2.2.1 Axioms
First, let define the probability space pΩ, F, Pq, where Ω is an event space (or sample space) equipped with a σ-algebra F and P is a probability measure. A probability measure is a real-valued function mapping P : Ω Ñ R and satisfying the followings axioms:
1. PpEq ě 0 @ω P Ω (non-negativity) 2. Ppωq ` Ppωc
q “ 1 @ω P Ω and ωcX ω “ H(unitarity)
3. for any set of mutually exclusive events tω1, .., ωi, .., ωnu P Ω(σ-additivity)
Pp 8 ď i“1 ωiq “ 8 ÿ i“1 Ppωiq
In the traditional Kolmogorov’s probability theory, a probability Ppωq associated to an event ω P Ω is defined to satisfy the above mentioned Kolmogorov’s axioms.
2.2.2 Random variables, CDFs and method of moments
Given a probability space pΩ, F, Pq, a random variable X is defined as a map X : ω P Ω Ñ Xpωq P IX Ă R, which relates basic events ω in the event space Ω to a value
Xpωq included in the random variable support IX, subset of the real line. If X is
discrete, then it is generally associated with a probability mass function fXpxq defined
as fXpxq “ PpX “ xq “ Pptω P Ω : Xpωq “ xuq. If X is continuous, then is associated
with a probability density function (PDF) fXpxq, where fXpxqis non-negative Lebesgue-
integrable function such that:
Ppa ď x ď bq “
b
ż
a
fXpxqdx (2.1)
where the PDF express how likely is to have the random va in the interval ra, bs. A cumulative distribution functions (CDFs) FXpxq is a non decreasing mapping
from P to [0,1] such that for a probability measure P and for each x P R, the followings FXpxq “ Ppp´8, xsq and fXpxq “ dFXdxpxq hold. An empirical CDF can be used to
estimate the CDFs of a random variable X given data. In particular, given a set of realisations tX1, .., Xi, .., XNu, the empirical CDF is defined as:
FXepxq “ 1 N N ÿ i“1 IxěXipxq (2.2)
where Xi is the ith realization of the random variable X and the indicator function
IxěXi is equal 1 if x ě Xi and 0 otherwise.
The nthmoment of a real-valued continuous density function associated to a random
variable is defined as:
µn“ `8
ż
´8
px ´ cqnfXpxqdx (2.3)
Given a population of X samples, it is possible to estimate the row moments and central moments of the underlying FXpx, pqwhere p is the parameter vector identifying
a specific distribution family. The first raw moment is named expectation (or sample mean) and the second central moment is named variance, both can be estimated from samples of FXpx, pqas follows: ErX s “ 1 N N ÿ i“1 Xi V arrX s “ 1 N ´ 1 N ÿ i“1 pXi´ ErX sq2 (2.4)
where for a sufficiently large number of samples the 2 moments will correctly estimate the true mean µ and standard deviation σ of the underlying distribution family. One of the limitations of classical probability theory, is that the measure Xpωq is a crisp (precise) value, which is obtained assuming exact knowledge of the underlying PDF and cumulative probability distribution function. Especially for cases affected by a lack of data, where imprecise information or expert judgement are utilised, and there is a poor understanding of all the relevant underlying process, strong initial assumptions may be needed to characterize Xpωq is a crisp (precise) way, i.e. using classical probabilistic methods.
2.2.3 Limitations: Do we have enough data?
From a pragmatic preservative, uncertainty can be conveniently classified into two categories, one is the aleatory uncertainty and the other is the so-called epistemic uncertainty. Aleatory uncertainty (also known as Type I or irreducible uncertainty), represents stochastic behaviours, inherent variability and randomness of events and variables. Hence, due to its intrinsic random nature it is normally regarded as irreducible. Some examples of aleatory uncertainty are future weather conditions, stock market prices or chaotic phenomenon. On the other hand, epistemic uncertainty (sometimes named Type II or reducible uncertainty), is commonly associated with lack of knowledge about phenomena, imprecision in measurements and poorly designed models. It is considered to be reducible since further data can decrease the level of uncertainty, although this might not always be practical or feasible. This classification
is often considered very useful as it allows do discuss and understand if there is hope for a better determined of a model output quantity of interest (e.g. by further data gathering to reduce the epistemic uncertainty) and to which extent this quantity is inherently variable (i.e. affected by aleatory uncertainty and thus not better defined but just quantified).
Generally speaking, traditional probability methods rely upon a good characteri- sation of variables as well as distribution moments to be estimated through samples data. This usually requires a considerable body of empirical information in order to properly define probability distributions or to accurately estimate expectation, variance or higher moments. Furthermore, prior assumptions can be necessary to define the shape distribution family FXpx, pq (e.g. Gaussian, Beta, Exponential), even if no
prior knowledge on the shape of family type is actually available. More importantly, probability theory provides only single measure for the uncertainty. This makes the uncertainty analyst unable to grasp how much of the uncertainty is due to inherent variability and to what extent the uncertainty is due to poor data quality (therefore suitable to be reduced in principle).
In recent decades, efforts were focused on the explicit treatment of imprecise knowl- edge, non-consistent information and both epistemic and aleatory uncertainty [9]. The methodologies have been proposed and discussed in literature by different mathemati- cal concepts: Dempster-Shafer theory of Evidence [136]- [37], interval probabilities [8], Bayesian approaches [106], Fuzzy-based approaches [150], info-gap approaches [16], probability boxes [49] and Credal sets [63] are some of the most intensively applied concepts and will be briefly reviewed in the following sections.