2 Introduction to Imprecise Probabilities
43
scenario, a value which will not be exceeded with high probability. This can be
achieved by taking some quantile or a risk measure in general (see Chap. 13) or
[3, Ch. 12] as the new objective. Similarly, for the case of constrained optimisation,
we may, again, demand that the violation will be unlikely, i.e. that the probability
of violation will be low. This would lead to a redefinition of the constraint as
P r(constraint is violated) ≤ α. In both cases, we need to specify a concrete number
that represents what exactly are these high and low probabilities, a priori.
In the aforementioned cases, the optimised objective is replaced by a real
function, so standard optimization algorithms can be used.
Let us consider that we, again, want to determine the earliest safe time to enter the
contaminated area from Example 2.1, now with the uncertain parameter a modelled
as a RV with exponential law defined earlier in this section.
The objective function (Eq. (2.3)) is unaffected by the uncertainty in the
parameter; thus, we can keep it as it is. The constraints will now have to be
reformulated, because U(t) is a RV.
If we admit that we cannot ensure the safety certainly, we can still aim for a low
probability of encountering the dangerous environment and replace the constraint
by bounding the probability of exceeding the safety limits, P r(u(t, A) > u s ), say
by α (= 0.01, 0.001, 0.0001 . . .). The optimisation problem derived from Eq. (2.3)
will be reformulated as
min
t≥0
t
s.t. Pr(U(t)>u s ) ≤ α.
(2.6)
The explicit solution, due to monotonicity, will be attained for the first t for which
the 1 − α quantile of U(t) will be equal to u s . Thus, the answer of the model is
• “The smallest safe time t for visiting the area is λ
ln(u s )
ln(1−α) ”.
Although probability theory provides a convenient modelling framework, it
is difficult to properly encode available information into probabilistic models.
To construct and manipulate the models, we often have to postulate additional
assumptions, which we may not be able to justify (independence of RVs, specific
low-dimensional distribution models, etc.). Also, even if our assumptions were
correct, if we were to construct the models using the methods of statistical inference,
we could only approach the true model asymptotically, as the number of samples
would approach infinity. But in engineering applications, we often have only a small
number of observations, which makes standard inference methods unreliable. Also,
the knowledge elicitation process requires that a domain expert exactly assigns
probabilities to each possible event, which is generally considered impossible.
The situation is, in a sense, analogical to that of providing point estimates, here
for the distributions, and may be solved either by modelling the uncertainty by a
hierarchical stochastic model, or, again, by introducing imprecision.
43
scenario, a value which will not be exceeded with high probability. This can be
achieved by taking some quantile or a risk measure in general (see Chap. 13) or
[3, Ch. 12] as the new objective. Similarly, for the case of constrained optimisation,
we may, again, demand that the violation will be unlikely, i.e. that the probability
of violation will be low. This would lead to a redefinition of the constraint as
P r(constraint is violated) ≤ α. In both cases, we need to specify a concrete number
that represents what exactly are these high and low probabilities, a priori.
In the aforementioned cases, the optimised objective is replaced by a real
function, so standard optimization algorithms can be used.
Let us consider that we, again, want to determine the earliest safe time to enter the
contaminated area from Example 2.1, now with the uncertain parameter a modelled
as a RV with exponential law defined earlier in this section.
The objective function (Eq. (2.3)) is unaffected by the uncertainty in the
parameter; thus, we can keep it as it is. The constraints will now have to be
reformulated, because U(t) is a RV.
If we admit that we cannot ensure the safety certainly, we can still aim for a low
probability of encountering the dangerous environment and replace the constraint
by bounding the probability of exceeding the safety limits, P r(u(t, A) > u s ), say
by α (= 0.01, 0.001, 0.0001 . . .). The optimisation problem derived from Eq. (2.3)
will be reformulated as
min
t≥0
t
s.t. Pr(U(t)>u s ) ≤ α.
(2.6)
The explicit solution, due to monotonicity, will be attained for the first t for which
the 1 − α quantile of U(t) will be equal to u s . Thus, the answer of the model is
• “The smallest safe time t for visiting the area is λ
ln(u s )
ln(1−α) ”.
Although probability theory provides a convenient modelling framework, it
is difficult to properly encode available information into probabilistic models.
To construct and manipulate the models, we often have to postulate additional
assumptions, which we may not be able to justify (independence of RVs, specific
low-dimensional distribution models, etc.). Also, even if our assumptions were
correct, if we were to construct the models using the methods of statistical inference,
we could only approach the true model asymptotically, as the number of samples
would approach infinity. But in engineering applications, we often have only a small
number of observations, which makes standard inference methods unreliable. Also,
the knowledge elicitation process requires that a domain expert exactly assigns
probabilities to each possible event, which is generally considered impossible.
The situation is, in a sense, analogical to that of providing point estimates, here
for the distributions, and may be solved either by modelling the uncertainty by a
hierarchical stochastic model, or, again, by introducing imprecision.