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«DECISION MAKING WITH CONSONANT BELIEF FUNCTIONS: DISCREPANCY RESULTING WITH THE PROBABILITY TRANSFORMATION METHOD USED Esma Nur CINICIOGLU School of ...»

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Yugoslav Journal of Operations Research

24 (2014), Number 3, 359-370

DOI: 10.2298/YJOR140401033C

DECISION MAKING WITH CONSONANT BELIEF

FUNCTIONS: DISCREPANCY RESULTING WITH THE

PROBABILITY TRANSFORMATION METHOD USED

Esma Nur CINICIOGLU

School of Business, Istanbul University, Istanbul, TURKEY

esmanurc@istanbul.edu.tr

Received: April 2014 / Accepted: October 2014

Abstract: Dempster−Shafer belief function theory can address a wider class of uncertainty than the standard probability theory does, and this fact appeals the researchers in operations research society for potential application areas. However, the lack of a decision theory of belief functions gives rise to the need to use the probability transformation methods for decision making. For representation of statistical evidence, the class of consonant belief functions is used which is not closed under Dempster’s rule of combination but is closed under Walley’s rule of combination. In this research, it is shown that the outcomes obtained using both Dempster’s and Walley’s rules do result in different probability distributions when pignistic transformation is used. However, when plausibility transformation is used, they do result in the same probability distribution.

This result shows that the choice of the combination rule and probability transformation method may have a significant effect on decision making since it may change the choice of the decision alternative selected. This result is illustrated via an example of missile type identification.

Keywords: Belief functions, Consonant belief functions, Plausibility transformation, Pignistic transformation.

MSC: 68T37.

1. INTRODUCTION Belief functions represent ignorance, and a wider class of uncertainty than the standard probability theory, which creates a flexible framework for any sort of application where information is gathered from semi-reliable resources. For that reason 360 E. N. Cinicioglu / Decision Making With Consonant Belief Functions belief functions establish a major appeal to operation researchers for potential application areas wherein uncertainty is involved. Belief functions has been used in a wide range of applications such as target identification (Delmotte and Smets 2004), data fusion (Appriou 1997), auditing (Srivastava et al. 2011), and data mining (Wickramaratna et al.

2009).

Belief functions theory originates to the early works of Dempster (1967 & 1968) on upper and lower limits of probability. This work was developed by Shafer (1976) and thus, belief functions are also named as Dempster-Shafer theory of belief functions, or theory of evidence.

For combination of independent belief functions, Dempster’s rule of combination is the classical one, and the most widely used rule in Dempster-Shafer (D-S) theory. However, the rule has been criticized on various matters. One of the criticisms made at DS combination rule is the fact that consonant belief functions are not closed under Dempster’s rule of combination. As a remedy to this problem, Walley (1987) proposed an alternative rule for combination of belief function representations of statistical evidence. Consonant belief functions are closed under Walley’s rule of combination; however, the drawback of Walley’s rule is that it is only defined for partially consonant belief functions. A detailed review and comparison of these two combination rules is done by Cinicioglu and Shenoy (2006). In their work they also showed that when plausibility transformation is applied to outcomes obtained by Dempster’s and Walley’s combination rule, they do result in the same probability distribution function. The fact that there is no decision theory of belief functions gives rise to the need to transform belief functions into probability distributions. Consequently, according to the result obtained by Cinicioglu and Shenoy (2006), a decision made with the expected utility theory would be indifferent of the combination rule (Dempster or Walley’s rule) used. On the other hand, though plausibility transformation is one of the available methods of probability transformation, it is not the only one. In this research, both pignistic and the plausibility transformation are used to transform the outcomes of both of the combination rules into a probability distribution. It is shown that when pignistic transformation is used on the outcomes obtained from both of the combination rules, the result is different probability distributions. So, the choice of the combination rule, and the resulting discrepancy, depending on the transformation method used, may have significant effect on decisions.

The remainder of the paper is as follows: In section 2, the basics of belief functions theory and consonant belief functions are introduced. Additionally, two combination rules, the classical rule of Dempster and Walley’s combination rule for consonant belief functions are demonstrated via an example of Missile Fall Down. In section 3, the need for probability transformation is explained, and the two transformation methods, plausibility and pignistic transformation are applied to the results of the combinations. The discrepancy resulting from the choice of the combination and/or transformation method is demonstrated. Finally, in section 4, we summarize and conclude.





E. N. Cinicioglu / Decision Making With Consonant Belief Functions 361

–  –  –

2.1. Belief Functions Belief functions theory is also called the theory of evidence since it deals with weights of evidence and with numerical degrees of support based on evidence (Shafer 1976). The main advantage of belief functions lies in their ability to represent ignorance and ambiguity. As shown by Ellsberg’s paradox (1961), the probability theory is unable to distinguish between a situation of complete ignorance and a situation where we have complete knowledge (Srivastava, 1997). Dempster-Shafer theory allows the description of partial or complete ignorance, since the belief not accorded to a proposition does not have to be accorded to the negation of that proposition (Cattaneo, 2011).

There are several equivalent ways of representing a belief function, namely basic probability assignment, belief function, plausibility function, and a commonality function.

Consider a set of mutually exclusive and exhaustive propositions, Θ = {θ1, θ2, …, θK}, referred to as frame of discernment. A proposition θi states the lowest level of discernible information. Any proposition that is not singleton, e.g.

{θ1, θ2}, is referred to as a composite.

A basic probability assignment (bpa) m for Θ is a function m: 2Θ → [0, 1] such that m(∅) =0 and Σ{m(A) | A ⊆ Θ}=1 (1) m(A) is a measure of the belief that is committed exactly to A. If m(A) 0, then A is called a focal element of m. Note that if is the complement of A, then m(A) + m( ) ≤ 1. Basic probability assignment differ from a probability function in that they can assign a measure of belief to a subset of the state space without assigning any belief to its elements. If all the focal elements are singletons, a belief function is reduced to a Bayesian probability function (Shafer 1976). Consequently, belief function calculus is a generalization of probability calculus, and any Bayesian model of uncertainty is also a belief function model (Shafer and Srivastava 1990) There are three important functions in DS theory, belief functions, plausibility functions, and commonality functions. They can all be defined in terms of the basic probability assignments.

A belief function Bel corresponding to a bpam is a function Bel: 2Θ → [0, 1] such that Bel(A) = Σ{m(B) | B ⊆ A} for all A ⊆ Θ (2) Bel(A) can be interpreted as the probability of obtaining a set observation that implies the occurrence of A.

A plausibility function Pl corresponding to a bpam is a function Pl: 2Θ → [0, 1] such that Pl(A) = Σ{m(B) | B∩A ≠ ∅} for all A ⊆ Θ (3) Pl(A) can be interpreted as the probability of obtaining a set observation that is consistent with some element of A.

A commonality function Q corresponding to bpam is a function Q: 2Θ → [0, 1] such that Q(A) = Σ{m(B) | B ⊇ A} for all A ⊆ Θ (4) Q(A) can be interpreted as the probability of obtaining a set observation that is consistent with every element of A. Since the m-values add to one, commonality

functions have the property:

362 E. N. Cinicioglu / Decision Making With Consonant Belief Functions

–  –  –

A famous example named “Betty’s testimony”, given by Shafer & Srivastava (1990), demonstrates the fact that belief functions base the beliefs on the evidence.

Suppose that I have a friend called Betty, and according to my subjective probability, Betty is reliable 90% of time, and she is unreliable 10% of time.

Accordingly, P(Betty = reliable) = 0.9, P(Betty = unreliable) = 0.1.

Betty tells me a tree limb fell on my car. Betty's statement must be true if she is reliable, but it is not necessarily false if she is unreliable. So, representing this situation

under a belief function framework, the following results are obtained:

Bel(limb fell) = 0.9 Bel(no limb fell) = 0 It does not mean that I am sure that no limb fell on my car, as a zero probability would. This zero value only means that Betty's testimony gives me no reason to believe that no limb fell on my car.

In the following section, consonant belief functions are introduced, and the use of consonant belief functions for representation of statistical evidence is demonstrated.

2.2. Consonant Belief Functions

A belief function is said to be consonant if its focal elements are nested, meaning that each is contained in the following one (Shafer, 1976). The nested structure of consonant belief functions restricts the number of focal elements the belief function may have. In a belief function which is not consonant, depending on the number of elements n, the belief function may have up to 2n – 1 focal elements. This property of consonant belief functions makes it preferable for representation of statistical evidence (Shafer 1976). However, when Dempster’s rule is applied for combination of consonant belief functions, then the resulting belief function is not consonant any more. For that reason, an alternative rule was proposed by Walley (1987) for combination of partially consonant belief functions.

An example of a consonant bpa m with the frame of discernment {x, y, z}, and the focal elements {x}, {x, z} and {x, y, z} is as follows: m({x}) = 0.5, {x, z} = 0.1 and {x, y, z} = 0.4. The corresponding belief, plausibility, and commonality functions are given in Table 1.

–  –  –

A type of consonant belief functions is partially consonant belief functions, where the state space is partitioned and with focal elements nested within each element of partition, (Walley, 1987). Partially consonant belief functions are the only class of DS belief functions that are consistent with the likelihood principle of statistics. A decision theory for partially consonant belief functions is proposed by Giang and Shenoy (2011).

An example for representation of statistical evidence that use consonant belief functions is provided below as “Missile Fall Down”.

Example: Missile Fall Down Suppose that an attack has been placed, and three foe missiles are thrown to the country of Neverland. The missile defense system of Neverland is able to shut down two of the three missiles. Neverland Security Defense Deputy has the information that all three foe missiles are of the same type, either type X309, type Y118, or type Z127.

However, they do not know which type these three foe missiles do belong. These three types use different technologies and hence have different likelihoods for being shut down by the missile defense system of Neverland. Let MT denote the missile type and let the state space of MT be denoted by ΩMT = {X309, Y118, Z127}. Let F denote the result of Neverland Missile Defense Systems’ response, f indicate the foe missile fell down, nf indicate it did not fell down (it was missed by the missile defense system of Neverland) ΩF = {f, nf}. The probabilistic likelihoods for each type of foe missile for being shut

down by Neverland forces are as follows:



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