2 Introduction to Imprecise Probabilities
41
• “The smallest safe time t for visiting the area is −
ln(u s )
a ” in the optimistic (Γ -
maximax) case, which provides the earliest time for which compliance with
regulations is possible, but not assured.
2.2.3 A Probability Distribution
Probability theory is a dominant framework to address uncertainty in science and
engineering. When an outcome of an experiment cannot be determined uniquely
from the available information, probability theory aims to formulate a law which
models the behaviour of repeated outcomes from identical trials.
Apart from modelling of the repetition of trials, probability theory also provides a
consistent reasoning framework, an extension to boolean logic [26]. With probability theory, we may encode our degree of faith in logical statements as probabilities
(e.g. x ∈ [1, 2] as P r(x ∈ [1, 2])) and utilise probability theory to also obtain
consistent degrees of faith for derived statements (e.g. P r(f (X) < 3)). There
are various ways to construct these models, ranging from statistical inference to
elicitation by domain experts.
If a model parameter is a random variable (RV), the model predictions themselves
are treated as RVs too. Especially when neither the random parameter nor the function are bounded, without further assumptions, we cannot construct any reasonable
bounded set of credible values of the argument to carry out the best–worst case
scenario inferences as in Sect. 2.2.2 (however, a heuristic construction is possible
via confidence intervals and credible sets). Without loss of generality, we can assess
the distribution of our predictions and back our further decisions on probabilistic
logic.
Let us consider Example 2.1 again. Now, we will assume that the parameter a is
a RV, A (denoted by a capital letter), distributed according to the exponential law
P A (·) with rate parameter λ and cumulative distribution function (CDF)
F A (x) := P r(A < x) = 1 − exp(−λx).
We can straightforwardly express the distribution of U(t) (again denoted by a capital
letter to emphasise that it is a RV) as
P r(U (t) ∈ E) = P A ({a : u(t; a) ∈ E}).
This will be further formalised in Eq. (2.7). Due to monotonicity, we get a CDF for
every U(t) , the F U(t) , as (Theorem 2.1)
41
• “The smallest safe time t for visiting the area is −
ln(u s )
a ” in the optimistic (Γ -
maximax) case, which provides the earliest time for which compliance with
regulations is possible, but not assured.
2.2.3 A Probability Distribution
Probability theory is a dominant framework to address uncertainty in science and
engineering. When an outcome of an experiment cannot be determined uniquely
from the available information, probability theory aims to formulate a law which
models the behaviour of repeated outcomes from identical trials.
Apart from modelling of the repetition of trials, probability theory also provides a
consistent reasoning framework, an extension to boolean logic [26]. With probability theory, we may encode our degree of faith in logical statements as probabilities
(e.g. x ∈ [1, 2] as P r(x ∈ [1, 2])) and utilise probability theory to also obtain
consistent degrees of faith for derived statements (e.g. P r(f (X) < 3)). There
are various ways to construct these models, ranging from statistical inference to
elicitation by domain experts.
If a model parameter is a random variable (RV), the model predictions themselves
are treated as RVs too. Especially when neither the random parameter nor the function are bounded, without further assumptions, we cannot construct any reasonable
bounded set of credible values of the argument to carry out the best–worst case
scenario inferences as in Sect. 2.2.2 (however, a heuristic construction is possible
via confidence intervals and credible sets). Without loss of generality, we can assess
the distribution of our predictions and back our further decisions on probabilistic
logic.
Let us consider Example 2.1 again. Now, we will assume that the parameter a is
a RV, A (denoted by a capital letter), distributed according to the exponential law
P A (·) with rate parameter λ and cumulative distribution function (CDF)
F A (x) := P r(A < x) = 1 − exp(−λx).
We can straightforwardly express the distribution of U(t) (again denoted by a capital
letter to emphasise that it is a RV) as
P r(U (t) ∈ E) = P A ({a : u(t; a) ∈ E}).
This will be further formalised in Eq. (2.7). Due to monotonicity, we get a CDF for
every U(t) , the F U(t) , as (Theorem 2.1)