Table of Contents
The belief system
The representation of uncertain probabilities will be based on a belief model similar to the one used in the Dempster-Shafer theory of evidence. The Dempster-Shafer theory was first set forth by Dempster in the 1960s as a framework for upper and lower probability bounds and subsequently extended by Shafer who in 1976 published A Mathematical Theory of Evidence [5] [6].
The first step in applying the Dempster-Shafer belief model is to define a set of possible situations, which is called the frame of discernment. A frame of discernment delimits a set of possible states of a given system, exactly one of which is assumed to be true at any one time. Fig.1 illustrates a simple frame of discernment denoted by Θ with 4 elementary states x_1,x_2,x_3,x_4∈Θ.
Fig. 1 Example frame of discernment
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The elementary states in the frame of discernment 2 will be called atomic states because they do not contain sub states. The power set of Θ, denoted by 2
Θ;, contains the atomic states and all possible unions of the atomic states, including Θ. A frame of discernment can be finite or infinite, in which cases the corresponding power set is also finite or infinite respectively.
An observer who believes that one or several states in the power set of Θ might be true can assign belief mass to these states. Belief mass on an atomic state x∈2
Θ is interpreted as the belief that the state in question is true. Belief mass on a non-atomic state x∈2
Θ is interpreted as the belief that one of the atomic states it contains is true, but the observer is uncertain about which of them is true. The following definition is fundamental of the Dempster-Shafer theory.
Belief Mass Assignment
Let Θ be a frame of discernment. If with each subset x∈2
Θa number m_Θ (x) associated such that:
m_Θ (x)≥0
m_Θ (x)=0
∑_(x∈2
Θ)▒〖m_Θ (x)〗=1
Then m_Θis called belief mass assignment on Θ. For each subset x∈2
Θ, the number m_Θ (x) is called belief mass of x [1] [2].
Belief Function
Let Θ be a frame of discernment, and let m_Θ be a BMA on Θ. Then the belief function corresponding with m_Θ is the function b∶ 2
Θ↦[0,1] defined by:
b(x)=∑_(y⊆x)▒〖 m_Θ (x), x,y ∈2
Θ 〗
Similarly, to believe, an observer’s disbelief must be interpreted as the total belief that a state is not true.
Disbelief Function
Let Θ be a frame of discernment, and let m_Θ be a BMA on Θ. Then the disbelief function corresponding with m_Θ is the function d: 2
Θ↦[0,1] defined by:
d(x)=∑_(y∩x=∅)▒〖 m_Θ (x), x,y ∈2
Θ 〗
Uncertain Function
Let Θ be a frame of discernment, and let m_Θ be a BMA on Θ. Then the uncertain function corresponding with m_Θ is the function u: 2
Θ↦[0,1] defined by:
u(x)=∑_█(y∩x=∅@y⊈x)▒〖 m_Θ (x), x,y ∈2
Θ 〗
Total uncertainty can be expressed by assigning all the belief mass to Θ. The belief function corresponding to this situation is called the vacuous belief function. A BMA with zero belief mass assigned to Θ is called a dogmatic BMA.
With the concepts defined so far, a simple theorem can be stated: “The sum of the belief, disbelief and uncertainty functions is equal to the sum of the belief masses in a BMA which according to sums up to 1. [7]” Which is fundamental of uncertain probability.
b(x)+d(x)+u(x)=1,x∈2
Θ,x≠∅
Relative Atomicity
Let Θ be a frame of discernment and let x,y ∈2
Θ. Then for any given y≠∅ the relative atomicity of x to y is the function a: 2
Θ↦[0,1] defined by:
a(x/y)=(x∩y)/(|y|), x,y ∈2
Θ,y≠∅
It can be observed that x∩y=∅⟹a(x/y)=0, and that y⊆x ⟹a(x/y)=1. In all other cases the relative atomicity will be a value from 0 and 1. The relative atomicity of an atomic state to its frame of discernment, denoted by a(x/Θ), can simply denoted as a(x), which is also called prior base rate.
Probability Expectation
Let Θ be a frame of discernment with BMA m_Θ, Then the probability expectation function corresponding with m_(Θ ) is the function E∶ 2
Θ↦[0,1] defined by:
E(x)=∑_y▒〖m_Θ (y)a(x/y), y∈2
Θ 〗
Opinion
Let Θ be a binary frame of discernment with 2 atomic states x and ¬x, and let m_Θbe a BMA on Θ where b(x),d(x),u(x) and a(x) represent the belief, disbelief and relative atomicity function on x in 2
Θrespectively. Then the opinion about x, denoted by ω_x, is the tuple defined by:
ω_x≡(b(x),d(x),u(x) and a(x)).
Fig. 2 graphically illustrate opinions defined in above equation.
Fig. 2 Opinion triangle
The horizontal bottom line between the belief and disbelief corners in Fig.2 is called the probability axis. The relative atomicity can be graphically represented as a point on the probability axis. The line joining the top corner of the triangle and the relative atomicity point becomes the director.
The projector is parallel to the director and passes through the opinion point. Its intersection with the probability axis defines the probability expectation value.
Opinions situated on the probability axis are called dogmatic opinions. They represent situations without uncertainty and correspond to traditional frequentist probabilities. The distance between an opinion point and the probability axis can be interpreted as the degree of uncertainty.
Opinions situated in the left or right corner, i.e. with either b=1 or d=1 are called absolute opinions. They represent situations where it is absolutely certain that a state is either true or false and correspond to ‘TRUE’ or ‘FALSE’ proposition in binary logic.
Ordering of Opinions
Let w_xand w_ybe two opinions. They can be ordered according to the below criteria of priority:
The opinion with the greatest probability is the greatest opinion.
The opinion with the least uncertainty is the greatest opinion.
The opinion with the least relative atomicity is the greatest opinion.
