2 Introduction to Imprecise Probabilities
69
Remark Here, the equality in the dual model occurs as a is a free variable in the
original problem.
Here, p j ’s (j = 1, 2, . . . , k) are the probability mass functions. Note that
k
j =1 p j g(x j ) is the expectation of gamble g with respect to the probability mass
functions p 1 , p 2 , . . . , p n . The feasible region forms a convex set of probability mass
functions. We call this convex set of probability mass functions the credal set. This
duality relation shows the interesting connection between the lower prevision and
credal sets M. It also shows that the natural extension of a lower prevision is the
lower expectation with respect to the credal set.
Theorem 2.7 (Lower Envelope Theorem) Let P be any lower prevision,
and let M be the corresponding credal set. Then, the following statements
are true:
1. P avoids sure loss if M is non-empty.
2. If P avoids sure loss, and then, its natural extension E is the lower
envelope of M, that is, it satisfies
E(g) = min
⎧
⎨
⎩
k
j =1
p j g(x j ) : p j ∈ M j = 1, 2, . . . k
⎫
⎬
⎭
for all g ∈ B
(2.22)
3. P is coherent if it avoids sure loss and for all g ∈ dom P
P (g) = min
⎧
⎨
⎩
k
j =1
p j g(x j ) : p j ∈ M j = 1, 2, . . . k
⎫
⎬
⎭
(2.23)
The above theorem is a direct consequence of the duality relation derived from
the natural extension for a finite set of outcomes and a finite domain of P . It shows
that the natural extension of the lower previsions is the lower expectation. This can
also be extended for infinite sets, and the generalisation can be proved using the
Hahn–Banach theorem. This allows us to characterise thee notions of avoiding sure
loss, coherence and natural extension of the lower prevision in terms of their dual
models [40].
We can derive this condition for avoiding sure loss using the example in
Table 2.5. We saw in the example that, we no longer incur sure loss with the
revised buying price. This can be verified with the help of Eq. (2.16). We can write
Eq. (2.16) as follows:
69
Remark Here, the equality in the dual model occurs as a is a free variable in the
original problem.
Here, p j ’s (j = 1, 2, . . . , k) are the probability mass functions. Note that
k
j =1 p j g(x j ) is the expectation of gamble g with respect to the probability mass
functions p 1 , p 2 , . . . , p n . The feasible region forms a convex set of probability mass
functions. We call this convex set of probability mass functions the credal set. This
duality relation shows the interesting connection between the lower prevision and
credal sets M. It also shows that the natural extension of a lower prevision is the
lower expectation with respect to the credal set.
Theorem 2.7 (Lower Envelope Theorem) Let P be any lower prevision,
and let M be the corresponding credal set. Then, the following statements
are true:
1. P avoids sure loss if M is non-empty.
2. If P avoids sure loss, and then, its natural extension E is the lower
envelope of M, that is, it satisfies
E(g) = min
⎧
⎨
⎩
k
j =1
p j g(x j ) : p j ∈ M j = 1, 2, . . . k
⎫
⎬
⎭
for all g ∈ B
(2.22)
3. P is coherent if it avoids sure loss and for all g ∈ dom P
P (g) = min
⎧
⎨
⎩
k
j =1
p j g(x j ) : p j ∈ M j = 1, 2, . . . k
⎫
⎬
⎭
(2.23)
The above theorem is a direct consequence of the duality relation derived from
the natural extension for a finite set of outcomes and a finite domain of P . It shows
that the natural extension of the lower previsions is the lower expectation. This can
also be extended for infinite sets, and the generalisation can be proved using the
Hahn–Banach theorem. This allows us to characterise thee notions of avoiding sure
loss, coherence and natural extension of the lower prevision in terms of their dual
models [40].
We can derive this condition for avoiding sure loss using the example in
Table 2.5. We saw in the example that, we no longer incur sure loss with the
revised buying price. This can be verified with the help of Eq. (2.16). We can write
Eq. (2.16) as follows: