17 Introduction to Evidence-Based Robust Optimisation
549
words m(θ 1 ) + m(θ 2 ) = 1. DST, instead, does not make any assumption on
the information available for each of the two events and contemplates also the
ignorance statement θ 1 ∪ θ 2 , or m(θ 1 ) + m(θ 2 ) ≤ 1. For this reason, in DST the
additive rule (m({θ 1 , θ 2 }) = m({θ 1 }) + m({θ 2 })) does not hold and is replaced by
a super-additive (m({θ 1 , θ 2 }) > m({θ 1 }) + m({θ 2 })) or sub-additive (m({θ 1 , θ 2 }) <
m({θ 1 }) + m({θ 2 })) rule. The immediate consequence is that m(A) + m(¬A) ≤ 1,
i.e. the probability on a set and on its negation do not sum to one.
17.2.2 Belief and Plausibility
At the beginning of this section we already informally introduced the idea of belief
Bel and plausibility P l with a simple example. Now that frame of discernment,
power set and bpa are defined, one can more formally calculate the belief and
plausibility associated to a particular quantity of interest f : U → R, where U
is the uncertain space and is here considered to be equal to the power set: U = 2 Θ .
Calling Φ a target region for the quantity of interest f , one is interested in measuring
the degree of belief in the realisation of f (x) ∈ Φ. If one defines the preimage:
A = {x ∈ U |f (x) ∈ Φ},
(17.20)
the total degree of belief associated to (17.20) can be computed by collecting all the
pieces of evidence that either fully or partially support that statement. Given the set
A defined in (17.20) and the focal elements ω, the cumulative functions Bel and P l
are defined as:
Bel(A) =
ω i ⊆A
m(ω i )
(17.21)
P l(A) =
ω i
A =∅
m(ω i )
(17.22)
where ω i is the generic F E inside F, m and the plausibility measure P l is the dual
function of the belief measure Bel:
P l(A) = 1 − Bel(A)
(17.23)
with A is the complement to A. From Eq. (17.21) one can see that the Bel function
is the sum of all the pieces of evidence that completely support the statement f ∈
Φ, whereas the P l function is the sum of all the pieces of evidence that partially
support the statement f ∈ Φ: this means that m(ω i ) is added to Bel only if all
possible realisations of x ∈ ω i belong to A, on the contrary m(ω i ) is added to P l
if at least one realisation of x ∈ ω i belongs to A. This suggests that the Evidential
Interval (EI ) [Bel P l] is a measure of the degree of ignorance on the probability
549
words m(θ 1 ) + m(θ 2 ) = 1. DST, instead, does not make any assumption on
the information available for each of the two events and contemplates also the
ignorance statement θ 1 ∪ θ 2 , or m(θ 1 ) + m(θ 2 ) ≤ 1. For this reason, in DST the
additive rule (m({θ 1 , θ 2 }) = m({θ 1 }) + m({θ 2 })) does not hold and is replaced by
a super-additive (m({θ 1 , θ 2 }) > m({θ 1 }) + m({θ 2 })) or sub-additive (m({θ 1 , θ 2 }) <
m({θ 1 }) + m({θ 2 })) rule. The immediate consequence is that m(A) + m(¬A) ≤ 1,
i.e. the probability on a set and on its negation do not sum to one.
17.2.2 Belief and Plausibility
At the beginning of this section we already informally introduced the idea of belief
Bel and plausibility P l with a simple example. Now that frame of discernment,
power set and bpa are defined, one can more formally calculate the belief and
plausibility associated to a particular quantity of interest f : U → R, where U
is the uncertain space and is here considered to be equal to the power set: U = 2 Θ .
Calling Φ a target region for the quantity of interest f , one is interested in measuring
the degree of belief in the realisation of f (x) ∈ Φ. If one defines the preimage:
A = {x ∈ U |f (x) ∈ Φ},
(17.20)
the total degree of belief associated to (17.20) can be computed by collecting all the
pieces of evidence that either fully or partially support that statement. Given the set
A defined in (17.20) and the focal elements ω, the cumulative functions Bel and P l
are defined as:
Bel(A) =
ω i ⊆A
m(ω i )
(17.21)
P l(A) =
ω i
A =∅
m(ω i )
(17.22)
where ω i is the generic F E inside F, m and the plausibility measure P l is the dual
function of the belief measure Bel:
P l(A) = 1 − Bel(A)
(17.23)
with A is the complement to A. From Eq. (17.21) one can see that the Bel function
is the sum of all the pieces of evidence that completely support the statement f ∈
Φ, whereas the P l function is the sum of all the pieces of evidence that partially
support the statement f ∈ Φ: this means that m(ω i ) is added to Bel only if all
possible realisations of x ∈ ω i belong to A, on the contrary m(ω i ) is added to P l
if at least one realisation of x ∈ ω i belongs to A. This suggests that the Evidential
Interval (EI ) [Bel P l] is a measure of the degree of ignorance on the probability