2 Introduction to Imprecise Probabilities
71
exists p j ∈ M, such that our expected reward on each gamble is higher than the
supremum buying price of each gamble. That is, M is non-empty.
2.6 Constructing the Laws
The classical interpretation of probability is the relative frequency of occurrence.
Approximate models of the underlying laws may be constructed from finite number
of observations as probability distributions. Conversely, the subjectivistic point
of view does not treat probabilities as objective quantities but rather as a tool
to describe our state of knowledge about quantities and our degrees of belief in
statements. The problem of what probability is and how it can be measured is
more rigorously addressed in [14, 32]. Statistical methods are usually employed
for model construction if observations are available. These seek a way to find a
mathematical form of the probability distribution of the uncertain quantity which
complies with the available information. If observations are not present, we need
to rely on an expert opinion to construct the models via the process of expert
knowledge elicitation. We will omit the expert elicitation process in this chapter.
Some ideas of what can be elicited in the subjective setting can be found in [15] for
precise models or in [40] for IP models.
In this section, we will briefly remind the basics of statistical inference. For
an exhaustive treatment, we refer the reader to [9]. We are going to revise the
basic principles to extract knowledge from the available data. Our main aim was
to introduce how the inferential procedures can be extended for the IP theory.
2.6.1 Statistical Inference with Precise Probabilities
Let us now recall the basic methods of statistical inference for data analysis.
Hereafter, we will assume that we have a set of measurements
x = {x 1 , . . . , x n },
independent and identically distributed (i.i.d.) samples, which were generated
according to some precise ground-truth distribution ˆ
P . Our final intention was to
provide probabilities of various events of interest based/conditioned on this dataset.
The common practice is to construct an approximation (model) P of the sampling
distribution ˆ
P and estimate the desired probabilities from P .
A simple way of inferring probability distributions from a set of samples is
given by non-parametric methods. Here, for an arbitrary event E, the probability
is estimated as P (E) =
1
n #{x i ; x i ∈ E}, the relative ratio of observations which
comply with E. Distributions inferred in this way are usually labelled as empirical
and constitute models with least additional assumptions. An example of an empirical
CDF is depicted in Fig. 2.7 with a label “empirical”.
Alternatives to the non-parametric methods search for an approximative distribution by inverting the model of the sampling process. They look for an answer
71
exists p j ∈ M, such that our expected reward on each gamble is higher than the
supremum buying price of each gamble. That is, M is non-empty.
2.6 Constructing the Laws
The classical interpretation of probability is the relative frequency of occurrence.
Approximate models of the underlying laws may be constructed from finite number
of observations as probability distributions. Conversely, the subjectivistic point
of view does not treat probabilities as objective quantities but rather as a tool
to describe our state of knowledge about quantities and our degrees of belief in
statements. The problem of what probability is and how it can be measured is
more rigorously addressed in [14, 32]. Statistical methods are usually employed
for model construction if observations are available. These seek a way to find a
mathematical form of the probability distribution of the uncertain quantity which
complies with the available information. If observations are not present, we need
to rely on an expert opinion to construct the models via the process of expert
knowledge elicitation. We will omit the expert elicitation process in this chapter.
Some ideas of what can be elicited in the subjective setting can be found in [15] for
precise models or in [40] for IP models.
In this section, we will briefly remind the basics of statistical inference. For
an exhaustive treatment, we refer the reader to [9]. We are going to revise the
basic principles to extract knowledge from the available data. Our main aim was
to introduce how the inferential procedures can be extended for the IP theory.
2.6.1 Statistical Inference with Precise Probabilities
Let us now recall the basic methods of statistical inference for data analysis.
Hereafter, we will assume that we have a set of measurements
x = {x 1 , . . . , x n },
independent and identically distributed (i.i.d.) samples, which were generated
according to some precise ground-truth distribution ˆ
P . Our final intention was to
provide probabilities of various events of interest based/conditioned on this dataset.
The common practice is to construct an approximation (model) P of the sampling
distribution ˆ
P and estimate the desired probabilities from P .
A simple way of inferring probability distributions from a set of samples is
given by non-parametric methods. Here, for an arbitrary event E, the probability
is estimated as P (E) =
1
n #{x i ; x i ∈ E}, the relative ratio of observations which
comply with E. Distributions inferred in this way are usually labelled as empirical
and constitute models with least additional assumptions. An example of an empirical
CDF is depicted in Fig. 2.7 with a label “empirical”.
Alternatives to the non-parametric methods search for an approximative distribution by inverting the model of the sampling process. They look for an answer