44
D. Krpelík and T. Basu
2.2.4 A Set of Probability Distributions
In the framework of IP, we will combine the approaches introduced in Sects. 2.2.2
and 2.2.3. The core idea is to consider a set of probability distributions among
which we do not make any further judgements about their likeliness of being the
true model. Including the likeliness, as in the case of hierarchical modelling, would
result in a collapse of the set of admissible distributions into a single one, a mixture
of distributions. With a purely imprecise model, we are again allowed to ask for
expected values and probabilities of events, such as in the precise case, but the
answers are now set-valued (often a simple interval). This is analogical to what had
happened when we asked for the value of a function with an imprecisely specified
parameter in Sect. 2.2.2.
A rigorous approach to IP will be described in the following sections. Let us now,
for the sake of an introduction, just consider what would happen if we only knew
that the parameter a from the Example 2.1 is an imprecise RV A, which follows one
of the distributions P A := {exp(λ) : λ ∈ [λ, λ]}; however, we cannot further specify
which one. We can ask for probabilities of events A ∈ E, derived events u(t; A) ∈
E and the expected values. Given that all these depend on the underlying probability
distribution, we can, as in Sect. 2.2.2, consider all the possible values they can attain
over the set of all the admissible distributions P A . For practical reasons, we can just
focus on the lower and upper bounds for P (.) and E [.].
Similarly, as in the case of precise distribution, we would like to assess the CDF
of U(t). In the IP framework, and using monotonicity properties of our example, we
can construct bounds on the inferred CDF of U(t).
F (U(t) < x) ∈
min
p∈P A
{1 − F p (u
−1
t (x))}, max
p∈P A
{1 − F p (u
−1
t (x))}
=
exp
−λu
−1
t (x)
, exp
−λu
−1
t (x)
=
x
λ
t
, x
λ
t
=:
P (U (t) < x), P (U(t) < x)
,
where P and P are, respectively, the so-called lower and upper probability
measures, which will be properly introduced in Sect. 2.4.
Similarly, we can do with the quantiles. These would again be given as extremes
over the set of admissible distributions. An example of such an inference is depicted
in Fig. 2.3.
The theory of imprecise probabilities provides a more general theoretical framework for modelling different types of uncertainties and the subsequent (consistent)
reasoning. Boolean logic measures statements as true or false, and probabilistic
logic measures each statement by a real number, · ∈ [0, 1], the probability of them
being true. The IP framework introduces a possibility for modelling ignorance. For
each statement, it can supply a probability of it being true and the probability of the
D. Krpelík and T. Basu
2.2.4 A Set of Probability Distributions
In the framework of IP, we will combine the approaches introduced in Sects. 2.2.2
and 2.2.3. The core idea is to consider a set of probability distributions among
which we do not make any further judgements about their likeliness of being the
true model. Including the likeliness, as in the case of hierarchical modelling, would
result in a collapse of the set of admissible distributions into a single one, a mixture
of distributions. With a purely imprecise model, we are again allowed to ask for
expected values and probabilities of events, such as in the precise case, but the
answers are now set-valued (often a simple interval). This is analogical to what had
happened when we asked for the value of a function with an imprecisely specified
parameter in Sect. 2.2.2.
A rigorous approach to IP will be described in the following sections. Let us now,
for the sake of an introduction, just consider what would happen if we only knew
that the parameter a from the Example 2.1 is an imprecise RV A, which follows one
of the distributions P A := {exp(λ) : λ ∈ [λ, λ]}; however, we cannot further specify
which one. We can ask for probabilities of events A ∈ E, derived events u(t; A) ∈
E and the expected values. Given that all these depend on the underlying probability
distribution, we can, as in Sect. 2.2.2, consider all the possible values they can attain
over the set of all the admissible distributions P A . For practical reasons, we can just
focus on the lower and upper bounds for P (.) and E [.].
Similarly, as in the case of precise distribution, we would like to assess the CDF
of U(t). In the IP framework, and using monotonicity properties of our example, we
can construct bounds on the inferred CDF of U(t).
F (U(t) < x) ∈
min
p∈P A
{1 − F p (u
−1
t (x))}, max
p∈P A
{1 − F p (u
−1
t (x))}
=
exp
−λu
−1
t (x)
, exp
−λu
−1
t (x)
=
x
λ
t
, x
λ
t
=:
P (U (t) < x), P (U(t) < x)
,
where P and P are, respectively, the so-called lower and upper probability
measures, which will be properly introduced in Sect. 2.4.
Similarly, we can do with the quantiles. These would again be given as extremes
over the set of admissible distributions. An example of such an inference is depicted
in Fig. 2.3.
The theory of imprecise probabilities provides a more general theoretical framework for modelling different types of uncertainties and the subsequent (consistent)
reasoning. Boolean logic measures statements as true or false, and probabilistic
logic measures each statement by a real number, · ∈ [0, 1], the probability of them
being true. The IP framework introduces a possibility for modelling ignorance. For
each statement, it can supply a probability of it being true and the probability of the