

-Bel (A) + Bel (Â) ≤ 1
Since Cr (A) + Cr (Â) ≤ 1 and similarly Bel (A) ≤ Cr(A) and Bel (Â) ≤ Cr (Â) it is obvious that According to this proposition we can rearrange the schema of support functions, credibility functions and belief functions as [4] Simple separable support Crdbl belief Support  support  func  func  func Func func Figure 3.A general view of proposed belief functions [4]

NUMERICAL EXAMPLE
Let"s assume a weather forecast problem.There are some measurements of temperature, wind, air humidity, etc. which give us an opinion about how the weather will be.Opinions about how the weather will be are rainy (A), cloudy (B) and clear (C).Membership measures of each opinion depending on the measures are given in Table 2.

A View to a Special Property of the Proposed System
In a Bayesian probability function space in which A i A j = 0 for i≠j, probability measures are used as credibility measures; thus, belief functions turn to be square of credibility functions [4].According to this arrangement in a head or toss experiment, we can say that probability of head (or toss) is also a credibility measure and it is equal to 0,5.In Bayesian probability space, measurement of Head U Toss must be equal to zero.After calculating belief measure for head (or toss) it can be easily seen that it is equal to 0,25.Obviously the residual belief measure will be allocated to Head U Toss option.In this formulation Bel(Head U Toss) can be considered as a reserved belief that is neither allocated to head nor allocated to toss but allocated to "Head or Toss" option.