2 Introduction to Imprecise Probabilities
37
several drawbacks, which make their application questionable in practical scenarios.
Interval arithmetic is often overly conservative and fails to capture correlations
among quantities of interest. Probability theory requires us to specify how likely the
occurrence of each possible outcome is, which can be impossible up to the required
level of precision needed to construct the mathematical models.
To overcome the issues with these formalisms, we will demonstrate the theory,
which results from their flourishing marriage. We will introduce imprecise probability (IP) theory.
The history of imprecision in probabilistic assessments dates back to Boole’s
work on inductive logic [8, Chap. 18]. Imprecise probabilities could also be identified in several attempts to obtain bounds on probabilistic assessments when precise
values were intractable (e.g. Markov’s, Jensen’s and Chebyschev’s inequalities).
Some early examples may also be found in the field of sensitivity analysis for statistical inference [6]. Nevertheless, by the mid-twentieth century, a separated theory
of imprecise probabilities began to emerge as a generalisation of probability theory.
This would, not exclusively, include the introduction of non-additive measures by
Choquet [10], generalisation of statistical inference by Dempster [16], Walley’s
work on statistical inference with imprecise probabilities [40], and development of
the theory of lower previsions [39].
In this chapter, we intend to show the basic ideas and structures behind IP theory
together with some examples of its application.
2.2 Some Models of Uncertainty
We intend to begin the chapter with a practical example to show how uncertainty
may be modelled and how it influences the quality of our predictions. This will be
demonstrated on a simple, analytically solvable decision problem. We will show
solutions given by various models and highlight their differences, but also their
similarities.
For the rest of this section, we will be interested in the following scenario.
Example 2.1 Suppose that there exists an area, which is polluted. Such
pollution will slowly deteriorate over time. Our question is, what is the earliest
time when we can send people there without risking their health?
Let this be the set of information available to us without a doubt:
• Pollution level is known at time t = 0, say u(0) = 1.
• The highest pollution level, which does not pose any danger to human
health, u s ∈ R, is also known.
• The pollution level decreases according to a known relationship,
(continued)
37
several drawbacks, which make their application questionable in practical scenarios.
Interval arithmetic is often overly conservative and fails to capture correlations
among quantities of interest. Probability theory requires us to specify how likely the
occurrence of each possible outcome is, which can be impossible up to the required
level of precision needed to construct the mathematical models.
To overcome the issues with these formalisms, we will demonstrate the theory,
which results from their flourishing marriage. We will introduce imprecise probability (IP) theory.
The history of imprecision in probabilistic assessments dates back to Boole’s
work on inductive logic [8, Chap. 18]. Imprecise probabilities could also be identified in several attempts to obtain bounds on probabilistic assessments when precise
values were intractable (e.g. Markov’s, Jensen’s and Chebyschev’s inequalities).
Some early examples may also be found in the field of sensitivity analysis for statistical inference [6]. Nevertheless, by the mid-twentieth century, a separated theory
of imprecise probabilities began to emerge as a generalisation of probability theory.
This would, not exclusively, include the introduction of non-additive measures by
Choquet [10], generalisation of statistical inference by Dempster [16], Walley’s
work on statistical inference with imprecise probabilities [40], and development of
the theory of lower previsions [39].
In this chapter, we intend to show the basic ideas and structures behind IP theory
together with some examples of its application.
2.2 Some Models of Uncertainty
We intend to begin the chapter with a practical example to show how uncertainty
may be modelled and how it influences the quality of our predictions. This will be
demonstrated on a simple, analytically solvable decision problem. We will show
solutions given by various models and highlight their differences, but also their
similarities.
For the rest of this section, we will be interested in the following scenario.
Example 2.1 Suppose that there exists an area, which is polluted. Such
pollution will slowly deteriorate over time. Our question is, what is the earliest
time when we can send people there without risking their health?
Let this be the set of information available to us without a doubt:
• Pollution level is known at time t = 0, say u(0) = 1.
• The highest pollution level, which does not pose any danger to human
health, u s ∈ R, is also known.
• The pollution level decreases according to a known relationship,
(continued)