A Discussion on Simple Support Functions

. Prediction about each and every incident happening in our daily life is impossible. But we can predict about some incidents. Probability is most helpful tool in predicting about outcomes or conclusions of such incidents. Such incidents happened in our life, always follow some known or unknown statistical probability distribution which may consist of simple or complicated probability density function. Therefore with help of probability distributions, we get some blurred idea about the functioning of incidents happening in our life. Using some commonly used probability distributions, we obtain conclusions which are helpful in decision making. Support functions viz. simple support functions are very useful in decision making. In this paper, we quote some results and applications regarding simple support function based on probability transformations


Introduction
In this section, we give some preliminary background about belief function, plausibility function, commonality function and simple support function.

Frame of Discernment
Dictionary meaning of "Frame of Discernment" is frame of good judgment insight.The word "discern" means recognize or find out or hear with difficulty.
If frame of Discernment  is then every element of  is a proposition.The propositions of interest are in one -to -one correspondence with the subsets of .The set of all propositions of interest corresponds to the set of all subsets of , denoted by 2  .

Definition 1.1 If  is frame of discernment, then a function is called basic probability assignment whenever
The quantity m(A) is called A's basic probability number and it is a measure of the belief committed exactly to A. The total belief committed to A is sum of m(B), for all proper subsets B of A.
(1) If  is a frame of discernment, then a function is called belief function over  satisfying above condition (1).(4) which expresses the extent to which one finds A credible or plausible [3].

Simple Support Functions
We have embedding of sets as follows: (5) The class of support functions includes all those belief functions that can be obtained by coarsening the frame of a separable support functions.

Definition 1.3 If s is the degree of support for A, where
, then the degree of support for is given by Suppose one body of evidence has the precise effect of supporting to the degree s 1 , while the another entirely separate body of evidence has the precise effect of supporting A to the degree s 2 , then degree of supporting A two bodies of evidence together is with .

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Volume 10 The relation between the weight of evidence ,w, precisely and unambiguously supporting A and the degree of support s for A in the corresponding simple function is (7) Where c is negative constant (figmentary taken as c=-1).Notice that the weight of evidence can take any non-negative value, including 0 and infinity.Evidence of zero weight produces a degree of support zero, while evidence of infinity weight produces a degree of support one or infinity. .In next section, we will study particular cases obtained by our new probability transformations and corresponding properties.

Bernoulli's Rule of Combination
Suppose one body of evidence has the precise effect of supporting to the degree s 1 , while the another entirely separate body of evidence has the precise effect of supporting A to the degree s 2 , then degree of supporting A obtained by pmf of first distribution , while another entirely separate body of evidence has precise effect of supporting A obtained by pmf of second distribution .Then degree of supporting A based on two bodies of evidence together i.e. total evidence is Remarks:-1.A's degree of support on the total evidence is greater than its degree of support on either of the seperate bodies of evidence.Indeed the deficit from unity of the degree of support provided by one of the bodies of evidence is reduced by a proportion equal to the degree of support by the other body of evidence.

The Weight of Evidence
The degree of support for the various propositions discerned by a frame  ought to be determined by the weights of the items of evidence attesting to those various propositions.for evidence underlying a simple support function, if the evidence points precisely and unambiguously to a single subset , then the degree of support for A ought to be completely determined by the weight of that evidence i.e. it must be a function of that weight.
The weight w of the evidence pointing to A can take any non-negative value, the degree of support must be between zero and one.Hence we require a function such that .The function g is closely determined by Bernoulli's rule of combination with the intuitive idea that weights combine additively.
If w 1 and w 2 be the weights of evidence underlying the simple support function S 1 and S 2 then the weight of evidence underlying their orthogonal sum will be .
If we set then . Hence f must be exponential.Volume 10 As , the constant c must be negative.Also as Solving for w, we get Note: -The weight of evidence can take any non-negative value, including zero and infinity.Evidence of zero weight produces a degree of support zero while evidence of infinite weight produces a degree of support one or certainty.

Heterogeneous Evidences
The essential feature of combining evidence that points towards a proposition A with evidence that points towards a different but compatible proposition B is that the total evidence provides a support not only for A and B separately, but also for the conjunction AB.
Suppose and we wish to combine S 1 and S 2 where S 1 is a simple support function focused on A with and S 2 is a simple support function focused on B with .By Dempster's rule, AB is supported to the extent .By Figure 1, the orthogonal sum has basic probability numbers Therefore (10) for all . In above, we assumed that neither A nor B contains the other.If A is subset of B then the evidence is not so heterogeneous and the expression for the orthogonal sum simplifies to (11) Here we listed results which are dependent on two simple support functions focusing on subset A. In next section, we generalize these results for more than two simple support functions focused on subset A [1].

Bernoulli's Rule of Combination
Suppose one body of evidence has the precise effect of supporting to the degree s 1 , while the another entirely separate bodies of evidence has the precise effect of supporting A to the degree The Bulletin of Society for Mathematical Services and Standards Vol. 10 s 2 and s 3 , then degree of supporting A obtained by pmf of first distribution , while another entirely separate bodies of evidence has precise effect of supporting A obtained by pmf of second and third distribution and .Then degree of supporting A based on three bodies of evidence together i.e. total evidence is with Remarks:-1.A 's degree of support on the total evidence is greater than its degree of support on either of the seperate bodies of evidence.Indeed the deficit from unity of the degree of support provided by one of the bodies of evidence is reduced by a proportion equal to the degree of support by the other bodies of evidence.Suppose one body of evidence has the precise effect of supporting to the degree s 1 , while the another entirely separate bodies of evidence has the precise effect of supporting A to the degree , then degree of A obtained by pmf of first distribution , while another entirely separate bodies of evidence has precise effect of supporting A obtained by pmf of second and third upto distribution .Then degree of supporting A based on k bodies of evidence together i.e. total evidence is Remarks:-1.A 's degree of support on the total evidence is greater than its degree of support on either of the seperate bodies of evidence.Indeed the deficit from unity of the degree of support provided by one of the bodies of evidence is reduced by a proportion equal to the degree of support by the other bodies of evidence.

The Weight of Evidence
The degree of support for the various propositions discerned by a frame  ought to be determined by the weights of the items of evidence attesting to those various propositions.for evidence underlying a simple support function, if the evidence points precisely and unambiguously to a single subset , then the degree of support for A ought to be completely determined by the weight of that evidence i.e. it must be a function of that weight.
The weight w of the evidence pointing to A can take any non-negative value, the degree of support must be between zero and one.Hence we require a function such that .The function g is closely determined by Bernoulli's rule of combination with the intuitive idea that weights combine additively.
If be the weights of evidence underlying the simple support function respectively then the weight of evidence underlying their orthogonal sum will be .then The Bulletin of Society for Mathematical Services and Standards Vol. 10

Definition 1 . 2 A
subset of a frame  is called a focal element of a belief function Bel over  if m(A)>0.The union of all the focal elements of a belief function is called its core.The quantity (3) is called commonality number for A which measures the total probability mass that can move freely to every point of A. A function is called commonality function for Bel Degree of doubt : We have.

Figure 2 :
Figure 2: Heterogeneous Evidence Suppose with S 1 is a simple support function focused on A with and S 2 is a simple support function focused on B with then defined is a simple support function focused on A. Notes :-1.If is a simple support function focused on A, then S is a belief function with basic Probability numbers and for all other .2.Here s can be obtained by probability mass function or probability density function of distribution.The Bulletin of Society for Mathematical Services and Standards Vol. 10

2 .
Bernoulli's rule of combination is a special case of Dempster's rule of combination.This can be justified as: Let simple support function S 1 has basic probability numbers and and simple support function S 2 has basic probability numbers and .The only upper right hand rectangle of measure fails to be committed to A. the orthogonal sum has probability numbers S is a simple support function A focused on with

2 .
Bernoulli's rule of combination is a special case of Dempster's rule of combination.This can be justified as: Let simple support function S 1 has basic probability numbers and , simple support function S 2 has basic probability numbers and and simple support function S 3 has basic probability numbers and .The only upper right hand parallelopiped of measure fails to be committed to A. the orthogonal sum has probability numbers S is a simple support function focused on A with Thus Bernaulli's rule of combination is generalized for k simple support functions as follows:

2 .
Bernoulli's rule of combination is a special case of Dempster's rule of combination.This can be justified as: Let simple support function S 1 has basic probability numbers and , simple support function S 2 has basic probability numbers and and simple support function S k has basic probability numbers and .The only upper right hand parallelopiped of measure fails to be committed to A. the orthogonal sum has probability numbers (12) (13) S is a simple support function focused on A with Thus we got generalized Bernaulli's rule of combination for k simple support functions.