2 Introduction to Imprecise Probabilities
39
for possible values of observed quantities. It is also often easier for experts to specify
some “credible bounds” for a parameter, instead of a precise value. In all these
cases, and many others, the uncertain variables of interest are only known to belong
to some set, with no further preferences of credibility among its elements. Let us
therefore return to the previous example and investigate what would happen if we
were to know only that the model parameter a from the Example 2.1 lies in a set
Ω A .
As will be the case also in the next types of uncertain parameter specifications,
the qualitative nature of the parameter will be carried, propagated, through the
model and provide an answer of similar quality. If we propagate an imprecise
parameter through a deterministic model, the model will, generally, give us imprecise answers. In Eq. (2.1), it will be a set of credible pollutant levels. Our least
informative assessment about u(t; a) is the image of the union of its arguments’
domains, u(t, Ω A ) := {u(t, a)|a ∈ Ω A }. For real-valued functions, considering
that these sets are intervals, and describing the uncertainty by the lower and upper
bounds on the quantity of interest (QoI) often suffice. Translated to u(t) from
Example 2.1, our assessment about u(t) will take the form
u(t) ∈
inf
a∈Ω A
u(t; a), sup
a∈Ω A
u(t; a)
= [u(t), u(t)].
(2.4)
Let us assume that we know that a ∈ Ω A =
a, a
. For the process in
Example 2.1, we will exploit that the function u(t; a) is monotone (decreasing) in a
for all t. Thus, the extremes will be attained on the set boundary. Given the explicit
solution (Eq. (2.2)), we may therefore judge that
∀t : u(t) ∈ [u(t; max{a ∈ Ω A }), u(t; min{a ∈ Ω A })]
= [exp(−at), exp
−at
].
(2.5)
Time evolution of the pollution is presented in Fig. 2.1 via the lower and upper
bounds.
A remark: Through imprecision, we are actually able to model a wider set of
problems without introducing any additional computational complexity. Consider a
dynamical process described by ordinary differential equation d t ˜
u(t) = −a(t) ˜
u(t)
with initial condition ˜
u(0) = 1. The explicit solution is ˜
u(t) = exp
−
t
0 a(x)dx
.
Let us further assume that the exponential rate is bounded at each time, i.e. ∀t :
a(t) ∈
a, a
. Because
˜
u(t) = exp
−
t
0
a(x)dx
< exp
−
t
0
adx
= exp
−at
= u(t),
39
for possible values of observed quantities. It is also often easier for experts to specify
some “credible bounds” for a parameter, instead of a precise value. In all these
cases, and many others, the uncertain variables of interest are only known to belong
to some set, with no further preferences of credibility among its elements. Let us
therefore return to the previous example and investigate what would happen if we
were to know only that the model parameter a from the Example 2.1 lies in a set
Ω A .
As will be the case also in the next types of uncertain parameter specifications,
the qualitative nature of the parameter will be carried, propagated, through the
model and provide an answer of similar quality. If we propagate an imprecise
parameter through a deterministic model, the model will, generally, give us imprecise answers. In Eq. (2.1), it will be a set of credible pollutant levels. Our least
informative assessment about u(t; a) is the image of the union of its arguments’
domains, u(t, Ω A ) := {u(t, a)|a ∈ Ω A }. For real-valued functions, considering
that these sets are intervals, and describing the uncertainty by the lower and upper
bounds on the quantity of interest (QoI) often suffice. Translated to u(t) from
Example 2.1, our assessment about u(t) will take the form
u(t) ∈
inf
a∈Ω A
u(t; a), sup
a∈Ω A
u(t; a)
= [u(t), u(t)].
(2.4)
Let us assume that we know that a ∈ Ω A =
a, a
. For the process in
Example 2.1, we will exploit that the function u(t; a) is monotone (decreasing) in a
for all t. Thus, the extremes will be attained on the set boundary. Given the explicit
solution (Eq. (2.2)), we may therefore judge that
∀t : u(t) ∈ [u(t; max{a ∈ Ω A }), u(t; min{a ∈ Ω A })]
= [exp(−at), exp
−at
].
(2.5)
Time evolution of the pollution is presented in Fig. 2.1 via the lower and upper
bounds.
A remark: Through imprecision, we are actually able to model a wider set of
problems without introducing any additional computational complexity. Consider a
dynamical process described by ordinary differential equation d t ˜
u(t) = −a(t) ˜
u(t)
with initial condition ˜
u(0) = 1. The explicit solution is ˜
u(t) = exp
−
t
0 a(x)dx
.
Let us further assume that the exponential rate is bounded at each time, i.e. ∀t :
a(t) ∈
a, a
. Because
˜
u(t) = exp
−
t
0
a(x)dx
< exp
−
t
0
adx
= exp
−at
= u(t),