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D. Krpelík and T. Basu
2.3 Probability Theory
In this section, we will review selected topics of probability theory. In the two
following sections, we will do the same for the theory of imprecise probabilities.
Since both these fields cover a vast range of topics, we have decided to focus on
what constitutes the underlying structure of these theories and how they relate to
each other. Special emphasis will be put on extending our (partial) specification of
the model to answer enquiries about derived quantities consistently. This means that
given some claims about some aspects of some RVs, we are interested in what other
claims can be deduced about transformed RVs.
We will show two complementary approaches for building an axiomatic theory
of probability. The first one is based on Kolmogorov’s formulation [27], in which
a probability distribution is represented by a positive additive measure (Sect. 2.3.1).
Such measure, which is a set function, directly encodes the modelled probabilities
of various assertions about the outcomes of random experiments, allows us to
assess an expected value of a RV and also extends the models to derived random
quantities. This approach to probability theory has become dominant across fields
as it offers an intuitive description of random outcomes and enables us to construct
efficient general algorithms for solving many practical problems (Monte Carlo
algorithms, Bayesian inference, etc.). The measure-theoretic formulation will then
be generalised for IP in Sect. 2.4.
Another approach for constructing an axiomatic base for probability theory
is based on a functional representation of random quantities [15, 44]. Here,
each probability distribution is represented by a functional, the prevision, which
corresponds to the expected value operator in the measure-theoretic approach. A
model is specified by assessing the expected values for several selected functions—
the RVs. This allows us to pose less assumptions on the models since the underlying
probability measure does not need to be specified exactly, but also pose limitations
in extending the assessments to derived quantities. These extensions will (mostly)
result only in bounds on the expectations of the derived RVs. The approach will be
fully generalised for imprecise probabilities in Sect. 2.5.
2.3.1 Measure-Theoretic Probability
Suppose that we are to perform an experiment. We will denote the set of all its
possible outcomes as the sample space, Ω. Let us further assume that an outcome
cannot be exactly determined prior to its actual observation—it is uncertain. But
even though the experimental outcome can be random (i.e. we are not able to
predict it using any finite algorithm), multiple repetitions of the same experiment
may follow some predictable law. Probability theory aims to describe these laws.
D. Krpelík and T. Basu
2.3 Probability Theory
In this section, we will review selected topics of probability theory. In the two
following sections, we will do the same for the theory of imprecise probabilities.
Since both these fields cover a vast range of topics, we have decided to focus on
what constitutes the underlying structure of these theories and how they relate to
each other. Special emphasis will be put on extending our (partial) specification of
the model to answer enquiries about derived quantities consistently. This means that
given some claims about some aspects of some RVs, we are interested in what other
claims can be deduced about transformed RVs.
We will show two complementary approaches for building an axiomatic theory
of probability. The first one is based on Kolmogorov’s formulation [27], in which
a probability distribution is represented by a positive additive measure (Sect. 2.3.1).
Such measure, which is a set function, directly encodes the modelled probabilities
of various assertions about the outcomes of random experiments, allows us to
assess an expected value of a RV and also extends the models to derived random
quantities. This approach to probability theory has become dominant across fields
as it offers an intuitive description of random outcomes and enables us to construct
efficient general algorithms for solving many practical problems (Monte Carlo
algorithms, Bayesian inference, etc.). The measure-theoretic formulation will then
be generalised for IP in Sect. 2.4.
Another approach for constructing an axiomatic base for probability theory
is based on a functional representation of random quantities [15, 44]. Here,
each probability distribution is represented by a functional, the prevision, which
corresponds to the expected value operator in the measure-theoretic approach. A
model is specified by assessing the expected values for several selected functions—
the RVs. This allows us to pose less assumptions on the models since the underlying
probability measure does not need to be specified exactly, but also pose limitations
in extending the assessments to derived quantities. These extensions will (mostly)
result only in bounds on the expectations of the derived RVs. The approach will be
fully generalised for imprecise probabilities in Sect. 2.5.
2.3.1 Measure-Theoretic Probability
Suppose that we are to perform an experiment. We will denote the set of all its
possible outcomes as the sample space, Ω. Let us further assume that an outcome
cannot be exactly determined prior to its actual observation—it is uncertain. But
even though the experimental outcome can be random (i.e. we are not able to
predict it using any finite algorithm), multiple repetitions of the same experiment
may follow some predictable law. Probability theory aims to describe these laws.